Two-dimensional Digital Filters

Two-Dimensional Digital Filters

ELECTRICAL ENGINEERING AND ELECTRONICS A Series of Reference Books and Textbooks

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Two-Dimensional Digital Filters Wu-Sheng Lu A n d m Rntoniou University of Victoria Victoria, British Columbia, Canada

Marcel Dekker, Inc.

New York Basel Hong Kong

Library of Congress Cataloging-in-Publication Data Lu, Wu-Sheng Two-dimensional digitalfilters / Wu-sheng Lu, Andreas Autoniou. p.cm. - (Electricalengineeringandelectronics ; 80) Includes bibliographical references and index. ISBN 0-8247-8434-0 (acid-free paper) 1. Electricfilters,Digital.2.Signalprocessing--Digital techniques. I. Antoniou, Andreas. 11. Title. III.Series. TK7872.FSA65 1992 621.3815’324-dc20 CIP

This book is printed on acid-free paper.

Copyright

0

1992 byMARCELDEKKER, INC. All Rights Reserved.

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PRINTED IN THE UNITED STATES OF AMERICA

92

To Catherine and Rosemary

This Page Intentionally Left Blank

Preface

The theory of two-dimensional (2-D) digital filters has been a subject for study during the past two decades and has matured so that it now forms an important part of multidimensional digital signal processing. The objective of this book is to present basic theories, techniques, and procedures that can be used to analyze, design, and implement 2-D digital filters. Applications of 2-D digital filters in image and seismic data processing are also presented that demonstrate the importance of using 2-D filtersin realworld signal processing. The prerequisiteknowledge is anundergraduatemathematicsbackground of calculus, linear algebra and complex variables, and l-D digital filters. Sufficient background information on l-Ddigital filters canbe found in the book Digital Filters: Analysis, Design, and Applications, Second Edition, by A. Antoniou. Chapter 1 introduces the 2-D digital filter as a linear, discrete system. eleCharacterization, flow-graph and networkrepresentations,andan mentary space-domain analysis are then introduced. The chapter also includes a brief discussion on stability and on the realization of 2-D digital filters. The Givone-Roesser model is introduced in Chapter 2 and is then used as atool in a more advancedspace-domain analysis. The concepts of controllability and observability are then defined. The chapter concludes with a section on state-space realization. V

vi

Preface

The 2-D z transform is studied in Chapter 3. The complex convolution is introduced as a tool for obtaining the z transforms of products of spacedomain functions and for the derivation of Parseval’s formula. The 2-D sampling theorem is then introduced as the fundamental link between 2-D continuous and discrete signals. Chapter 4 introduces the important concept of the transfer function as an alternative, z-dornain’representationof 2-D digital filters. Some of the properties of the discrete transfer function and its application to spacedomain and frequency-domain analysis are then described. Chapter 5 is devoted to a detailed stability analysis in both frequency and state-space domains. Also included in this chapter is a 2-D Lyapunov stability theory developed recently. It is shown that theproblem of stability in 2-D digital filters, unlike its l-D counterpart, can get quite involved. Various design techniques are described in Chapters 6 to 9. Chapter 6 deals with the approximation problem for nonrecursive filters. Methods described in the chapter include the window method and a method based on the McClellan transformation. Chapter 7 presents several transformation methods for the design of recursive filters. In Chapter 8 a least pth and a minimax optimization method are used to design recursive filters. A design approach based on two-variable network theory that leads to stable filters is then presented.The chapter concludes with a design method based on the singular-value decomposition (SVD). The design of nonrecursive filters by optimization is considered in Chapter 9. Several direct and indirect realization methods are considered in detail in Chapter 10. These include nonrecursive realizations based on the McClellan transformation, nonrecursive and recursive realizations based on matrix decompositions, and recursive realizations based on the concept of the generalized-immittance converter. The finite wordlength effects of 2-D digital filters are considered in Chapter 11. An analysis on quantization errors and their computation is first discussed. State-space structures with minimized roundoff noise are then examined. Two types of parasitic oscillations, namely, quantization and overflow limit cycles, are examined and conditions for theirelimination are presented. Chapter 12 introduces the 2-D discrete Fourier transform and methods for its efficient computation in terms of the l-D and 2-D fast Fourier transforms as tools in the implementation of 2-D digital filters. Several systolic structures for the implementation of 2-D nonrecursive and recursive filters are then examined. Chapter 13, which concludes the book, introduces applications of 2-D filters to image enhancement, and restorationand seismic data processing. Most of the concepts and techniques are illustrated by examples and

Preface

’

vii

a selected set of problems is includedat the end of each chapter.The book can serve asa textbook for a one-semester courseon 2-D digital filtersfor first-year graduate students. The book should also be of interest to filter designers and engineers who intend to utilize 2-D filters in dealing with a variety of digital signal processing problems. We wish to thank Len T. Bruton of the University of Calgary for providing materials that have been used to write Secs. 8.4 and 13.4; Chris Charalambous for allowing us to use the designdata presented in Examples 8.2 and 9.1; Majid Ahmadi for reading parts of the manuscript and for supplying useful comments; Catherine Chang for her assistance in the preparation of the artwork and for typing Chapters 8 to 13 of the manuscript; Eileen Gardiner and Ruth Dawe for their persistent encouragement and involvement asthe acquisition editors and Joseph Stubenrauch for his assistance as the production editor of Marcel Dekker, Inc.; the Natural Sciences and Engineering Research Council of Canada and Micronet, Networks of Centres of Excellence Program, for supporting the research that led to some of the new results presented in Chapters 8 to 10; and the University of Victoria for general support. In addition, we wish to thank our wives Catherine Chang and Rosemary C. Antoniou for their sacrifices and constant support. Wu-Sheng Lu Andreas Antoniou

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Contents

Preface

V

Introduction

1

1 Fundamentals 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10

Introduction 2-D Discrete Signals The 2-D Digital Filter as a System Characterization Representation in Terms of Flow Graphs and Networks Introduction to Space-Domain Analysis Stability Realization Multiple-Input-Multiple-Output Filters Multidimensional Filters Reference Problems

2 State-SpaceMethods 2.1 Introduction 2.2 The Givone-RoesserModel

5

5 5 10 12 14 16 27 29 32 34 35 35 38 38 39 ix

Contents

X

2.3 Space-DomainAnalysis 2.4 ControllabilityandObservability 2.5 Realization

References Problems 3 TransformMethods 3.1 3.2 3.3 3.4

Introduction The 2-D z Transform The 2-DFourierTransform The SamplingProcess References Problems

4 The Application of the z Transform 4.1 4.2 4.3 4.4 4.5

Introduction The TransferFunction Stability Space-DomainAnalysis Frequency-DomainAnalysis References Problems

5 StabilityAnalysis

5.1 5.2 5.3 5.4 5.5

Introduction Stability Analysis in Frequency Domain Stability Analysis in State-Space Domain 2-DLyapunovStabilityTheory Stability of Low-Order Filters References Problems

6 ApproximationsforNonrecursiveFilters 6.1 6.2 6.3 6.4

Introduction Properties of2-D Nonrecursive Filters Design Based on Fourier Series DesignBasedonTransformations References Problems

48 50 52 52 53

55 55 56 65 68 74 74 77

77 77 82 84 86 103 103 106

106 107 118 121 124 127 129 132

132 133 137 163 173 174

xi

Contents 7Approximations 7.1 7.2 7.3 7.4 7.5 7.6

for. RecursiveFilters

Introduction Transformations Method of Hirano and Aggarwal Design of Circularly Symmetric Filters ConstantinidesTransformations Design of Filters Satisfying Prescribed Specifications References Problems

8 Design of Recursive Filters by Optimization

8.1 8.2 8.3 8.4 8.5

Introduction Design by Least pth Optimization MinimaxMethod Design Based on Two-Variable Network Theory Design of Recursive Filters Using Singular-Value Decomposition References Problems

9 Design of Nonrecursive Filters by Optimization 9.1 Introduction 9.2 Minimax Design of Linear-Phase Nonrecursive Filters 9.3 Design of Linear-Phase Nonrecursive Filters Using SVD

References Problems 10 Realization

10.1 Introduction 10.2 Nonrecursive Realizations Based on the McClellan

176 176 177 190 204 218 219 226 227

230 230 231 236 244

251 263 264

266 266 267 271 282 283

284 284

Transformation

285

Decompositions

289 294 295 300 313 315

10.3 Nonrecursive Realizations Based on Matrix 10.4 Direct Realization of Recursive Filters 10.5 LUD Realization of Recursive Filters 10.6 Indirect Realization of Recursive Filters

References Problems

xii

Contents

11FiniteWordlength

Effects

11.1 Introduction 11.2 Quantization Errors and Their Computation 11.3 State-Space Structures with Minimized Roundoff Noise 11.4LimitCycles References Problems 12 Implementation

12.1 Introduction 12.22-D Discrete Fourier Transform 12.3Computation of 2-D DFT 12.4SystolicImplementation References Problems 13 Applications

317

317 318 325 334 339 340 342

342 343 347 350 357 358 360

13.1 Introduction 13.2 Applications of Linear 2-D Filters to Image Enhancement 13.3ImageRestoration 13.4 Applications of Linear 2-D Fan Filters to Seismic Signal Processing References Problems

360

Index

388

360 370 379 386 387

Introduction

The field of two-dimensional (2-D) digital signal processing has been growing rapidly in recent years. Images such as satellite photographs, radar and sonar maps, medical X-ray pictures, radiographs, electron micrographs, and data from seismic, gravitational, and magnetic records are typical examples of 2-D signals that might need to be processed. The types of processing that can be applied may range from improving the quality of signals to extracting certain useful features from them. For example, a picture degraded by wideband noise might be improved by removing the noise without blurring the edges, or a seismic record might be made more readable by removing a certain large-amplitude, low-frequency signal known as ground-roll interference. A confinuow 2 - 0 signal is a physical or contrived quantity that is a continuous function of two real independentvariables. An example of such a signal might be the light intensity in the case of a photograph or image as a function of distances in the x and y directions. A discrete 2 - 0 signal is a sampled version of a continuous 2-D signal and is normally in the form of a 2-D array of numbers. Like a one-dimensional (l-D) discrete signal, a 2-D discrete signal can be represented by a frequency spectrum that can be modified, reshaped, or manipulated through filtering. This type of processing can be carried out by using 2-D digital filters. Two-dimensional digital filters, like their l-D counterparts, are discrete shift-invariant or shift-dependent, systems that can be linear o~r.nonlinear, causal or noncausal, and stable or unstable. JThey can be characterized in terms of difference equations or state-space equations in two independent 1

Introduction

2

variables and in terms of transfer functions or matrices of transfer functions, which are rational functions of polynomials in twovariables. Time-domain analysis in l-D digital filters is replaced by space-domain analysis in 2-D digital filters, since neither of the two independent variables needs to be time but frequency-domain analysis continues to be referred to by the same name, although frequency is sometimes referred to as spatialfrequency to emphasize that frequency may not bear an inverse relation with time. As in l-D digital filters, the transfer function yields the amplitude and phase responses, which are surfaces over a 2-D frequency plane rather thancurves plotted over a frequency axis. Two-dimensional digital filters can be classified as recursive or nonrecursive, depending on whether the outputof the filter depends on previous values of the output;alternatively, they can be classified as infinife-impulse response (IIR) orjinite-impulse response(FIR) filters, dependingon whether their impulse response is of infinite or finite duration. These types of 2-D digital filters are consistent with their l-D counterparts and have analogous properties. For example, FIR filters can be designed to have linear phase, whereas IIR filters can be designed to be more economical in terms of the amount of computation required for a given degree of selectivity. The design of 2-D digital filters, like that of l-D digital filters, involves several steps, as follows: Approximation Realization Implementation Study of quantization effects Approximation is the process of generating a rational transfer function that satisfies required specifications imposed on the amplitude, phase, or space-domain response. It can be accomplished by applying transformations to l-D analog or digital filters, by using optimization methods, or by applying transformations in conjunction with optimization methods. A prerequisite property here is that the transfer function generated should represent a stable digital filter. Realization is the process of converting the transfer function obtained through the approximation step into a signal-flow graph, digital-filter network., or state-space representation. Zmplementation is the process of convering the signal-flow graph, digitalfilter network, or state-space representation into a computer program or a dedicated piece of equipment. In this way software and hardware digitalfilter implementations can be obtained. When a 2-D digital filter is implemented in terms of either software on a general-purpose computer or dedicated hardware, numbers representing

Introduction

3

transfer-function coefficients and signals must be stored and manipulated in registers of finite length. When the approximation step is carried out, transfer-function coefficients are calculated to a high degree of precision; consequently, they must be quantized before the implementation of the digital filter. The net effect of coefficient quantization is to introduce inaccuracies in the amplitude response of the filter that tend to increase as the word length of the hardware is reduced. On the other hand, internal signals generated as products when signals are multiplied by coefficients are almost always too long to fit in the available registers and must again be quantized. The effect of signal quantization is to introduce noise at the output of the filter, which degrades the signal-to-noise ratio. Signal quantization can lead to other problems as well, such as the generation of spurious parasitic oscillations, known as limit cycles. While the effects of coefficient and signal quantization are insignificant if a general-purpose computer is to be used, owing to the high precision of the hardware, particular attention must be paid to these effects when fixed-point arithmetic or specialized hardware with reduced word length is to be employed. In such applications, the design process is not considered complete until the effects of quantization are studied in detail. The characterization, properties, and design of 2-D digital filters are usually simple extensions or generalizations of the characterization, properties, and design of l-D digital filters. Nevertheless, notable exceptions arise where the extension or generalization is not simple, and it may on occasion be quite complicated. An example in this regard is stability analysis. In the l-D case, the stability of the digital filter is linked to the poles of the transfer function, which are isolated points in the l-D complex plane. As a result, powerful mathematical tests that have been used in the past to determine whether the zeros of a polynomial in z are located in a specified region of the z plane have been used to develop stability criteria, such as the Jury-Marden stability criterion, which simplify the stability analysis of l-D digital filters. In the 2-D case, on the other hand, the stability of the digital filter is closelylinked to contours in the 2-D complex plane for which the denominator polynomial of the transfer function is zero and may also on occasion be linked to contours in the 2-D complex plane for which the numerator polynomial is zero. These difficulties arise because polynomials in two variables are notgenerally factorable intofirstorder polynomials, owing to the fact that the Fundamental theorem of algebra does not extend to polynomials in two variables. A major consideration in the case of 2-D digital filters relates to the issue of computational complexity. With one moredimension added to the field of data, thecurse of dimensionality is brought to bearand the amount of computation tends to increase as the square of the order of the filter. Consequently, in 2-Dfilters it is particularly important to design the lowest-

4

Introduction

order filter that will satisfy the specifications. Another important consideration in 2-D digital filtersrelates to the ultimate receiverof the processed 2-D signal. In the case of hearing, phase distortion is quite tolerable and l-D filters are often designed without regard to phase-response linearity. On the other hand, phase distortion tends to be as objectionable as amplitude distortion when the finalreceiver is the humaneye.Particular attention must, therefore, be paid to thephase response when2-D digital filters are to be designed for image processing applications.

1 Fundamentals

1.l INTRODUCTION A 2-D digital filter is a discrete system that can be used to process 2-D

discrete signals. Like any other system, it can be linear or nonlinear, shiftinvariant or shift-dependent, causal or noncausal, and stable or unstable. It can be characterized by a difference equation or by a transfer function and can be analyzed in the space domain or frequency domain. In this chapter, 2-D continuous and discrete signals are introduced as extensions of their l-D counterparts. The characterization and fundamental properties of 2-D digital filters are then introduced.Space-domain analysis, the process of finding the response of the filter to a given input signal, is examined and is then used to develop theconcepts of causality and stability in 2-D digital filters. The chapter concludes with a brief introduction to the realization of 2-D digital filters, which is the process of converting a mathematical description of a filter into a network.

1.2 2-D DISCRETE SIGNALS A 2-D continuous signal is a physical or contrived quantity that depends

on two independent continuous variables fl and t2. It can be represented by a function x(t,, tz) and each of the two variables f, and f2 may represent

6

Chapter l

time, distance, or any other physical or contrived variable. Two examples of 2-D continuous signals are the light intensity of an image as a function of distance in the x and y directions and the depthof an ocean as afunction of distance in the east and north directions. A 2-D discrete signal, on the other hand, is a physical or contrived quantity that depends ontwo real independent integervariables n, and n,. It can be represented by a function x(n,T,, n,T,), where TI and T, are constants and n,T, or n,T, may represent time, distance, etc. A shorthand notation for a 2-D discrete signal, which will be used frequently in this book, is x(n,, n,). Although x(n,, n,) can in principle be complex, it is normally real and can be represented in terms of a 3-D plot, as depicted in Fig. 1.1. Two-dimensional discrete signals are usually obtained by sampling corresponding continuous signals. Two such examples are the arrays of numbers representing a digitized image and the depth of an ocean at discrete points in the north and east directions. In many applications, 2-D signals arise that are continuous with respect to onevariable and discrete with respect to the other variable. Such signals are sometimes said to be mixed. An example of a 2-D mixed signal is the set of acoustic waveforms that might be produced by an explosion, as measured by transducers placed at discrete intervals near the surface in

Figure 1.1 A 2-D discrete signal.

Fundamentals

7

the ground along a given direction. In such a signal the continuous variable is time and the discrete variable is distance.

1.2.1Region

I

of Support

A useful concept in the description of 2-D discrete signals is the region of support. Consider a signal x(nl, nz) and letA be a connected region in the (n,, nz)plane such that x(nl, n2) = 0 for all points in A. The complement of A with respect to the (nl, n2) plane, represented by S, is said to be a region of support of the signal. If A is the largest connected region in the (nl, n2) plane such that x ( n l , nz) = 0 for all points in A , the complement of A is said to be the minimum region of support. If a 2-D signal is zero everywhere except the first quadrant, it is said to have first-quadrant support, and if a 2-D signal is zero everywhere in the left half plane, it is said to have right half-plane support. A 2-D signal is said to be of finite extent if its minimum region of support is finite.

1.2.2 Quantization As in the l-D case, the amplitude of a 2-D discrete signal can be quantized by representing it in terms of a finite number of distinct levels so as to facilitate the storage of the signal in a digital memory or mass-storage device. Signal quantization depends on the type of arithmetic used for the representation of the signal (e.g., fixed-point or floating-point) and can be carried out by rounding or truncating the numbers that represent the levels of the signal.

1.2.3 Periodicity A 2-D discrete signal can be periodic with respect to either or both of the space-domain variables nl and nz. A 2-D signalthat is periodic with respect to both space-domain variables satisfies the relation

m , nz) for any pair of integers k, and k2. The constants Nl and NZare theperiods of the signalin the two directions. A periodic signalis illustrated in Fig. 1.2a. If a 2-D signal is periodic with respect to both space variables with periods N , and N Z , then any connected domain of x(nl, n2) comprising exactly Nl x NZpoints is said to be a 2-D period of the signal. Although the 2-D period of a signal is usually rectangular, as in Fig. 1.2a, many geometrical shapes are possible, as illustrated in Fig. 1.2b. x(%

+ klN1, n2 +

=

8

Chapter 1

Figure 1.2 Two-dimensionalperiodic discrete signals: (a) with rectangular period; (b) with nonrectangular period.

9

Fundamentals 1.2.4 Separability

A 2-D discrete signal is said to be separable if it can be expressed as a product of two l-D discrete signals; that is, x(n17

'2)

=

xl(nl)x2(d

The importance of separability arises from the fact that if the output of a 2-D digital filter can be expressed as a product of two l-D discrete signals, then the design of the filter can be broken down into the design of two l-D filters. Although 2-D signals are usually nonseparable, in certain circumstances it is possible to express a 2-D signal in terms of a linear combination of products of l-D signals; that is,

where ai for i = N L , . . . , NH are constants. In such a case it would be possible to design the 2-D filter by simply designing a set of l-D filters and then interconnecting them to form the 2-D digital filter. In the case where the 2-D signal has a finite support, the decomposition in Eq. (1.1) is always possible, as will now be demonstrated. Let x(n17

nz) = 0

for any pair (nl, n2) not in the rectangle defined by A = {(nl, nz): N L 1

5 1z1 5 N H 1 , N L 2 5 n2 5

Nm}

Such a signal can be expressed in terms of Eq. (1.1) by letting xdnl) =

~(1117

i)

and

1

for n2 = i

as can be easily verified. The preceding decomposition of a 2-D signal, though simple to understand, is not the most effective for the applications to be considered later. An alternative and more efficient decomposition can be obtained by expressing the (NHl - NL1 + 1) X (Nm - NL2 + 1) matrix defined by

x=

n2>}

10

Chapter 1

for (nl, nz) E A as a singular-value decomposition [l] of the form r

x = i2 uiuiv: =l where ui for i = 1, 2, . . . , r are said to be the singular values of X, r is the rank of A , ui and vi are column vectors, and v: is the transpose of vi. Evidently, ui and vi can be regarded as l-D signals and Eq. (1.2) can be considered to be equivalent to Eq. (1.1). The issue of separability will be reexamined later, in Chapters 8 and 9, when the design of 2-D digital filters is undertaken.

1.3 THE

2-D DIGITAL FILTER AS A SYSTEM

A 2-D digital filter is a discrete system that will receive an input signal x(nl, n2) and produce an output signal y(nl, n2), as depicted in Fig. 1.3.

The output is related to the input by some rule of correspondence. This fact can be represented mathematically by the relation where '3is an operator.This relation can be used to define the fundamental properties of a 2-D digital filter as a system, such as linearity, shift invariance, and causality.

1.3.1 Linearity A 2-D digital filter is linear if its response satisfies the principles of homogeneity and additivity given by

and

+ x2(n17 n2>1 = a x l ( n l Y ' 2 ) + ax2(nlY %) for all possible values of a and all possible input signals xl(nl, n2) and x2(n,, n2). The above two conditions can readily be combined into the a[x1(n17

x h n2) 0

L

2-D Digital Filter

Figure 1.3 The 2-D digital filter as a system.

-

Y h . n2) 0

11

Fundamentals

general superposition relationgiven by %[a,x,(n,,

+ a2x2(n*,n2)l

+

n2)

(1.4) If Eq. (1.4) is violatedfor some signalsor constants of proportionality, the 2-D digital filter is nonlinear. n2)

= a,%(n,,

n2)

a2*2(n1,

1.3.2 Shift Invariance An initially relaxed 2-D digital filter in which x(n,, n2) = 0 and y(n,, n2) =

0 for all nl < 0 or n2 < 0 is said to be shift-invariant if y(n1 - k,,

- k2) = W n 1 - kl, n2 - k2) (1.5) for all input signals x(n,, n2) and all possible integers k, and k2. If the condition for shift invariance is not satisfied, then the filter is said to be shift-dependent. n2

1.3.3 Causality A 2-D digital filter is said to be causal if its output for nl Ik, and n2 5 k2 is independent of the input for values of n, > kl or n2 > k2 for all possible values of kl and k2. In mathematical terms, a 2-D digital filter is causal if and only if

(1.6) k2 for all possible signals x,@,, n2) and x2(n,, n2) %(nl,

for n, Ikl and n2 I such that

122)

=

%x2(n1, n2)

xl(n1, n2) = x2(n1,

n2)

for n, I kl and n2 Ik2. Conversely, if Eq. (1.6) is not satisfied for some pair of possible signalsxl(nl, n2) and x2(nl, n2) such that x,(n,,

n2)

= x2(n1,

n2)

for n1 Ikl and n2 Ik2 and n2) f

x,(%, n J

for some pair of values of n, and n2 such that nl > k, or n2 > k2, then the filter is noncausal. If Eq. (1.6) holds only with respectto variable n, (n2), the 2-D digital filter is said to be causal only with respect to variable nl (n2).

Causality has physical significance only inthe case where n,T, or n2T2 represents real time. Nevertheless,the concept of causality is useful even in the case where neither of the two variables is time.

Chapter 1

12

1.4 CHARACTERIZATION Two types of 2-D digital filters can be identified, namely, recursive and nonrecursive, and like their l-D counterparts, they can be characterized in terms of difference equations.

1.4.1

Nonrecursive Filters

In a nonrecursive 2-D filter the output y(n,, n2) is a function of x(n, - i, n2 - j) for --CO < i, j < +m. Assuming linearity and shift invariance,.the output of a nonrecursive 2-D digital filter can be expressed as a weighted sum of all the possible values of the input; that is m

m

where aij are constants. Assuming causality, the output for n, S kl and n2 5 k2 is independent of the input for values of n, > kl or n2 > k2 for all possible values of k, and k2, and therefore we have a.. = 11

0

for --COSiorj

-

=

Pkvdnl,

- Pkln2(nl?

Pnlk2(nl

It2

k l ? n2)

-

k2)

(1.18)

The 2-D unit pulse obtained is shown in Fig. 1.10. Other elementary signals of interest are the 2-D discrete sinusoid and exponential. The 2-D sinusoid is given by x(n,, n2) = sin(olnl

+ 02n2)

(1.19)

whereas the 2-D discrete exponential can be generated by forming the complex signal x(n,, n2) = cos(olnl

+ 02n2)+ j sin(olnl + 02n2)

+w n z )

=

ei(wnl

=

eiwntejwnz

(1.20)

The last two signals play the same key role in frequency-domain analysis as their l-D counterparts. Note that the exponential signal is separable while the sinusoid can be expressed as a linear combination of products of signals by using a trigonometric identity.

1.6.2 Induction Method Space-domain analysis is normally carried out through the use of powerful state-space or transform methods. In certain simple examples, however,

Chapter 1

22

5

(b) Figure 1.9 Unit-line pulses: (a) unit-line pulse along n, axis of width k,; (b) unitline pulse along n, axis of width k,.

it can also be carried out through induction, as is illustrated below by some examples. Although the approach is not the most efficient,it does provide understanding of the fundamental principles involved. Some more advanced methods for space-domain analysis will be examined in Chapter 2.

Fundamentals

23

k1- 1 Figure 1.10 A k ,

X

k2 rectangular unit pulse.

Example 1.2 (a) Find the impulseresponse of the filter described in Fig. 1.5a. (b) Find the impulse response of the filter characterized by the equation

Y ( h n2) = 4 4 , n2) + bY(4 - 1, n2 - 1) Assume that the filter is initially relaxed, that is, y(n,, n2) = 0

for n, < 0 or n2 < 0

Solution. (a) With x(n,, n2) = 6(n,, n2), Eq. (1.9) becomes v(n,, n2) = 6(n,, n2)

+ ealv(nl - 1, nz)

If n2 = 0, then

v(0, 0) v(1, 0) 4 2 , 0)

1 + ea%(- 1, 0) = 1 = 0 + ealu(O, 0) = eal = 0 + ealv(l, 0) = eZa1 =

... v(n,, 0) = P'"' If n2 > 0, then

+ ea%(- 1, nz) = 0 v(1, n2) = 0 + ealv(O, n2) = 0 v(0, n2) = 0

(1.21)

24

Chapter 1

Hence u(n,, n2) = en1a181(n1, n2)

With u(nl, n2) determined as above, Eq. (1.10) becomes

y(n17 122) = en1a161(n17 n2) + e"2y(n17n2 Thus

+ ea2y(nl, -1) = e"1a18,(n,,0) y(nl, 1) = e"1"161(nl,1) + eazy(nl,0) = 0 + eaZe"1a16l ( n 1, 0) y(nlj 2) = 0 + eu2y(nl,1) = 0) y(n,, 0) = e"1a161(nl, 0)

eZa2e"la181(n1,

... y(nl, n2) = enlal@za28 n 0)

for n2 2 0

Since

y(nl, n2) = 0

for n2 < 0

we obtain the impulse response of the filter as

h(nl, n2) = en1a1e"2a281(nl, 0)6,(0, - p a l p z a z U ( 1 7 n2> Evidently, we can write n2)

=

[e"1"181(n17

n2)

0)][e"2a262(0,

that is, the impulse response of the filter is separable. Under these circumstances, the filter is said to be separable since it can be viewed as a cascade arrangement of two distinct l-D digital filters. (b) On replacing x(n,, n2) in Eq. (1.21) by 6(nl, nz) we obtain

y(nl, %) = n2) + - l 7 n2 - l ) This equation can be solved recursivelyalong a set of parallel straight lines. Starting with point (0, 0), we can write y(0, 0) y(1, 0) y(0, 1) y(2, 0) y(1, 1) y(0, 2)

= 6(0, 0)

0 =0 = 0 =0 =0 =

+ by( - 1, - 1) = 1

+ by(0, -1)

=0

+ by( -1, 0) = 0 + by(1, -1) = 0 + by(0, 0) = b + by( - 1, 1) = 0

25

Fundamentals

y(3, 0) = y(2, 1) = y(1, 2) = y(0, 3) =

0 0 0 0

+ by(2,

+ + +

-1) = 0 by(1, 0) = 0 by(0, 1) = 0 by( - 1, 2) = 0

... Proceeding in the same way, one can show that

{

b" y(n17n2) = 0

for nl 5 0 and n2 2 0, and nl = n2 = n otherwise

1.6.3 Convolution Summation By expressing an arbitrary input in terms of a weighted sum of unit impulses, it is possible to express the response of a 2-D linear, shift-invariant digital filter in terms of its impulse response. An arbitrary input x(n,, n2) can be expressed as

where xij(nl,n2) =

{p

for i = n1 and j = n2 otherwise

Alternatively, xij(nl, n2) = x ( i , j)8(nl - i, n2 - j )

and hence 5

0

Now let us assume that a 2-D digital filter is characterized by Y(%,

122)

= *(n1,

n2)

Chapter 1

26

and that its impulse response is given by %) =

It2)

If the filter is linear and shift-invariant, then Eq. (1.22) yields

C C is x

=

"CS

x

h(i, j)x(nl - i, n2 - i)

(1.23b)

j = -m

If the filter is causal, then any two inputs xl(n17n2) and x2(n,, n2) such that n2)

for n, Ii and n2 Ij , and

=

n2>

'

xl(n17 n2) n2) for some pair of values of nl and n2 such that n1 > i or n2 > j, will produce the same response for n, 5 i and n2 5 j (see Sec. 1.3.3); that is

or

From the definition of the two input signals, the first term is zero. However, the second term is zero if and only if h(n,, n2) = 0 for n, < 0 or n2 < 0 and therefore Eq. (1.23) can be expressed as

x

-

(1.24)

Fundamentals

27

Now if x(n,, n,) has a first-quadrant support, we obtain

(1.25b) This relation is called the 2-0 convolution summation and is often represented in terms of the shorthand notation x @ h or h @ x . The convolution summation can be carried out as depicted in Fig. 1.11. The impulse response given in Fig. l . l l b is folded about the i and j axes, as illustrated in Fig. l . l l d , and is then shifted in the positive directions of The values in Fig. 1.11~ are then the two axes, as illustrated in Fig. 1.11~. multiplied by the corresponding values in Fig. l . l l a t o form Fig. 1.lle. The sum of all the products in Fig. l . l l e gives the response of the filter for nl and n,, as illustrated in Fig. 1.llf.

1.7 STABILITY A 2-D digital filter is said to be stable in the bounded-input, bounded-

output (BIBO) sense, if a bounded input such that Ix(nl, n,)l

S

M
1

and

1z21> 1

in order to guarantee the convergence of the z transform. From Eqs. (1.21) and (4.3b), the transfer function of the filter is obtained as

and hence Eq. (4.7) gives the unit-step response of the filter as

where contours rland

r2are chosen as

rl= {z, : lzll= 2)

and

r2 = {z2: Iz2(= 2)

in order to guarantee convergence. The preceding integrand can be written as

86

and hence for nl

Chapter 4 2

0 and n2 2 0, we have

2 - 0.5"'

if n1 2 0, n2 2 0, n1 2 n2 if nl 2 0, n2 2 0, n2 2 n, otherwise

- 0.5"'

4.5 FREQUENCY-DOMAINANALYSIS The most fundamental application of 2-D digital filters involves the manipulation of the frequency spectrum of a 2-D discrete signal. To determine the effect of a 2-D digital filter on a 2-D signal, a frequency-domain analysis must be carried out, which is concerned with the response of the filter to a sinusoidal input.

4.5.1 Sinusoidal Response Consider a 2-D digital filter characterized by a transfer function X

X

and assume that the input of the filter is given by

x ( n l , n2) = sin(nlolT,

+ n202T2)

The Application of the z Transform

X

87

sin[(n, - kl)wlTl

+ (n2 - k2)02T2]

From Eq. (4.8), we can write

Therefore %) sin[(nlolT1

+ n2°2T2)

+

O2)1

(4*12) This result shows that theeffect of a 2-D digital filter on a sinusoidal input is to introduce a gain M(w17 02)and a phase shift €)(col, 02)as in the l-D case. In the preceding analysis, the input x(n17n2) was assumed to be sinusoidal over the entire(n17n2) plane. Nevertheless, the result obtained holds y(n17 '2)

=

e(017

Chapter 4

88

for the case where the input issinusoidalin the first quadrant of the (nlyn2) plane and zero elsewhere;that is, for sin(n,o,T, + n202T2) provided that the filter is allowed to reach steady state. To demonstrate this fact, let us consider a 2-D filter whose transfer function can be expressed as x(n1, n2) = 4n,,

n2)

L

H(z1, 22) =

2 H1,(z,)Hz(z2)

(4.13)

I= 1

Let us assume that the poles of H,,(z,) and H2,(t2)are located inside the unit bicircle U2 =

{(q,z2) : lZll

=

1, lz21 = l }

to ensure that the filter is stable, and that none of the poles are multiple for the sake of simplicity. The z transform of x(nl, n2) is given by

C 2 m=O z 1 D

x(zl, z2) =

sin(nlo,Tl

+ n202~2)z;n1z~"z

n2=0

- z1z2[z1sin 02T2 + z2 sin olTl- sin(o,Tl + 02T2)] (zl- ,+")(z2 - e j w T z ) ( z l - e - i o l T 1 ) ( z 2 - e - j w T z ) - z:z2 sin 02T2 + zlzfsin olTl - zlz2 sin(olTl

+ w2T2)

01(21)02(22)

where 1

-

"

z1 -

1 ) e-jhT1

and 1

-

-=

ejuZTz

z2 - e-iozTz

The response of the filter is given by YhY

n2)

=

( 2 1 ~ jf)r2 ~ fr1 H(zl, z2)X(z1,

dz1 dz 2

and if contours rl and r2are taken to be in u2(see Sec. 4.3), we obtain Y(nlY n2) = YI(n1, n2) + Y h l , n2) (4.14) where y,(nl, n2) and ys(nl, n2) are output components due tothe residues

Application The

of the z Transform

89

at the poles of H(z,, z2) and X(zl, z2), respectively. After some manipulation, we obtain

and

where N,and M , denote the orders and pli and qri denote the ith and jth poles of HII(zl)and H2[(z2),respectively. Since \pli\< 1 and (qlj(< 1, we have lim

yr(nl, n2) = 0

ny=,ny-t=

that is, yt(n1, n2) is a transient component and, therefore, Eqs. (4.14) and (4.15) yield

4.5.2 Frequency Response The transfer function evaluated onthe unit bicircle as a function of (W,, w2) is said to be the frequency response of the 2-D digital filter. The quantities M ( w l , w2) and 0(w,, w2) as functions of (wl, w2) are said to be the amplitude response and phase response, respectively. They can be represented by 3-D plots or by contour plots. As anexample, the amplitude and phase responses of the 2-D filter characterized by the transfer function

PI

90

Chapter 4

where coefficients b, are the elements of matrix B given by

- - 128.4300 78.4280 B =

-

83.0110 -25.5190 3.4667 -0.1645

140.2000 136.7600 -22.7050 -13.2490 3.4048 0.3218

101.4700 -33.3130 -7.2275 -29.3020 23.8380 -4.8317

-36.0290 -7.4214 -33.4870 61.5510 -25.8710 3.3632

6.0739 -0.4134 1.4820 0.7907 28.7070 -6.3054 -28.7340 4.2941 7.8851 -0.7930 -0.6450 0.0652

can be represented by the 3-D plots shown in Fig. 4.2a and b or by the contour plots of Fig. 4.3a and b. A pair of parameters that are sometimes of interest are the “group delays” of the 2-D digital filter. These are defined as 71

=

-

ae(017 O Z )

a@,

and

7z

= -

%)

ae(017

do2

where the units of 7 , and T~ are seconds only if nlTl or nzTz represents time. In many applications, in particular in image processing, the phase response is required to be linear in order to prevent signal distortion. In such applications the group delays must be constant. The group delays corresponding to the phase response shown in Fig 4.2b are plotted with respect to the (ol, oz)plane in Fig. 4.4. It is easy to verify that the frequency response of a 2-D filter and, consequently, its amplitude and phase responses are periodic with periods equal to the sampling frequencies os,= 2 d T I and os, = 2dTZ, that is, H(ej(uu+klo,l)T~ & ( o z + k z w s z ) T ~ ) = H(ejwT1 e j o ~ h 7

)

for all integers kl and kz. The rectangular area defined by the fundamental period, namely,

B

=

{

(W1,

%l os1 os2 oz): - - I o1 5 -, -2 2 2

9

5025-

2

is often referred to as the baseband. The frequencies 4 2 and wsz/2 are referred to as the Nyquist frequencies.

4.5.3 Symmetries If the coefficients of the transfer function are real, the amplitude response of the filter is symmetrical with respect to the origin, that is, M(w,, oz)= M(-ol, -mz)

and

M(o,, -wz) = M ( - W , , oz)

Sometimes, certain symmetries are imposed by the application at hand. In

The Application of the z Transform

91

/

-R \

-R

Figure 4.2 Three-D plots: (a) amplitude response; (b) phase response.

certain applications the contours of the amplitude responsecan be rectangular, that is, M(wl, w2) = constant for

lo1[=

U

and

lo2[S

b, lo2[= b and lo1(S a

In other applications, in particular if the frequency spectrum of the 2-D signal is homogeneous with respect to the two frequency variables, a 2-D filter with circularly symmetric amplitude response may be required. In such an application the contours should be circular, that is, ~ ( w , ,w2) =

constantfor

V-

=

constant

Another typeof symmetry that has been considered in the literature is the case where the contours of the amplitude response are ellipses whose axes are aligned in the direction of the w1 and w2 axes [6,7] or whose orientation is arbitrary [8].

92

Chapter 4

Figure 4.2

Continued.

A broader type of symmetry is the case where the amplitude response is symmetrical with respect to both frequency axes, that is

This type of symmetry is said to be quadrantal. Evidently, every filter whose amplitude response has circular or rectangular symmetry also has quadrantal symmetry. On the other hand,if the contours are ellipses that are not aligned in the direction of the o1or o2axis, the amplitude response does not have quadrantal symmetry.

4.5.4

Idealized Filters

The design of 2-Ddigital filters can often be simplified by identifying a number of idealized types of filters on the basis of the shape of the amplitude (e.g., lowpass, highpass, etc.).

The Application of the z Transform

93

3

2

1

0

-1 \-/

-2 -3

-3

(a 1

-2

-1

0

1

2

3

(b)

Figure 4.3

Contour plots: (a) amplitude response; (b) phase response.

94

Chapter 4

Figure 4.4 Plots of group delays for the phase response in Fig. 4.2b: (a) group delay T ~ (b) ; group delay T ~ .

A filter with an amplitude response Mm,,

02)

=

1 0

for (wl, w2) E R I otherwise

is said to be a rectangularly symmetric lowpass filter since it will pass lowfrequency components located in the rectangular area R I and reject highfrequency components not in R I . The area R1is called the passband, the area outside R , is called the stopband, and the boundary of R , is called the passband boundary. The frequencies wcl and wc2 are said to be the cutoff frequencies of the filter. A filter with an amplitude response

W%

02)

=

0 1

for ( q , w2) E R , otherwise

95

The Application of thez Transform

.

1c

1c

is said to be a rectangularly symmetric highpass filter. A rectangularly symmetric bandpassfilter is one thatwill pass frequency components located between two rectangles, that is,

W%

02)

=

1 0

for (ol, 0,) E R1 r l R, otherwise

where R2 =

((01, 0 2 )

:1011

4 WC3

and

1021 5 o c d

where wc3 > ocl and oC4> wc2. Similarly, a rectangularly symmetric bandstop filter is one that will reject frequency components located between two rectangles, that is,

M(%

02)

=

0 1

for (ol, w2) E RI rl R, otherwise

The amplitude responses of the fourtypes of rectangularly symmetricfilters defined above are illustrated in Fig. 4.5. Corresponding circularly symmetric ideal filters can readily be identified by replacing rectangular areas R1 and R, by discs Cl and C, defined as

c1 = ((01,

02)

:

m,

5 0,1}

96

Chapter 4

Figure 4.5 Amplitude responses of rectangularly symmetric filters: (a) lowpass; (b) highpass; (c) bandpass; (d) bandstop.

97

The Application of the z Transform

\ -lF

1F' n

98

Chapter 4

.

- A

\

-K

A

'

K

n

n

n

(b)

Figure 4.6 Amplitude responses of circularly symmetric symmetli c filters: (a) lowpass; highpass; (c) bandpass; (d) bandstop.

The Application of the z Transform

.

-x

x

x

99

100

Chapter 4

and c 2

= ((o1,oz) :

5 oc2)

where oC2> W,,. The amplitude responses of the four types of circularly symmetric filters defined above are illustrated in Fig. 4.6. It is worth noting at this point that theamplitude response of a rectangularly symmetric lowpass filter can be expressed as

M(%

U21

=

M1(@11MZ(U2)

where

and

Consequently, such a filter can be designed by designing two l-D lowpass filters and then connecting them in cascade [g]. Similarly, in the case of a rectangularly symmetric bandpass filter, we can write

M(%

W21

=

M1(4M2(%)

where 1

for (W,, 02)E R , otherwise

1 0

for (W,, 02)E R, otherwise

0 and M2(w21

=

{

A s a result a bandpass filter can be obtained by connecting in cascade a highpass filter with cutoff frequencies W, and o , and a lowpass filter with

and oC4 where oC3 oC1 and oC4> oC2, Hence, like cutoff frequencies oC3 rectangularly symmetric lowpass filters, rectangularly symmetric bandpass filters can be designed in terms of l-D filters. In effect, these filters are separable and are, as a consequence, easier to design than corresponding circularly symmetric filters. Another type of filter that has been found useful in the processing of geophysical data is the so-called fan filter. An ideal fan filter may have an

101

The Application of the z Transform

-a

1 - R

.

a

(a> Figure 4.7 fan filter.

Amplitude response of fan filters: (a) bandpass fan filter; (b) bandstop

amplitude response.

or

(4.17) where Se is a sector given by

102

Chapter 4

c

-7r \

-a

(b)

.

Figure 4.7 Continued.

as depicted in Fig. 4.7a and b. The amplitude responses in Eq. (4.16) and Eq. (4.17) describe a bandpass and a bandstop fan filter, respectively. Note, however, that a filter which is bandpass with respect to the w1 axis is a bandstop filter with respect to the w2 axis and vice versa. Practical 2-D filters differ from ideal ones in that transitions between passbands and stopbandsare gradual, the gain in a passband is never exactly equal to unity, and the gain in a stopband is never exactly equal to zero. Furthermore, the contoursof the amplitude response in a rectangularly or circularly symmetric filter are only approximately rectangular or circular.

The Application of thez Transform

103

REFERENCES 1. J. L. Shanks, Two-dimensional recursive filters, SWZEECO Rec., pp, 19E119E8, 1969. 2. J. L. Shanks, The designof stable two-dimensional recursive filters, Proc. UMR-M. J . Kelly Commun. Conf., Univ. Mksouri, pp.15-2-1-15-2-2, Oct. 1970. 3. J. L. Shanks, S. Treitel, and J. H. Justice, Stabilityandsynthesis oftwodimensional recursive filters, ZEEE Trans. Audio Electroacoust., vol. AU-20, pp. 115-128, June 1972. filters, ZEEE Trans. Audio 4. T. S. Huang, Stability of two-dimensional recursive Electroacoust., vol. AU-20, pp. 158-163, June 1972. 5. P. A. Ramamoorthy and L. T. Burton, Design of two-dimensional recursive filters, in Topics in Applied Physics, edited by T. S. Huang, vol. 42, pp. 4183, New York: Springer Verlag, 1981. 6. R. M. Mersereau, W. F. G . Mecklenbrauker, andT. F. Quatieri, Jr., McClellan transformationfor two-dimensional digital filtering: I-Design, ZEEE Trans. Circuits Syst., vol. CAS-23, pp. 405-413, July 1976. 7. M. S. Reddy andS. N. Hazra, Designof elliptically symmetric two-dimensional FIR filters using the McClellan transformation,ZEEE Trans. Circuit Syst., vol. CAS-34, pp. 196-198, Feb. 1987. of 2-D digital filters 8. D. T. Nguyen and M.N. S. Swamy, Approximation design with elliptical magnitude response of arbitrary orientation, ZEEE Trans. Circuits Syst., vol. CAS-33, pp. 597-603, June 1976. 9. A. Antoniou, M. Ahmadi, and C. Charalambous, Design for factorable lowpass 2-dimensional filters satisfymg prescribed specifications,ZEE Proc., vol. 128, pp. 53-60, April 1981.

PROBLEMS 4.1

Obtain the transfer function of the filters represented by (a) the network in Fig. P1.5a and b. (b) the network in Fig. P 1 . 5 ~and b.

4.2

Obtain the transfer function of the filters represented by (a) the network in Fig. P1.6. (b) the difference equation given in Problem 1 . 7 ~ .

4.3

Obtain the transfer function of the filters represented by (a) the difference equation given in Problem 2.2b. (b) the difference equation given in Problem 2 . 2 ~ .

4.4

Obtain the transfer function of the filter described by the state-space equation given in Problem 2.5.

Chapter 4

104 4.5

Check the BIB0 stability of the filters characterized by (a) the state-space equation in Problem 2.5 (b) the transfer function 1

H(21’22) = zlz2

+ 0.22, + 0.5

(c) the transfer function

q z 1 , 22) =

2122

+ 2 2 - 0.5

zlz2 + 0.82, - 0.52, - 0.4

4.6

Find the unit-step response of the filter described by (a) the transfer function in Problem 4.5b. (b) the transfer function in Problem 4 . 5 ~ .

4.7

The transfer function given by Eq. (4.3a) can be written as

where and

By using the singular-value decomposition technique, show that the transfer function can be expressed as

where r = rank (A) and

4.8

Consider the same transfer function as in Problem 4.7. Show that if B ( z l , 2,) is separable, then H(tl, 2,) can be expressed as a sum of t separable subfilters, that is,

of the z Transform

Application The

where F k ( t l ) and Gk(z2) for 1 4 k filters. 4.9

105 5

r are l-D causal recursive

Find the steady-state sinusoidal response of the filters represented by the transfer functions

(b)

m 1 9 22)

=

2122

- 0.421 - 0.422' 4.10 Write MATLAB programs that generate the amplitude responses of the following idealized filters: * (a) a circularly symmetric lowpass filter with normalized cutoff frequency w, = 0.57~. (b) a circularly symmetric highpass filter with normalized cutoff frequency W, = 0.57~. 2122

4.11 Write MATLAB programs that generate the amplitude responses of

the following idealized filters: * (a) a circularly symmetric bandpass filter with wcl = 0.47~ andwC2 = 0.67~. (b) a circularly symmetric bandstop filter with wC1 = 0.47~ andwc2 = 0.6 7~. *Each of the programs in Problems 4.10 and 4.11 may be written as a MATLAB function with the cutoff frequency and the size of the sampled amplitude matrix as parameters. The recommended size of the sampled amplitude matrix is 91 X 91 for workstations and 71 X 71 for personal computers.

5 Stability Analysis

5.1 INTRODUCTION A digital filter must be stable in order to ensure that any bounded input

will produce a bounded output. Otherwise, the filter is not useful for any practical application. When a filter is beingdesigned, it is possible to obtain a transfer functionthat satisfies the required amplitude-or phase-response specifications while the filter is unstable. On the other hand, a filter that has been designed to satisfy the necessary conditions for stability onthe basisofinfinite-precisionarithmeticmaybecomeunstablewhen the transfer-function coefficients are represented in terms of finite-precision arithmetic. Consequently,it is of paramount importancein both the design and application of digital filters to have simple, robust, and efficient stability tests and criteriathat can be used to determine with certainty whether a given filter is stable or not. In thischapter the stability properties of 2-D digital filtersare examined and the necessary and sufficient conditions for stability are identified. Then frequency-domain tests and criteria are described, which can be used to check whether a 2-D digital filter is stableor not without actually finding its singularities. Subsequently,the stability properties of 2-D digital filters are reexamined inthe state-space domain, and corresponding stability conditions and tests are obtained. These results lead to the generalization of the classical Lyapunov stability test for l-D systems to the case of2-D 106

Stability Analysis

107

digital filters. The chapter concludes with some simple stability tests for low-order filters.

5.2 STABILITY ANALYSIS IN FREQUENCY DOMAIN Bounded-input, bounded-output (BIBO) stability can be studied in the frequency domain by examiningthe constraints imposed on the singularities of the transfer functionby the requirement that the output of the filter be bounded when the input is bounded. Consider a quarter-plane (causal) 2-D digital filter characterized by the transfer function

where

and

If two polynomials m(z;l, til) and b ( z ; ' , z~l)have no common factors, they are said to be factor coprime. On the other hand, if there are no points (zi', z, l ) at which the two polynomials assume the value of zero, the two polynomials are said to be zero coprime. Two polynomials in one variable are factor coprime if and only if they are zero coprime, as can be easily verified. Nevertheless,two polynomials intwo variables can be zero coprime without being factor coprime, as can be seen in Example 4.lb. In the following analysis, R(Zi', 2,') and b ( Z i ' , z~l),given by Eqs. (5.2) and (5.3), are assumed to be factor coprime but may or may not be zero coprime.

5.2.1 Stability Properties The 2-D filter.characterizedby the transfer function in Eq. (5.1) is stable if and only if its impulse responseh(nl, n2)is absolutely summable, as was demonstrated in Sec. 1.7. This important result is restated as Theorem 5.1 for the sake of convenience.

10s

Chapter 5

Theorem 5.1

B I B 0 stability {h(n,,nz)} E l, where {h(n,, n2)}is said to belong to l, if

The arrows + and t stand for "implies" and "is implied by, " respectively. While Eq. (5.4) with p = 1 is a necessary and sufficient condition for BIBO stability, the evaluation of the double summation is highly impractical since the impulse response is notin general easy to obtain from the transfer function. An alternative approachto establish whether a 2-D digital filter is stable or not is to use Shanks' stability theorem, which was stated as Theorem 4.1 in Sec. 4.3. This theorem is repeated below for the sake of convenience. Theorem 5.2 (Shanks) A 2-D digitalfilter represented by the transferfunction in Eq. (5.1) is BIBO stable if

D(z;',

2 )',

+O

2 )',

E 77

I, Iz,'~

I1 )

for (Zi',

(5.5)

where p is the unit bidisc defined as

F

= {(ZT',

2 )',

: lzill

I

Shanks' condition, whilesufficientfor BIBO stability, isnotalways necessary and the question, therefore, arises as to how close it is to being necessary. It turns out that the stability of 2-D digital filters depends on the location of nonessential singularities of the second kind. These singularities have been defined in Sec. 4.3 as points in the (zi', z ~ ' )plane at which the numerator and denominator polynomialsof the transfer function become zero while being factor coprime. To illustrate the role of singularities of this type, let us consider the transfer functions

and

Both of these transfer functions satisfy the following conditions: (i) the numerator and denominator polynomials are factor coprime, (ii)

Stability

109

D(zil, z;l)

+

o for ( 2 ~ 1 z1;,) E P except at ( z c ' , 2)'; = (I, I), and (iii) each transfer function has a nonessential singularity of the second kind at point (z; l , z; l ) = (1,l).The first filter can be shown to be stable by demonstrating that its impulse response is absolutely summable [l], whereas the second filter can be shown to be unstable by demonstrating the opposite. Evidently, if a filter satisfies Shanks' condition except that it has nonessential singularities of the second kind on the unit bicircle U2 given by UZ

= { ( Z i l , z;')

: 1~;~1

I,.(z;'l

=

I}

a more detailed analysis is necessary. Some additional stability properties which are sometimes useful are summarized in terms of the following three theorems. Their proofs are given by Goodman [l]. Theorem 5.3 H ( z l , zz) has no nonessential singularities of the first kind on p e BIBO stability. A nonessential singularity of the first kind is a point (z;l, z; l ) such that b ( z ; ' , z ; l ) = 0 and N ( z i ' , 2;') f 0. The symbol -H stands for "does not imply." The "does not imply" part of this theorem follows from the preceding two examples. Theorem 5.4

-

D(z;',

where -

UZ - U2

=

2 )';

f 0 on

{ ( Z ; ~ , z;l)

P - U2

: ( z ~ l2;,')

BIBO stability

E p and ( z ~ l2;,')

E

V}

The "does not imply" part of the theorem follows from the preceding examples. From Theorems 5.3 and 5.4, it follows that if a 2-D digital filter is BIBO stable, then b ( Z ; ' , z;l) # 0 on p except on U2 where all the zeros of D(Z,', z;l) must be nonessential singularities of the second kind of H(zl, 22).

Theorem 5.5

{h(n,, nz)}E lZ

B I B 0 stability

This theorem follows from Theorem 5.1 if it is noted that

{h(% nz)} E l1 {h(% nz)} E l2 In the rest of this chapter it is assumed that the transfer function has no nonessential singularities of the second kind on U Z , for the sake of

Chapter 5

110

simplicity. For this classof filters, Shanks' condition is both necessary and sufficient for stability; that is,

-

D ( z ; ' , z~l)# 0 on

v

BIBO stability

(5.6)

The stability of such filterscan, in principle,be verified by demonstrating has no zeros in p.Unfortunately, however, this type that b ( z i ' , z~l) of analysis is not easy in general and researchers havespent considerable effort during the past 20 years in developing alternative necessary and sufficient conditionsfor BIBO stability that are simpler to check. Someof these results are stated in terms of the following three theorems and form the foundation of a number of stability criteria, as will be demonstrated in Sec. 5.2.2. Theorem 5.6 (Huang [Z])

if 1. D(z,', 0)

2.

D(z,',

+O

ZZ')

A 2-D digital filter is BIBO stable if and only

for lzi'I

5

I.

4 O for Izill

=

(5.7a)

1and

1 ~ I I. ~ ~ 1

(5.7b)

Proof. The condition in Eq. (5.5) implies the conditions in Eqs. (5.7a) and (5.7b). To demonstrate the converse, let

and note that for fixed z,l with Jz;'I I1, the condition in Eq. (5.7b) implies that the preceding integral is well defined and, therefore, continuous, and represents the number of zeros of D(Zil, z;') in the unit disc Jz,'J 5 1 (see Silverman [3]). On the other hand, the condition in Eq. (5.7a) implies that n,(O) = 0. Since n&') is simultaneously continuous and an integer-valued function, we have n,(z,') = 0 for 122 5 1, which implies that the condition in Eq. (5.5) holds. Theorem 5.7 (Strintzis [4]) A 2 - 0 digital fiZter is BIBO stable if and only

if I1 .

(5.8a)

some p with 1pI I1.

(5.8b)

1. D(a, z~l)# 0 for )z;~) I1 and some a with la1 2.

D(zT', 9)

# 0 for lzilI

3. D ( z i ' , ZT')

+O

for

I1and

(ZY', 2 ~ 1 ) E

V.

(5.8~)

This theorem can be provedby using an argument similarto that used to prove Theorem 5.6. The special case where a = 9 = 1 is of particular interest and is stated in terms of the following corollary.

Stability Corollary 5.1 A 2-D digital filter is BIBO stable if and only if 1. D(1, 21') # 0 for

1) # 0 for

2.

D(2;1,

3.

D(2;1, 211)

IZFlI I1.

(5.9a)

I1.

(5.9b)

12;q

# 0 for

(2;1,

21')

E

v.

(5.9c)

Theorem 5.8 (DeCarlo, Murray, and Saeks[S]) A 2 - 0 digitalfilter is BIBO stable if and only if 1. D(+, 2-1) # 0 for 2. D ( q 1 , 211) # 0 for

(5.loa) (5.10b)

12-11 I

1.

&l,

221)

E

v.

Proof. Let kl and k, bethe numbers of zeros of D(z;l, 1)and ) < 1 and 12; < 1, respectively, and assume thatthe D(1, z ~ l in conditions in Eqs. (5.10a) and (5.10b) hold. From Rudin [6], the condition in Eq. (5.10b) implies that b(t-',2-l) has kl + k2 zeros in [.z-~\ < 1. Hence the conditions in Eqs. (5.10a) and (5.10b) imply that D(1, z ~ l # ) 0 for lz;lI I1 and

D(Z;',

I) # O

for

IzF'I

I1

By Corollary 5.1, it follows that the filter is BIBO stable. To prove the "only if" part of the theorem, we note that the filter is unstable if either of the two conditions is violated. m Some additional stability properties of 2-D digital filters and multidimensional discrete systems in general can be found in a paper published by Jury [7] and in the work of Bose [8].

5.2.2

Stability Criteria

Stability tests or criteria are efficient algorithms that can be used to determine whether a system is stable or not without actually finding the singularities of the transfer function. Classical stability criteria that have been used extensively in l-D continuous systems in the past are theRouthHurwitz and Nyquist stability criteria [9]. A related and equally useful stability criterion that has been used extensively in l-D discrete systems and digital filters is the Jury-Marden criterion [lo]. Theorems 5.2 and 5.6 to 5.8 have shown that there are two types of conditions that need to be satisfied to assure the stability of a 2-D digital filter, namely, conditions where the polynomial involved is a function of one variable, e.g., the condition in Eq. (5.8a), and conditions where the polynomial involved is a function of two variables, e.g., the condition in

Chapter 5

112

Eq. (5.5). Conditions of the first type can be checked by using the JuryMarden stability criterion, which will now be described. Given a real polynomial p ( z ) of the form p ( z ) = pozn

+ plzn-l +

+ Pn,

PO

>0

(5.11)

the so-called Jury-Marden table can be constructed by first forming two initial rows as {c11 c12

Cl,n+J

=

{PO

P,)

p1 *

{dl1 d l , . * dl,n+J = { p n P n - 1 . *PO} and then computing a number of pairs of subsequent rows as

cij = det

[

Ci-1.1

Ci-l,n-j-i+3

4-1.1

di-l,n-j-i+3

I

fori

=

2,3,. . . , n

-

1

and d [J. . = c.r , n - j - i + 3 Once the Jury-Marden table is constructed, stability conditions involving polynomials in one variable can be readily checked by applying the following theorem. Theorem 5.9 The zeros of polynomial p ( z ) , designated us zi for i = 1,2, . . . , n, are located on the open unit disc (i.e., lzil < 1 ) if and only if 1. p ( 1 ) > 0.

2. ( - l y p ( - l ) 3. ICil[ > I c ~ ,

> 0.

~ - ~ + ~for ~ i

= I, 2,

. . . ,n

- 1.

The preceding table along with Theorem 5.9 constitute the Jury-Marden stability criterion. An alternativel-D stability criterion can be obtained by using the SchurCohn matrix given by

K where 1 S i , j

In

=

{kij}

(5.12)

and

In this case, stability conditions involving polynomials in one variable can be checked by noting the following theorem.

Stability Analysis

113

Theorem 5.10 The zeros of polynomial p ( z ) are located on the open unit disc ifand only if its Schur-Cohn matrix K in Eq. (5.12)is positive definite.

The construction of matrix K and the application of Theorem 5.10 constitute the Schur-Cohn l-D stability criterion. Matrix K can be checked for positive definiteness (a) by diagonalizing it and checking its diagonal elements, (b) by computing the determinants of its principal minors, or (c) by finding its eigenvalues. If the diagonal elements in (a) or the determinants in (b) or the eigenvalues in (c) are all positive, then K is positive definite (see Noble [111 for details). The Jury-Marden and Schur-Cohn stability criteria necessitate that or z , l be changed into polynomials in z1 or 2, by polynomials in '2, multiplying them by z y or z y . For example, the polynomial in Eq. (5.9b) can be put in the appropriate form by multiplying it by z y ; that is, With this modification, it is evident that the condition in Eq. (5.9b) is satisfied if and only if the zeros of p ( z l ) are located on the openunit disc. If the stability of a 2-D digital filter is to be checked by using Theorem 5.6, the condition in Eq. (5.7a) can be readily checked by using the JuryMarden or Schur-Cohn criterion since it involves a polynomial in one variable. The condition in Eq. (5.7b), however, involves a polynomial in two variables and a more advanced method is, therefore, needed. It turns out that a generalization of the Schur-Cohn criterion is applicable, as will now be demonstrated. Let us multiply the polynomial in Eq. (5.7b) by z y to obtain a polynomial in z2 as c(z2) =

zyD(z;1,

(5.13)

ZT1)

We note that if the condition in Eq. (5.7a) holds, then the condition in Eq. (5.7b) is the same as c(z2) f 0

for l Z i l I = 1 and 1z212 1

(5.14)

By writing c(z2) in the form

where coefficients c k are polynomials of N2 Schur-Cohn Hermitian matrix C(zi') =

Zil,we

{~ij}

can construct the NZ x (5.15)

Chapter 5

114

where 1 5 i, j

S

NZ,and

The condition in Eq. (5.14), and therefore the condition in Eq. (5.7b), can now be checked by applying the following theorem. Theorem 5.11 The condition in Eq. (5.14) holds if and only if the SchurCohn matrix C ( z ; l ) in Eq. (5.15) is positive definite for lzilI = 1. Further progress in this analysis can be made by noting the following results, which are due toSiljak [12]. Matrix C ( z ; l ) is positive definite for lz;lI = 1 if and only if C(l) is a positive-definite matrix

(5.16a)

and f(z;') = det C ( z ; l)

>0

for Iz;lI = 1

(5.16b)

where polynomial f ( z ; l) always has the form

on the unit circle 1z;lI = 1. By defining a polynomial g ( z l ) as K

(5.17) the condition in Eq. (5.16b) can be transformed into thesimpler but equivalent conditions

and g ( z l ) has no rootsonthe

unit circle Izi'1 = 1

(5.18b)

Evidently, the condition in Eq. (5.18b) involves a polynomial in one variable and can, therefore, be easily checked by finding the roots of g ( z l ) using a reliable software package like MATLAB [l31 or by applying a recursive procedure described by Siljak [14].

115

Stability Analysis

The preceding results can be summarized as BIB0 stability S

Eq. (5.7a) Eq. (5.7b)

Eq*(5*7b) e

{

Eq. (5.16a) Eq. (5.16b)

and, therefore, a 2-D digital filter canbe checked for stabilityby checking the one-variable conditionsin Eqs. (5.7a), (5.16a), (5.18a) and (5.18b)in this order. Example 5.1 A 2-D digital filter is characterizedby the transfer function

where

-

D(z;',

= 1 - Z i ' - 0.72;'

2;')

+ 0.25~;~ + 0.672;'z,'

- 0* 1621-' 22

Check the filter for stability. Solution. The polynomial D(Zi', 0) = 1 - ti'

+ 0.252i2

has two zerosat 2,' = 2 and hencethe condition in Eq. (5.7a) is satisfied. Polynomial c(t2) given by Eq. (5.13) can be formed as 22D(2i1, 2,')

~(22)

=

(1 - ti'

+ 0.252;2)22

-

(0.7 - 0.672,'

+ 0.16~1~)

Since the highest power of z2 is one, the Schur-Cohn matrix for Iz;lI = 1 given by Eq. (5.15) is a 1 X 1matrix and can be expressed as

+ 0.252i2)(1 - 21 + 0.252:) - (0.7 - 0.672;' + 0.162i2)(0.7 - 0.6721 + 0.162:) = 1.0980 - 0.6738(21 + 2,') + 0.1380(2; + 2 i 2 )

C(Zi') = (1 -

Zil

Hence C(l) = 0.0254 > 0

116

Chapter 5

and the condition'in Eq. (5.16a) is satisfied. FromEqs. (5.16b) and (5.17) the polynomial g(zl) can be obtained as g(Z1) = =

2;

det C(zil)

0.1380 - 0.673821 + 1.09802: - 0.67382;

+ 0.1380~:

and so g(1) = 0.0264 > 0 that is, the condition in Eq. (5.18a) is satisfied. Finally,the zeros of g(zl) can be obtained as z1 = 2.0,

z1 = 1.8388,

z1 = 0.5438,

z1 = 0.50

and, as can be seen, g(zl) has no roots on the unit circle IZi'1 = 1, that is, the condition in Eq. (5.18b) is satisfied. Therefore, the filter is stable. The preceding procedure can also be used to check the two-variable condition D(z;', 2c1) # 0 for ( t i l , 22') E LIZ (5.19) which appeared in Corollary 5.1 and Theorems 5.7 and 5.8. For this condition an alternative approach proposed by Chiasson, Brierley, and Lee [l51 is also applicable. This approach is based on the resultant method for checking whether two polynomials have common zeros (see Walker [16]) and it is sometimes easier to apply. If p ( z ) and q(2) are two polynomials given by p ( 2 ) = pn2" + pn-12n-1

+

'

+ Po,

pn f 0

and q(2) =

qm2m

I-

qm-lZm-l

+ .. - + 40,

qm

f

0

then p ( z ) and q(z) have no common zeros if and only if det R # 0 where R denotes the so-called resultant matrix of polynomials p ( z ) and q(z) given by

Stability

117

To illustrate the application of the resultant method for the checking of the condition in Eq. (5.19), let us define the reciprocal polynomial of D ( z i ' , ZZ') as

D(z;',

z;1)

= Z,N1Z,N2D(Zl,

22)

Evidently, if D ( z i l , 2;')

then

= O

-

D@,,z2) = 0

at

(Z;',

2 , ' )

at (zl, z2) E

E

UZ

v2

and in addition

-

D * ( z ~22) , = b(tT,2;) = b ( z i ' , 2;')

=

O

at

(Zi',

ZZ')

E U2

In other words, the condition in Eq. (5.19) will be violated if and only if D(Zi', 2;') and D ( Z i ' , zz') have one or more common zeros on v2.On the basis of this observation, the following step-by-step procedure [l61 can be used to check the condition in Eq. (5.19). STEP

1: Write polynomials b ( Z i l ,

2;')

and D(zT ',2,')

in the form

respectively and obtain the determinant of the resultant matrix R as

118

Chapter 5 = 0 and

2:Find the roots of the l-D algebraicequationr(zi-1) represent them by z
forall i in the range 1 Ii

IK

then the condition in Eq. (5.19) holds. Otherwise, proceed to Step 3. 3:If lz (N - 1)/2. 2. It is symmetricalwithrespect to the nT axis, that is, wl(- n T ) = w,(nT).

Its amplitude spectrum, on the other hand, consists of a main lobe and several side lobes, such that the area of the side lobes isa small proportion of the area of the main lobe, as illustrated in Fig. 6.1. The parameters of a window functionare its order N,the width of its mainlobe, and its ripple

Figure 6.1 Spectrum of typical window function.

Chapter 6

140

ratio, which is the ratio of the maximum side-lobe amplitudeto the mainlobe amplitude, usually as a percentage [8]. The application of a window function consistsof multiplying the impulse response obtained by applying the Fourier series by the window function to obtain a modified impulse response as

hw(nT) = wl(nT)hl(nT) Since the windowfunctionisof function given by

(6.15)

finite duration, a finite-order transfer (N- l)n

2

Hw(z) =

wl(nT)h,(nT)z"'

(6.16)

n= - ( N - l ) / Z

is obtained.Further; if h,(nT) is an even function with respect to nT, then w,(nT)h,(nT) is also an even function with respect to nT by virtue of property (2) above and, therefore, the filter obtained haszerophase response. Through the application of the l-D complex convolution,the frequency response of the modified filter can be expressed as

Hm(ei"T)

=

2 2T

1

2dT

0

Hl(e@T)W1(e""-")*)dil

(6.17)

As demonstrated elsewhere [8], the application of the window function has two effects on the amplitude responseof the filter. First, the amplitudes of Gibbs' oscillations in the passbands and stopbands are directly related to the ripple ratio of the window. Second, transition bandsare introduced between passbands and stopbands whose widths are directly related to the main-lobe width of the window. The most frequently used window functionsare as follows: 1. Rectangular window: =

{

for 1. S ( N otherwise

-

1)/2

2. von Hann and Hamming windows: a

WH()2T)=

+ (1 - a) cos-

21~n N - l

for 1.

S

( N - 1)/2

otherwise

where a = 0.5 inthe von Hann window and a = 0.54 in the Hamming window.

141

Approximations for Nonrecursive Filters

3. Blackman window: w&T)

=

0.42

47rn + 0.08 COS + 0.5 COS- N21~n - l N - l

for In1

5

(N

-

1)/2

otherwise

I O

The parameters of these windows for N = 21,41, and 61 are summarized in Table 6.1. As can be seen, the ripple ratio depends on the window but is relatively independent ofN. The rectangular window corresponds to direct truncation of the Fourier series and, as can be seen in Table 6.1, its ripple ratio is quite large. An alternative window function which has the attractive property that its ripple ratio can be adjusted continuously from the low value of the Blackman window to thehigh value of the rectangular window bychanging a window parameter is the Kaiser window function. This is given by

where

01

is an independent parameter and

p

=

a[l -

(&)2]1n

Function Zo(x) is the zeroth-ordermodified Bessel function of the first kind and can be evaluated by using the rapidly converging series ZO(X)

=

1+

5

k=l

['(I)*I2 k! 2

Table 6.1 Parameters of Window Functions

Type of window

Rectangular von Hann Hamming Blackman

Ripple ratio (%)

Main-lobe N width

=

20,lN &,IN &,IN 60, JN

21.89 2.67 0.93 0.12

21

N =N4 = 161 21.77 2.67 0.78 0.12

21.74 2.67 0.76 0.12

Chapter 6

142

As in other window functions, the main-lobe width of the Kaiser window function can be adjusted by changing the value of N. Another variable window function is one developed by Dolph for the design of antenna arrays [g]. This window is based on the properties of Chebyshev polynomials and is oftenreferred to as the Dolph-Chebyshev window. For odd values of N, it is given by [10,11]

where

r =

amplitude of side lobes amplitude of main lobe

where ck(x) is Chebyshev polynomial given by =

{

'

cos@cos - x ) cosh(kcosh-' x )

for 1x1 I1 for 1x1 > 1

The most significant properties of the Dolph-Chebyshev window are as follows:

1. The amplitudes of the side lobes are all equal. 2. The main-lobe width is minimum for a given ripple ratio. 3. The ripple ratio can be independently assigned. 6.3.3Design

of 2-DFilters

Since the frequency response H2(ejmth,ejwzTz) of a 2-D digital filter is a periodic function with respect to variables (wl, w2) with fundamental period (2dT1, 2dT2),we can write X

X

Approximations for Nonrecursive Filters

143

where

provided that the preceding double summation converges. With the substitutions dolT1= z1and dWzT2 = z2, we obtain m

m

Hence, as with l-D filters, if an expression is available for the frequency response, a transfer function can be obtained, which happens to be noncausal and of infinite order. A finite-order transfer function can be obtained by letting

in Eq. (6.18) but as with l-D filters, the truncation of the Fourier series tends to introduce Gibbs’ oscillations. A causal transfer function can be obtained by introducing delays of (Nl - 1)T1/2 and (N2 - 1)T2/2 in the impulse response with respect to the nlTl and n2T2 axes. This is accomplished by modifying the transfer function as

If the required frequency response is an even function with respect to o1and 02,then h2(n1T1, n2T2) turns outto bean even function with respect to nlTl and n2T2. Consequently, the filter represented by H2(z1,z2) has

zero phase response and the filter represented by H;@,,G) has linear phase response.

6.3.4 2-D Window Functions The principles described in Sec. 6.3.2 can readily be applied for the elimination of Gibbs’ oscillations in digital filters designed by using the 2-D Fourier series. This is accomplished by using 2-D window functions. A 2-D windowfunction, representedby w2(nlT1,n2T2),is a 2-D discrete function with the following space-domain properties: 1. It has a finite region of support, that is,

144

Chapter 6

2. It is symmetrical with respect to the nlTl and n2T2axes, that is, W2(-nlTl, -n2T2) = wz(n1T1, n2T2) On the other hand, its amplitude spectrum consists of a 3-D main lobe centered at the origin of the (col, co2) plane and a numberof 3-D side lobes such that the volume of the side lobes is a small proportion of the volume of the main lobe. The application of 2-D window functions is analogous to thatin the case of l-D filters. The impulse response obtained by applying the 2-D Fourier series is multiplied by the 2-D window function to obtain amodified impulse response as h,(nlTlY n2T2) = wz(n1T1, n,T2)h2(nlTl, n27-2) (6.22) and since the window function has finite supportby virtue of property (1) above, a finite-order transfer function given by

HurZ(z1, 22) (N1-l)n

= cnt = -(NI

(NZ- 1)n

c

l)L? nz = -(NZ- 1)n

W2(nlTlY nzT,)h,(n,T,,

n27-2)Z;"'ZFn2

(6.23)

is obtained. If h2(n1T1,nJ2) is an even function with respect to nlTl and n2T2, then h,(nlTl, n2T2) is also an even function with respect to nlTl and n2T2by virtue of property (2) and, consequently, the filter represented by Hm(zl, z2) has zero phase response. Through the application of the 2-D complex convolution given in Sec. 3.2.5, the frequency response of the modified filter can be expressed as

If w,(nlTl) and wlB(n2T2) are l-D windows .of the type described in Sec. 6.3.2, then the function has the required properties and, therefore, it can serve as a 2-D window function. Another way to construct satisfactory 2-D window functions was proposed by Huang [l].In this approach, a l-D window wl(nT) is transformed into a 2-D window w2(nlT1,n2T2)by letting

The validity of Huang's approach is demonstrated in the following analysis.

Approximations forFilters Nonrecursive

145

Theorem 6.1 Let Wl(z) and Wz(zl, z2) be the z transforms of wl(nT) and w2(nlTl, %T2), respectively, andassumethat w2(nlTl, n2T2) is obtained from wl(nT) byusing Eq. (6.26). ZfAl(zl) andA2(zl,z2) are allpussfinctions such that =

A 2 ( z l > z2)

=

for all z1and z2, then m

where the symbol @ represents convolution.

Proof. From Eq. (6.26),wenote that wz(n1T1, 0) = wl(nlT1)

and since

and

and so

From the l-D complex convolution, we can write

(6.27)

Chapter 6

146

and, similarly, from the 2-D complex convolution (see Sec. 3.2.5), wehave

f ['2 1 ~fj Wz(v,, v2v2)A2(4?, x] dv, dv, v1 f [ f W2(v,,v,)v,l dv,1v,' dv, 21rj rl 27rj rz

= 2nj rl

r2

V1

=

It will now be demonstrated that Eq. (6.26) provides a mechanism for generating good 2-D windows for the broaderclass of 2-D circularly symmetric filters with arbitrary piecewise-constant amplitude responses. Assume that Tl = T2 = T and let (6.30) be thefrequency response of an arbitrary l-D ideal bandpass digital filter, where g is a constant, oa 2 0, and oa 5 n / T , and let

be the frequency response of a corresponding 2-D circularly symmetric bandpass digital filter. If w,(nT) is a l-D window function with a mainlobe width B such that B l

f I m l , 231

that is, half-plane symmetry does not imply quadrantal symmetry. The design of these filters can be accomplished by using two quadrant pass filters. Consider an ideal 2-D lowpass filter with a square passband and passband edges wlP = oZp= 1d2 - E, as depicted in Fig. 7.10a (E is a small positive constant). A separable realization of this type of characteristic can be achieved by cascading two l-D lowpass filters, as described in the previous section. Let the transfer function of the filter obtained be On applying the transformations

&l,

01, 229 02)

=m , ,

e,)fr,(z,,

02)

where the coefficients of R,(z,, e,), l?&,, €l2), and fi(zl, e,, z2, e,) are complex, in general. The effect of the preceding transformations is to shift the amplitude response of the original 2-D filter by 8, and 0, in the directions of the positive W, axis and the positive w2 axis, respectively. If 8, = 0, = d2, the passband of the original filter is shifted into thefirst quadrant of the (m1, 0,) plane, as shown in Fig. 7.10b. The filter obtained is said

Approximations for Recursive Filters

201

Figure 7.10 (a) Amplitude response of ideal lowpass filter; (b) effect of transformation on the amplitude response.

202

Chapter 7

to be a first-quadrant pass filter. By shifting the passband of the original 2-D filter into second, third, etc., quadrant,second-, third-, etc., quadrant pass filters can be obtained. The transfer functions of these filters are given by (7.23a) (7.23b) (7.23~) (7.23d) where the subscript of fi identifies thd quadrant to which the passband has been moved. These filters can now be used to synthesize two quadrant zero-phase pass filters. On using Eqs. (7.23a) and (7.23~) orEqs. (7.23b) and (7.23d), we have

or

where fi(zl, z2)is the transfer function obtained by replacing the coeffiz2) by their complexconjugates.Note that fi,*(zr1, cientsin l?&,, q") is the complex conjugate of fil(zl, and z2)so

Hence each of the two terms in Eqs. (7.24a) or (7.24b) is real when z1 = ejolT1and z2 = ejoZT2 and, furthermore, each will yield a passband in one quadrant. Thus the filters characterized by Eqs. (7.24a) and (7.24b) have passbands in the first andthird quadrants and the second and fourth quadrants, respectively, as illustrated in Fig. 7.11 By using the method described in Sec. 7.3.1, arbitrary quadrantally symmetric filters whose passbands or stopbands are combinations of rectangular domains can readily be designed. By connecting two quadrant pass filters in cascade with these designs, filters whose characteristics have half-plane symmetry can also be designed. For example, by connecting the filter designed in Example 7.1 in cascade with the filter characterized by Eq. (7.24a), the chracteristic of Fig. 7.12 is obtained.

Approximations for Recursive Filters

-

"1

(b) Figure 7.11 Two quadrant pass filters: (a) Eq. (7.24a); (b) Eq. (7.24b).

203

204

Chapter 7

Figure 7.12 Filter with half-plane symmetry.

7.4 DESIGN OF CIRCULARLY SYMMETRIC FILTERS Transformations can also be usedfor the design of 2-Dnonseparable filters having piecewise-constant amplitude responses with circular symmetry. Noteworthy methods of this class are those described by Costa and Venetsanopoulos [2] and Goodman [6]. In these methods, transformations that can produce rotation in the amplitude response of either an analog or a digital filter are used to achieve circular symmetry in the amplitude response. The two methods leadto filters that are, in theory, unstable but by using an alternative transformation proposed by MendonGa et al. [9], this problem can be eliminated.

7.4.1 Method ofCosta and Venetsanopoulos In the method of Costa and Venetsanopoulos, a set of 2-D continuous transfer functionsis obtained by applyingthe transformation of Eq. (7.12a) for several different values of the rotation angle p to a l-D continuous lowpass transfer function.A set of 2-D discrete lowpass transfer functions is then deduced throughthe application of the bilinear transformation.The

Approximations Filters for Recursive

205

design is completed by connecting the 2-D digital filters obtained in cascade. The steps involved are as follows: STEP

1: Obtain a stable l-D continuous lowpass transfer function

(7.25) where zai and pIli for i = 1, 2, . . . , are the zeros and poles of HA1(s), respectively, and KOis a multiplier constant. STEP 2:

Let

Pk

for k = 1,2, . . . , K be a set of rotation angles given by

Pk=

{

[(2k - 1)12K + 111~ for even K [ ( k - 1)lK + 111~ for odd K

(7.26)

STEP 3: Apply the transformation of Eq. (7.12a) to HA1(s) to obtain a 2D continuous transfer function as HAZk(sl,

=

”SI

sin p*+= cos pr

(7.27)

for each rotation angle Pk identified in Step 2. STEP 4:

Apply the double bilinear transformation of Eq. (7.14) to HAzk(s1,

s2) to obtain

Assuming that Tl = T2 = T , Eqs. (7.25) and (7.27) yield

where N-M K1

=

KO($)

206

Chapter 7

In order to illustrate the design procedure, consider the design of a 2-D filter using the second-order Butterworth transfer function

and assume that osl = os, = 10 rad/s and K = 6 . From Step 2, the rotation angles are 195", 225", 255", 285", 315", and 345". The amplitude response of the 2-D digital subfilter obtained in Step 4 for k = 4, that is, for a rotation angle of 285", is illustrated in Fig. 7.13a. The amplitude response obtained by cascading the subfilters with rotation angles 195", 225", and 255" consists of a set of concentric ellipses as shown in Fig. 7.13b and that obtained by cascading the subfilters with rotation angles 285", 315", and 345" consists of a set of concentric ellipses as shown in Fig. 7.13~.Now, connecting all these subfilters in cascade yields an overall amplitude response whose contours are concentric circles as depicted in Fig. 7.13d. The 2-D digital filter designed by the preceding method will be useful in practice only if the transfer function in Eq. (7.29) is stable for each rotation angle &. From Eq. (7.29), we can write

where

The filter designed will be stable if and only if each Hi(zl, z2) represents a stable first-order filter. At point (zl, z2) = (- 1, - l ) , both the numerator and denominator polynomials of Hi(zl, z2)assume the value of zero, as can be readily seen has a nonessential from Eqs. (7.30) and (7.32), and hence each Hi(zl, z2) singularity of the second kind on the unit bicircle

Consequently, the stability methods of Chap. 5 cannot be used to check the stability of the filter designed. Nevertheless, Goodman [6] has shown that in theory Hi(zl, z2) represents an unstable subfilter in general by demonstrating that the impulse response of the subfilter is not absolutely summable. In practice, however, the numerator polynomials of H i ( z l , z2)

ApproximationsFilters for Recursive

207

are unlikely to be precisely equal to zero, owing to coefficient quantization, and the subfilter may or may not be stable depending on chance. The nonessential singularity of each Hi(zl,z2) can be eliminated and, furthermore, each subfilter can be stabilized by letting b;2i

=

b12i

=

b22i

+ ~b11i + Eb21;

(7.33a) (7.33b)

where E is a small positive constant. With this modification, the denominator polynomial of each Hi(zl, z2) is no longer zero and, furthermore, from Sec. 5.5, the stability of the subfilter can be guaranteed if

1. IbLil < IbLl

From Eqs. (7.30) and (7.33), condition 1 is satisfied if

where pi is the real part of Tpai/2.Now l 4

L

since E > 0, and if we assume that (7.35) we have 1

and on letting

we obtain

208

Chapter 7

1

0.8 0.6 -

0.4

-

0.2

-

0-

-0.2

-

-0.6 -

-0.4

-0.8 -1 -1

d

-0.5

0

0.5

,

,

,

,

-0.5

0

0.5

1

l

0.4 -

0.2 -

0-

::l -0.4

-1 -1 (b)

Figure 7.13 Contour plots for rotated filters: (a) subfilter for rotation angle of 285"; (b) subfilters for rotation angles of 195", 225", and 255"incascade; (c) subfilters for rotation angles of 285", 315", and 345" in cascade;(d) all subfilters in cascade. (In these plots the Nyquist frequencies are normalized to unity.)

209

Approximations for Recursive Filters 1-

0.8

-

0.6 0.4 0.2

-

0-0.2 -0.4 -

-0.6

-

-0.8 -

'

-1 -1 (c>

-0.5

0

0.5

1

Chapter 7

210

On the other hand, b21i

-

blli

-

b21i

lb;2i - bi2il - /b22i- blzi

-

b11i

+ E(bZli- blIi)1 =

l&l


0,

sin P k > 0

and c > 0

Chapter 7

214

then the application of this transformation followed by the application of the double bilinear transformation yieldsstable 2-D digital filters that are free of nonessential singularitiesof the second kind. If, in addition c=-

1 4lax

then local-type preservation canbe achieved on set sl, given by

-

sl, =

{(Wl,

W2)

:0

1

2

0, W2

= 0,

w,w2



where N- h4 K1

=

KO($)

and all!=

-cosPk-sinpk-

aIzi= -cospk

+ sinPk-

(I-+-$)

zai, aZli=cosPk-sinPk-

zoi, aZzi= cospk + sinPk-

(I

--$).ai

ecursive Approximations for

215

The 2-D discrete transfer function obtained can readily be shown to be free of nonessential singularities of the second kind by examining its behavior at z1 = ? l, z2 = & l. An equivalent design can beobtained by applying the allpass transformation of Eq. (7.41) with ck =

1 + cos P k - sin 1 - cos P k - sin

111+ ek = 1-

dk

=

Pk - (4c/TZ) Pk + (4c/T2) + sin Pk - 4c/TZ - sin Pk + 4c/TZ

cos P k cos P k cos pk + sin P k cos P k - sin P k

+ 4c/T2 + 4c/T2

in Goodman's method.

7.4.4

Realization of Highpass, Bandpass, and Bandstop Filters

Although the design of circularly symmetric lowpass filters can be achieved by using any one of the preceding methods, the design of circularly symmetric highpass filters is not as straightforward, as will now be demonstrated. Consider two zero-phasefilter sections that have been obtained by rotating a l-D continuous highpass transfer function by angles -P1 and PI, where 0" < P1 < 90". Idealized contour plots for the two sectionsare shown in Fig. 7.15a and b. If these two sections are cascaded, the amplitude response of the combination is obtained by multiplying the amplitude responses of the two sections at corresponding points of the (q, oz)plane. The idealized contour plot of the cascade filter is obtained as shown in Fig. 7.15~.As can be seen, the resulting contour plot does not represent the amplitude response of a 2-D circularly symmetric highpassfilter and, therefore, the design of these filters cannot be achieved by cascading rotated sections as with circularly symmetric lowpass filters. Their design can, however, be accomplished through the use of a highpass configuration suggested by Mendonca et al. [9], which comprises a combination of zerophase cascade and parallel sections. If the preceding rotated sections are connected in parallel, a filter is obtained whose contour plot is shown in Fig. 7.15d. By subtracting the output of the cascade filter from the output of the parallel filter, a filter is obtained whose contour plot is shown in Fig. 7.15e.Evidently, this plot resembles the idealized contour plot of a 2-D circularly symmetric highpass

216

Chapter 7

1 1

(4

2

1

(4

Figure 7.15 Derivation of highpassconfiguration: (a) contour plot of section rotated by angle PI; (b) contour plot of section rotated by angle - p,; (c) contour plot of sections in (a) and (b) connected in cascade; (d) contour plot of sections in (a) and (b) connected in parallel; (e) contour plot obtained by subtracting the amplitude response of the cascade filter in (c)from that of the parallel filter in (d).

Approximations for Recursive Filters

217

filter. The configuration obtained is illustrated in Fig. 7.16a and is characterized by fil

=

fil

=

+

f p l

H-P1

- Hp1H-P'

where = Hl(z1, z ~ ) H ~ ( z ; Z' ,T ' )

H p 1

and H-Pl

=

H 1(21 7 z?1)H1(z;17

z2)

represent zero-phase sections rotated by angles p1 and - pl, respectively. As in lowpass filters, the degree of circularity can be improved by using several rotation angles. For two different rotation angles p1 and p2, the transfer function of the 2-D highpass filter is obtained as fi2

=

fil

+ H z - &,R2

where

R2 = f p z + H - P 2 - f p z H - P z with H e z = H z ( ~ 1 z2)H~(zi1, , ZT~)

and H-Sz

=

H2(z1,z ~ ' ) H 2 ( z ; lz2) ,

Similarly, for N rotation angles, fiN

=

fiN-1

fiN

is given by the recursive relation

+RN

-

fiN-1fiN

H - ~ N

-

~ N H - P N

(7.45)

where

RN= f

p + ~

and can beobtainedfrom H N - l and f i N - 2 , and so on.The configuration obtained is illustrated in Fig. 7.16b, where the realization of H N - l is of the same form as that of f i N . The rotatedsections should be designed using the transformation in Eq. (7.43) in order to avoid the problem of nonessential singularities of the second kind. However, the use of this transformation leads to another problem: the 2-D transfer function obtained has spurious zeros at the Nyquist points. These zeros are due to thefact that the transformation in Eq. (7.43) does not have type preservation in the neighborhoods of the Nyquist points but their presence does not appear to beof serious concern. It should also be mentioned that the complexity of the configuration in

218

Chapter 7

I

I

(6)

Figure 7.16 2-D highpass configuration: (a) one rotation angle; (b) N rotation angles.

Fig. 7.16tends toincrease rapidly with the number of rotations and, therefore, their number should be kept to a minimum. With the availability of circularly symmetric lowpass and highpass filters, bandpass andbandstop filters with circularly symmetric amplitude responses, like those in Fig. 4 . 6 ~and d, can readily be obtained. A bandpass filter can be obtained by connecting a lowpass filter and a highpass filter with overlapping passbands in cascade, whereas a bandstop filter can be obtained by connecting a lowpass filter and a highpass filter with overlapping stopbands in parallel.

7.5 CONSTANTINIDES TRANSFORMATIONS A l-D discrete transfer function can be transformed into another

l-D discrete function by using a set of transformations due to Constantinides [13]. These transformations are also useful in the design of 2-D digital

Approximations Filters for Recursive

219

filters and, as was demonstrated by Pendergrass, Mitra, and Jury [3], their application leads to a variety of amplitude responses. They are of the form (7.46) where I and m are integers and a: is the complex conjugate of a,. A s can readily be demonstrated [14], these transformations map the unit circle I t 1 = 1 of the t plane, its exterior, and its interior onto the unit circle (21 = 1 of the Z plane, its interior, and its exterior, respectively. Therefore, passbands and stopbands in the original transfer function give rise to passbands and stopbands, respectively, in the derived transfer function. Furthermore, a stable transfer function results in a stable transfer function. By choosing the parameters I, m,and a, in Eq. (7.46) properly, a set of four specific transformations can be obtained that can be used to transform a lowpass transfer function into a corresponding lowpass, highpass, bandpass, or a bandstop transfer function. These transformations are summarized in Table 7.2, where subscript i is included to facilitate the application of the transformations to 2-D discrete transfer functions. Let aiand wi for i = 1 , 2 be the frequency variables in the original and transformed transfer function, respectively. If H N ( z I z2) , is a lowpass transfer function with respect zi with a passband edge Ofi,then the application of the lowpass-to-lowpass or lowpass-to-highpass transformation of Table 7.2 will yield a lowpass or highpass transfer function with respect to tiwith a passband edge of wfi. On the other hand, if the lowpass-to-bandpass or lowpass-to-bandstop transformation of Table 7.2 is applied, a bandpass or bandstop filter is obtained with passband edges w f I iand wfzi. On applying any two of these transformations to HN(zl,z2), the 2-D transfer function

F,) =

HN(z1, 22)

I

z1=fi(t1),

zz=fi(iz)

is obtained. Some of the possible amplitude responses are illustrated in Fig. 7.17a-f.

7.6 DESIGN OF FILTERS SATISFYING PRESCRIBED SPECIFICATIONS Practical 2-Ddigital filters differ from their ideal counterparts in that their passband gain is only approximately equal to unity, their stopband gain is only approximately equal to zero, and transitions between passbands and stopbands are gradual. Practical filters can be made to approach their ideal counterparts as closely as desired by increasing the order of the transfer function, but since this would increase the complexity of the filter, and

t4 h,

0

Table 7.2 Constantinides Transformations

Transformation

Type

Parameters

-

LP to LP

2.

'

LP to HP

zi= -2

LP to BP

=

z. =

zi

-

1

2, - a, 1 - aizi

a. =

'

sin[(flpi - wpi)Ti/2]

sin[(flp, + opi)Ti/2]

zi - ai -1 - aizi

2aiki k. - 1 -zi + ki + 1 ki + 1 2a.k. ki - 1 2 2, + --! k, + 1 ki + 1

a. =

'

COS[(O,,~.

+ ~,,i)Ti/2]

COS[(O,,~.

- ~,ii)Ti/2]

flpiTi ( o p t . - o p 1 i ) T i ki = tan -cot 2 2 -2

LP to BS

zi =

zi

1 + k,

1 + ki

201, zi 1+ki

+ -z:l1 +-kkii -

1 - ki - 2ai -2, + -

l - -

a. =

'

cos[(wpt.

+ OPIi)Ti/2]

cos[(opt. - OP,,)Ti/2]

Approximations for Recursive Filters

221

I "2

Figure 7.17 Application of Constantinides transformations to 2-D digital filters: (a) circularly symmetric lowpass filter; (b) LP to LP for z, and z,; (c) LP to HP for z, and 2;, (d) LP to HP for z, and LP to LP for z,; (e) LP to BP for z, and z,; (f) LP to BP for z, and LP to LP for 2.,

222

Chapter 7

hence the amount of computation needed for its implementation, the lowest-order transfer function that satisfies the required specifications with a small safety margin should be used. The problem of designing filters that satisfy prescribed specifications has to a large extent been solved for the case of l-D filters, and by extending the available methods [14], 2-D digital filters that satisfy prescribed specifications can also be designed.

7.6.1

LowpassFilters with RectangularPassbands

Let us assume that a 2-D lowpass digital filter characterized by H(zl, 2,) is obtained by cascading two l-D lowpass digital filters characterized by H,(z,) and H2(z2).The amplitude response of the cascade filter is given by

M(%, 0 2 )

=

M1(01)M2(02)

(7.47)

where

Mi(oi) =

IHi(ejwiTi)l,

i = 1, 2

If we assume that (1

SPi) 5 Mi(oi) I1 for loi[S opi 0 IMi(oi) ISai for loil 2 wai

-

(7.48a) (7.48b)

then Eqs. (7.47) and (7.48) yield (1 - S,,,)(l - S,,,) IM(w,, 0,) I1 for lo1['oPland lo21IW, 0 IM(o,, W*) IS,, for loll 2 W,] 0 IM(ol, 0,) ISa2 for lo,[ L W,

(7.49a) (7.49b) (7.49c)

Now if the minimum passband gain and the maximum stopband gain of the cascade arrangement are assumed to be (1 - A,,) and A,, respectively, and that S,, = S,, = S,, and S,, = S,, = S,, Eq. (7.49a) gives

(1 - S,,),

=

1 - A,

and since S, 0, r is the rank of A, ui and vi are the ith eigenvector of A A T and ATA, respectively, +i = u,lnui,yi = u,lnvi, and {+i : 1Ii Ir} and { y i : 1 I i 5 r} are sets of orthogonal L-dimensional and M-dimensional vectors, respectively. An important property of the SVD can be stated as

where $i E RL,qi E R', that is,

and norm IlXll may be either theFrobenius norm, lL?

M

or the L2 norm, that is,

where hi is an eigenvalue of X% and X = {xlm} E R L x M The . preceding is a minimal relation shows thatfor anyfixed K (1IK Ir),:X mean-square-error approximation to A. Since all the entries of A are nonnegative, it follows that all the entries of and y1 are nonnegative [16,17]. Nevertheless, some elements of +i and yi for i 2 2 may assume negative values.

8.5.2 Design Approach In a quadrantally symmetric filter, H(zl, z2) has a separable denominator [ 6 ] .Therefore, H(zl, z2) can be expressed as

Now note that Eq. (8.30) can be written as

A = where

+nT

+

E1

(8.33)

Design of Recursive Filters

by Optimization

253

On comparing Eq. (8.33) with Eq. (8.32) and assuming that K = 1 and that + l , y1 are sampled versions of the desired amplitude responses for the l-D filters characterized by Fl(z,) and G1(z2),respectively, a 2-D digital filter can be designed through the following steps:

1. Design l-D filters Fl and G1 characterized by Fl(zl) and G1(z2). 2. Connect filters Fl and G1in cascade. Step 1 can be carried out by using a quasi-Newton optimization algorithm, such as Algorithm 1 of Sec. 8.2.2 or by using a minimax algorithm, such as Algorithm 2 in Sec. 8.3.2. When filters Fl and G1are designed, we have 15 I

IFl(ei"")l = +l,,

5

L

and

1 Im

IGl(ejmum)(= ylm,

S

M

where +lr and ylm denote the Zth component of and mth component of yl,respectively. The transfer function of the cascade filter obtained in Step 2 is given by H1(z17

z2)

=

Fl(~l)Gl(~Z)

where (IH1(ejmW, e")l)

(+lnlm)

=

$1~;

and from Eq. (8.33)

II IA - H1(ejWW', e")l

1l = IIA - +nTll

=

IlElll

The approximation error associated with transfer function Hl(zl, z2) can be reduced by realizing more of the terms in Eq. (8.30) by means of parallel filter sections. From Eq. (8.30), we can write (8.34) where

Since +2 and y2may have some negative components, a careful treatment of the second term in Eq. (8.34) is necessary. Let $1 and ;y be the absolute values of the most negative components of +z and y2, respectively. If e, = [l1

l]' E RL

and

e, = [l1 *

- l]=E RM

Chapter 8

254

then all components of

62

and 9 2 = Y2 + s e , are nonnegative. Now let us assume that it is possible to design l-D linearphase or zero-phase filters characterized by F&), G1(z2), F2(z1),and G2(z2) such that =

+2

+ +Fe+

where

and qzmare the Zth component of i2 and mth component of p, Here 621 respectively, and al,a2are constants that are equal to zero if zero-phase filters are to be employed. Let a1 =

-m1,

a2 =

- m 2

with integers n,, n2 2 0

and define F2(z1)= F2(z1)- +;z;”l G 2 ( 4 = G2(~2) - ~ 2 ~ 1 “ ’

and

Moreover, if we form

then

(8.35)

255

Design of Recursive Filters by Optimization which in conjunction with Eq. (8.34) implies that IIA - (H2(ejmkt, ej-

)I II = IIA 5

-

I+nT

+

+2rTI II

IIA - (+lYT + +2Y2T)ll =

= minllA

llE2ll

(8.36)

+’ $, + f)ll

-

d?i.qi

Evidently, through the preceding technique it is possible to realize the second term in Eq. (8.34) by means of a parallel subfilter, thereby reducing E ~ According . to Eq. (8.36), the twothe approximation error from to section 2-D filter obtained has an amplitude response that is a minimal mean-square-error approximation to the desired amplitude response. corresponds to thelargest singular valueul,the subfilter Since Fl(zl)Gl(z2) characterized by Fl(zl)Gl(zl)is said to be the main section of the 2-D represents a correction to the filter. On the otherhand, IF2(ejmw’)G2(ej-)l is said amplitude response and the subfilter characterized by F2(z,)G2(z2) to represent a correction section. Through the use of data +i and yi (i = 3, . . . , K , K I r) given in Eq. (8.30), vectors and qi can be found, andcorrection sections characterized by Fi(z1)Gi(z2) can then be designed in a similar manner. When K sections are designed, HK(zl,z2)can be formed as K

~ K ( z 1~,

2 = )

C~i(z~~i(z2) i=l

and from Eq. (8.36) we have

S

I/

1 1 ~ ~ 1=1 min A +. ^ . ,.y,

2

+iyTll

i= 1

In effect, a 2-D digital filter comprising K sections is obtained whose amplitude response is a minimal mean-square-error approximation to the desired amplitude response. The method leads to an asymptotically stable 2-D filter, provided that all l-D subfilters employed are stable. This requirement is easily satisfied in practice. The general structure of the 2-D filter obtained is illustrated in Fig. 8.5, where the various l-D subfilters may be either linear-phase or zero-phase filters, as was shown earlier. The structure obtained is a parallel arrangement of cascade low-order sections, and, consequently, the traditional advantages associated with parallel andor cascade structures apply. These

256

Chapter 8

)M

Figure 8.5 General structure of 2-D filter.

include low sensitivity to coefficient quantization, efficient computation due to parallel processing, and a relatively low number of required multipliers. If linear-phase subfilters are tobe employed, the equalities in Eq. (8.35) must be satisfied. This implies that the subfilters must have constant group delays. Causal subfilters of this class can be designed as nonrecursive filters by using the weighted-Chebyshev approximation method [18,19] or by using the Fourier series method along with the Kaiser window function [20]. In such a case, the 2-D filter obtained can be used in real-time applications. The application for theSVD for thedesign of 2-D nonrecursive filters will be studied in detail in Chap. 9. If a record of the data to beprocessed is available, the processing can be carried out in nonreal time. In such a case, the subfilter in Fig. 8.5 can be designed as zero-phase recursive filters. The resulting structure is depicted in Fig. 8.6, where and Fi(z;l)[Gi(z2) and -di(zzl)] contribute equally to the amplitude response of the 2-D filter. The design can be completed by assuming that thedesired amplitude response for subfilters F,, G,, Ti, and Gi for i = 2, . . . , K are +in, y:n, +;'2, and y'" for i = 2, . . . , K,respectively. When a circularly symmetric 2-D filter is required, the design work can be reduced significantly. Matrix A defined in Eq. (8.29) is symmetric and, therefore, Eq. (8.30) becomes "

(8.37)

257

Design of Recursive Filters by Optimization

Figure 8.6 Structure using zero-phase recursive filters.

where s1 = 1 and si = +- 1 for 2 Ii Ir. This implies that each parallel section requires only one l-D subfilter to be designed and asa consequence the design work is reduced by 50 percent.

8.5.3 Error Compensation When the main section and the correction sections are designed by using an optimization method, approximation errors inevitably occur that will accumulate and manifest themselves as the overall approximation error in the design. The accumulation of error can to a certain extent be reduced by using the following compensation technique. When the design of the main section is completed, define an errormatrix

El = A - Fl(ejmpf)Gl(ejmm) and then perform SVD on El to obtain El = S22+22YT2 +

*

-

+ &z+r2Y;

(8.38)

Data +22 and yu in Eq. (8.38) can be .used to ,deduce F2(z1)and G2(z2) as in Sec. 8.5.2; thus, the first correction section can be designed. Now form error matrix E2 as

E2 = El - ~ ~ ~ F ~ ( e j ~ p ~ ) G ~ ( e j - ) =

A - [F,(ej"P')G,(ej-)

+ s,F2(ejmpf)G2(ejmm)]

258

Chapter 8

and perform SVD on E, to obtain

E, = ~33+33~,T,+

+ ~r3+r3~2

As before, data +33 and y33can be used to design the second correction section. The procedure is continued until the norm of the error matrix becomes sufficiently small for the application at hand. Example 8.4 Design a circularly symmetric, zero-phase 2-D filter specified by

assuming that oSl=

W,

=

27r.

Solution. By taking L = M = 21 and assuming that the amplitude response varies linearly with the radius in the transition band, the sampled amplitude response is given by a 21 x 21 matrix as A = [A'0

(8.39)

where

1

A1

1 1

0.75 0.75 0.5 0.5 0.25 0 0.25

... ...

1 0.75 0.5 0.25 0 0.75 0.5 0.25

... ...

...

...

0

12x 12

The ideal amplitude response of the 2-D filter is shown in Fig. 8.7. It is worth noting that although the vector = yl) obtained from the SVD of A is a typical sampled amplitude response for a l-D lowpass filter, the data given by the SVD error matrices E, and E, lead to the necessity of designing l-D filters with arbitrary amplitude response. For example, given A as in Eq. (8.39), the square root of is obtained as [3]

+in

=

[1.04151.02631.02631.00050.96250.91200.83000.7075 0.55140.37050.1866

0

. . . O]*

Now if a sixth-order approximation isobtained for transfer function F,(,?,),

Design of Recursive Filters by Optimization

259

- A

IC

A

Figure 8.7 Ideal amplitude response of circularly symmetric lowpass filter (Example 8.4).

the SVD of E, gives

6:"

=

[1.05560.95880.96150.81140.82000.62110.3735

0.0 0.2030

0.46680.64360.77950.83680.83760.83760.83760.8376 0.83760.83760.83760.8376IT which represents an irregular amplitude response. By using the procedure in Sec. 8.5.2 along with the errorcompensation technique in Sec. 8.5.3 and a l-D optimization algorithm based on Algorithm 3, a 2-D zero-phase filter comprising the main section and two cor-

260

Chapter 8

Figure 8.8 Amplitude response of (a) mainsection; (b) mainsectionplus one correction; (c) main section plus two corrections (Example8.4).

rection sections has been designed. The filter coefficients obtained with sixth-order l-D transfer functions for the various subfilters are given in Table 8.4. The amplitude responses of (a) the main section, (b) the main section plus one correction section, and (c) the main section plus two correction sections are depicted in Fig. 8.8a-c. Design of l-D digital filters by using optimization methods can sometimes yield unstable filters. This problem can be eliminated by replacing poles outside the unit circle of the t plane by their reciprocals and simultaneously adjusting the multiplier constant to compensate for the change in gain (see Sec. 7.4 of Antoniou [20]).

261

Design of Recursive Filters by Optimization

2 -K

-K

z

-x

x

x

262 W

F

09

0 F

0 N

+

W

m W

9 4

+

N

N L

F

m

m

r

N

2 + S m

09 4

+

N

N L

W W

r

m

2 + 3

3

2 +

N

N L

I

,

I

Chapter 8

Design of Recursive Filters by Optimization

263

REFERENCES 1. G. A. Maria and M. M. Fahmy, An l, design technique for two-dimensional digital recursive filters, IEEE Trans. Acomt., Speech, Signal Process., vol. ASSP-22, pp. 15-21, Feb. 1974. filters, 2. C. Charalambous, Design of 2-dimensional circularly-symmetric digital IEE Proc., vol. 129, pt. G,pp. 47-54, April 1982. 3. A. Antoniou and W.-S. Lu, Design of two-dimensional digitalfilters by using the singular value decomposition, IEEE Trans. Circuits Syst., vol. CAS-34, pp. 1191-1198, Oct. 1987. 4. R. Fletcher, Practical Methods of Optimization, 2nd ed., Chichester: Wiley, 1987. 5. D. G.Luenberger, Linear and Nonlinear Programming, 2nd ed., Reading, Mass.: Addison-Wesley, 1984. 6. P. K. Rajan and M. N. S. Swamy, Quadrantal symmetry associatedwith twodimensional digitaltransfer functions, IEEE Trans. CircuitsSyst., vol. CAS29, pp. 340-343, June 1983. 7. C. Charalambous, A unified review of optimization, IEEE Trans. Microwave Theory and Techniques, vol. "'IT-22, pp. 289-300, March 1974. 8. C. Charalambous and A. Antoniou, Equalisation of recursive digital filters, IEE Proc., vol. 127, pt. G,pp. 219-225, Oct. 1980. 9. C. Charalambous, Acceleration of the least pth algorithm for minimax optimization with engineering applications, Math Program., vol. 17, pp. 270297, 1979. 10. P. A. Ramamoorthy and L. T. Bruton, Design of stable two-dimensional analog and digitalfilters with applications in image processing, Circuit Theory Appl., vol. 7, pp. 229-245, 1979. 11. P. A. Ramamoorthy and L. T. Bruton, Design of stable two-dimensional recursive filters, Topics in Applied Physics, vol. 42, T. S. Huang, ed., pp. 41-83, New York: Springer Verlag, 1981. 12. H. Ozaki and T. Kasami, Positive real function of several variables andtheir applications to variable networks, IRE Trans. Circuit Theory, vol. (3-7, pp. 251-260, 1960. 13. H. G. Ansel, On certain two-variable generalizations of circuit theory, with applications to networks of transmission lines and lumped reactances,IEEE Trans.Circuit Theory, vol. CT-11, pp. 214-223, 1964. 14. T. Koga, Synthesis of finite passive networks with prescribed two-variable reactance matrices, IEEE Trans. Circuit Theory, vol. CT-13, pp. 31-52,1966. 15. G.W.Stewart, Introduction to Matrix Computations, New York: Academic Press, 1973. 16. R. E. Twogood and S. K. Mita, Computer-aided design of separable twodimensional digital filters, IEEE Trans. Acomt., Speech, Signal Processing, vol. ASSP-25, pp. 165-169, Feb. 1977. 17. P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed., New York: Academic Press, 1985.

Chapter 8

264

18. J. H. McClellan, T. W. Parks and L. R. Rabiner, A computer program for

designing optimum FIR linear phase digital filters, ZEEE Trans. Audio Electroacoust. , vol. AU-21, pp. 506-526, Dec. 1973. 19. A. Antoniou, New improved method for the design of weighted-Chebyshev, nonrecursive,digitalfilters, ZEEE Trans. Circuits Syst., vol.CAS-30,pp. 740-750, Oct. 1983. 20. A. Antoniou, Digital Filters: Analysis and Design, New York: McGraw-Hill, 1979 (2nd ed. in press).

PROBLEMS 8.1 Verify Eq. (8.4). 8.2

Consider Algorithm 1 with C k obtained by using the DFP or BFGS formulas. Show that if Sk is positive definite, then J(xk

+

< J(xk)

for any a > 0. (In other words, the hereditary property of the DFP and BFGS formulas assures the descent property of the algorithm.) 8.3 Design a circularly symmetric highpass filter of order (2,2) with up,= up, = 0.12 rad/s, W,, = W, = 0.081~ rad/s, and W,, = W, IT rad/s by the least pth optimization. 8.4

Verify Eqs. (8.9) and (8.10).

8.5 Design a circularly symmetric highpass filter of order (2, 2) with the same specifications as in Problem 8.3 by using Algorithm 2. 8.6

Repeat Problem 8.5 by using Algorithm 3.

8.7

Design a circularly symmetric highpass quarter-plane filter of order (5, 5) with W, = 0.31~ rad/s and W,, = W, - 1.21~ rad/s.

8.8 Design a circularly symmetric highpass nonsymmetric half-plane filter radls and W,, = W,, = 1.21~ rad/s. of order (5, 5) with W, = 0.31~ 8.9 By applying the SVD method to the idealized amplitude response

where T = 1 S , obtain a 2-D transfer function of the form given in Eq. (8.32) with K = 3. The orders of the l-D transfer functions in Eq. (8.32) should be equal to orless than 6.

Design of Recursive Filters

by Optimization

265

8.10 Repeat Problem 8.9 with K = 6 . Compare the design results with

those obtained in Problem 8.9 8.11 Repeat Problem 8.9 with the error compensation method described

inSec. 8.5.3. Compare the designresultswiththoseobtainedin Problem 8.9.

Design of Nonrecursive Filters by Optimization

9.1 INTRODUCTION In Chap. 8, several optimization methodsthat can be used for the design of recursive filters have been described in detail. With some modifications, these methods can also be applied for the design of nonrecursive digital filters [l-51. The most serious problem in applying optimization methods for the design of nonrecursive filters relates to the fact that these filters havelow selectivity. Consequently, even a moderately demanding application would require a high-order filter that would, in turn, entail a large number of filter coefficients. The magnitude of this problem can to some extent be reduced by reducing the number of independent filter coefficients. As was shown in Chap. 6, a linear phase response implies that the impulse response of the filter satisfies a set of symmetry conditions [seeEq. (6.9)], and, as a result, the number of independent filter coefficients is reduced by half. If, in addition, the amplitude response of the filter has certain types of symmetry, then the number of independent filter coefficients can be reduced further. This chapter begins with a study of some general symmetry properties of 2-D nonrecursive filters. Then a minimax optimization method for the design of linear-phase nonrecursive filters due to Charalambous [3] is described. The second half of the chapter dealswith application of the SVD method of Sec. 8.5 for the design of linear-phase nonrecursive filters with arbitrary amplitude responses. 266

Design of Nonrecursive Filters

by Optimization

267

9.2 MINIMAX DESIGN OF LINEAR-PHASENONRECURSIVE FILTERS The minimax method of Sec. 8.3 yields some fairly good recursive designs, as was demonstrated in [2] of Chap. 8. With some modifications that address the basic differences between recursive and nonrecursive filters, the minimax method of Sec. 8.3 can readily be applied for the design of nonrecursive filters. 9.2.1 Symmetry Properties

Consider a nonrecursive 2-D filter with a transfer function given by NI-l NZ-l

H(z1, 22) =

2 2

n l = 0 nz=O

h(nlT1, n2T2)~;"'~2"~

The filter has a linear phase response if its impulse response satisfies Eq. (6.9), as was shown in Sec. 6.2.1. The frequency response of such a filter is given by Eq. (6.10). Many practically useful digital filters possess certain symmetry properties in the frequency domain. Forinstance, circularly symmetric lowpass, bandpass, and highpass filters have octagonal symmetry. That is, the amplitude response of a filter of this class satisfies the relations IH(ejolT1,ejo2h)l = IH(e-jwT1 ejozTz)I = IH(ejolT1e-jozTz >I (9.1)

for 1 S nl

NI - 1 ,lSn2S-

I

NZ- 1 2

268

Chapter 9

Another important class of filters is the class of fan filters which have quadrantal symmetry. The amplitude responseof these filters satisfiesEq. (9.1), which implies that h(n1,n2) = h(n1, N2 - 1 - n2) = h(N1 - 1 -

n1, n2)

For a digital filter with octagonal symmetry and Nl = N2 = N with N odd, the frequency response canbe written as [3] H(ejwT1, ejozTz) = M(" 1 7 2)e-j(N-l)(olTI+ozTz)R (9.5) where

+ c0s(n2"1T*) cos(n1w2T2)] +

(N - 1)R

c.

n=O

a(n, n) cos(n"lTl) cos(no2T2) (9.6)

On the other hand, for a digital filter with quadrantal symmetry with Nl and N2 odd, the frequency response can be written as H(ejw TI,ejw Tz) = M(",,~ 2 ) ~ - j I ~ N ~ - l ~ o ~ ~ ~ + ~ N z - l ) ~ z T z l R where (NI- l)R (NZ- l)R

M(",,"2)

=

c c

nl=0

nz=O

+l,

n2) cos(nl"1~1) C&2"27-2)

(9.7)

9.2.2 MinimaxOptimizationAlgorithm The design of nonrecursive filters can be transformed into an unconstrained minimax optimization problem by defining the objective function as E(x, "1, "2) =

"1,

"2) - MI(",, "2)

where M ( x , col, 02) is the actual amplitude response of the filter to be designed and MI(",,02) represents the ideal amplitude response. The former quantity given is byEq. (9.6) or (9.7) depending upon the symmetry properties of the filter. As in Sec. 8.3, the preceding error function can be minimized in the minimax sense by finding a vector x that solves the optimization problem minimize F ( x ) X

269

Design of Nonrecursive Filters by Optimization where F(x) = max fi(x) lcism

and &(x) = IE(x, coli, wJ,

i = 1, . . . , m

Although Algorithm 3 of Sec. 8.3.2 is directly applicable to thepresent design problem, Charalambous [3] has found that better results can be achieved by using the conjugate direction method of Powell [6] for the unconstrained minimization required in Step 2 instead of a quasi-Newton method. Conjugatedirection methods are preferred because they are more efficient than other methodswhen the number of independent variables is large and the objective function is a positive-definite quadratic function with respect to x. Function @(x, 5, A) can be shown to have these properties by using Eqs. (9.6) and (9.7). An excellent analysis of the properties of conjugate direction methods can be found in Fletcher [7, pp. 63-69]. Example 9.1 Design a 2-D nonrecursive circularly symmetric lowpass filter with W,, = 0 . 4 1W,~ ~= 0 . 6 1 and ~ ~ oSl= W, = W, = IT by using the optimization algorithm described in Sec. 9.2.2. Solution. A solution of this problem obtained by Charalambous [3] is as follows. Since the frequency response of the filter to be designed is circularly symmetric, it satisfies Eqs. (9.1) and (9.2). Consequently, sampling points need to be chosen only in the [0-45"] sector of the (m1, 02) plane. Furthermore, it is appropriate to choose the sampling points on arcs of circles encircling the origin. If n,, and n, are the numbers of circles in the passband and stopband regions, respectively, then the radii of the circles may be determined as

, rnp+nc-i=

W,

+

{

[

i = 1,2,.

. . ,np

1- cos (n, - i - 1)2n, - 3]}(l i = 1,2,. . . , n , - 2 ~

ma17

With W,, = 0 . 4 1W,~ ~= 0 . 6 1n,, ~ ~= 8 and n, = 9, the radii of the circles described above can be calculated as

Chapter 9

270

for the passband region, and

O . ~ I T ,0.60871~, 0.63461~, 0.67641~, 0.73231~, O . ~ I T ,0.87631~, 0.95821~, IT for the stopband region. On each circle in the sector between 0" and 45", seven equally spaced sample points can be chosen, and the sample points (wl, w2) can be obtained as (w11, w22)

(wlk,

= (0, 0)

wzk) = (Ti COS Y j , Ti sin ~ i ) ,

where

Yj =

IT(i - 1) 24

for i = 1 , 2 , . . . , 17, j = 1 , 2 , . . . , 7 , and k = 2, . . . , 120. As can be seen from Fig. 9.1, additional sample points are needed to cover the points whose distance to the origin is larger than IT. On each of the circles with radius

for i = 2, 3, . . . , 7, (8 - i ) equally spaced points can be chosen in the sector between (i - 1)180"/24 and 45". With these additional 21 points and the origin included, there are a total of 141 sample points over the [O-45'1 sector of the (m1, w2) plane. By defining

ab,

x =

y)

N + l 2

a(i, i )

a(i,

y)

N - l N - l 2 ' 2

N-2i+1 2

Design of Nonrecursive Filters by Optimization

271

1

cos[(N

1

-

cos W,

+ cos

l)o,/2]

+ cos[(N - l)w2/2]

W2

...................................

...................................

cos io,cos i o 2 I

cos io,cos[(N - l)w,/2]

L I

+ cos[(N - 1)02/2]cos io*

...................................

...................................

cos[(N - 1)0,/2] cos[(N - 1)02/2]

Equation (9.6) can be written as M(o1, 0 2 , x) =

C=(%

w2)x

With a given total number of circles n,, the number of circles inthe passband np and the number of circles in the stopband nu should be chosen as

nu = n, - np where int[v] denotes the largest integer not greater than v. are Someresults obtained using a starting point x(0) = 0 and E = summarized in Table 9.1. The amplitude response of a 17 X 17 design is depicted in Fig. 9.lb.

9.3

DESIGN OF LINEAR-PHASENONRECURSIVEFILTERS USING SVD

In the design of recursive filters by means of the SVD (see Sec. 8.5), high selectivity can be achieved by using low-order recursive designs for the parallel l-D subfilters. However, zero phaseis required for each subfilter. This necessitatesdata transpositions at the inputs and outputs of subfilters, and, as a result, the usefulness of these designs is limited to nonreal-time applications wherethe delay introduced in the processing is unimportant.

Chapter 9

272

"*

t

Sample points

/l

Figure 9.1 Design of circularly symmetriclowpass filter (Example 9.1): (a) sample points of filter; (b) amplitude response achieved using a filter of order 17 x 17. (From [3].)

In this section, it is shown that the SVD of the sampled frequency response of a 2-D digital filter with real coefficients possesses a special structure: every singular vector is either mirror-image symmetric or antisymmetric about its midpoint. Consequently, the SVD method can be applied along with l-D nonrecursive-filter techniques for the design of linear-phase 2-D filters with arbitrary amplitude responses that are symmetrical with respect to the origin of the (q,02)plane.

Table 9.1 Results for the Filter in Example 9.1

Passband edge

0.4

Stopband edge

Order Maximum error 5 x 5 7 x 7 9 x 9 130.05023 x 13 X 17 170.02282

0.26780 0.12643 0.11433

273

Design of Nonrecursive Filtersby Optimization

\

-x

- R

lr

-

-x

(b) Figure 9.1 Continued

9.3.1 Symmetry Properties of the SVD of a Sampled Frequency Response A 2-D nonrecursive digital filter with support in the rectangle defined by - Ni/2 Ini5 NJ2, i = 1,2, can be characterized by the transfer function

where h(n,, nz) is the impulse response. If h(n,, nz)is real and W , , n2) = h( - 4 , -n2) then the frequency response of the filter given by

Chapter 9

274

is a real function that is symmetrical with respect to the origin of the (wl, w2) plane such that

X(%

02)

=

X(- 0 1 ,

(9.9)

-02)

where -T Iwl, w2 IIT. Now assume that matrix A = {alm}represents a desired arbitrary frequency response such that Eq. (9.9) is satisfied, that is, X(ITP~,

where 1 l 5 L and 1 5 frequencies such that p,= - 1 + 2

and -1

Ip, 5

1, -1

X ( -TF,? - m m ) = alm m IM. The quantities p,, ,U are normalized =

-

urn= - 1 + 2

(EL), I, U

I1. The

:(7 ;)

SVD of A gives

r

A =

2 aiuivT

(9.10a)

9 fii?T

(9. lob)

i= 1

=

i= 1

where ai are the singular values of A such that a, 2 a, 2 . . . 2 or,ui is the ith eigenvector of AAT associated with the ith eigenvalue a:, vi is the ith eigenvector of ATA associated with the ith eigenvalue U:, r is the rank of A, and Bi = ufnui,ti = ufnvi. A very important propertyof nonrecursive filters can now be stated in terms of the following theorem. Theorem 9.1 If the frequency response of a nonrecursivefilter satisfies Eq. (9.9), then vectors ui and vi in the SVD of Eq. (9.10b) are either mirrorimage symmetric or antisymmetric simultaneoucslyfor j = 1, 2, . . . , r.

Proof. Let H = {hi,j:1 Ii, j real elements such that

5

h.. = 1.1

2N) be a 2N

X

2N arbitrary matrix with

2N+l-i,2N+l-j

and, for the sake of simplicity, assume that the matrix is square and of even dimension and has distinct singular values. If matrices i, f , and H are defined by

Design of Nonrecursive Filters

by Optimization

275

respectively, where the dimensions of i,, I,, H,, and Hz are N X N , and the dimensions of P and H are 2N X 2N, thenmatrix H can be decomposed as H = iHf = -VT = [U, ui u2,]Z[v, vi * V2NIT (9.11)

-

where ui and vi are eigenvectors of and H T H , respectively. Assume that ui is a normalized eigenvector of HH*, that is, there exists a ui such that (9.12)

a a = u i = upi

with Ilui((= 1. Substituting Eq. (9.11) into Eq. (9.12), we have mi(fHi)Tui = (fHHlf)ui =

U.U. I 1

and H H ~ ~= U U ~~U ,

(9.13)

If we let

then

and, therefore, Eq. (9.13) becomes (9.14) We can now write

-

- HIHT + H2HF HIHT + H2HT - HzHT + H,HT H2H; + HIHT]

[

A B A] [B where A is a positive semidefinite and B is a symmetric matrix. Therefore, Eq. (9.13) can be expressed as

(9.15)

Chapter 9

276

from which two equations can be obtained as Axil Bxil

+ Bx, + Ax,

= aixil = aixiz

(9.16a) (9.16b)

By writing Eq. (9.16) in another matrix notation, we have (9.17)

On comparing Eq. (9.17) with Eq. (9.15), we note that both vectors

[:l

[",:l

and

are eigenvectors of matrix HHT associated with the same eigenvalue a,. Therefore, the two vectors must be linearly dependent, that is, they must satisfy the relation

which implies that xil = xi2

xil = -xi2

or

If xil = xi2 = xi, we can write

and so ui =

['i]

IXj

which means that uiis mirror-image symmetric. On the other hand, if xil = -xi2 = xi, we have

which implies that ui is mirror-image antisymmetric. Now from Eq. (9.11)

v

= iJlTi(Uy2-1

Design of Nonrecursive Optimization by Filters

277

and since i is symmetric and U is orthogonal, that is, (UT)-l = U,matrix V can be expressed as V = iHTm-1 (9.18)

If U,

=

["l Ix,

then Eq. (9.18) implies that

where y, = a;l(H1

+ H2)=xi.If

then Eq. (9.18) implies that

where yi = a;'(H1 - HJTxi. This shows that the two vectors U, and vi have the same symmetry properties simultaneously, that is, they are both either mirror-image symmetric or antisymmetric.

9.3.2 Design Approach A 2-D nonrecursive filter having an arbitrary amplitude response satisfying Eq. (9.9) can readily be designed by using a parallel arrangement of K 2-D nonrecursive sections each comprising two l-D subfilters in cascade, as will now be demonstrated. Such an arrangement can be represented by the transfer function K

~

(

~

~1 29 = )

C ~i(zl)~i(z2)

i= 1

(9.19)

where Fi(zl) and Gi(z2) are the transfer functions of two cascaded l-D subfilters. If these subfilters are nonrecursive filters with support in the rectangle defined by -Ni12 5 ni 5 Ni12, i = 1, 2, we have

C ~(nl)~;"' -N I R

NlR

Fi(z1) =

"I=

(9.20)

278

Chapter 9

and

and if I;l(zl) and Gi(z2)are assumed to represent zero-phase or d2-phase filters, then their frequency responses are given by

(9.22)

=

ri(w2)ejei

(9.23)

If h(nl) and gi(n2)are mirror-image symmetric, then Oi = 0 in Eqs. (9.22) and (9.23) and ai(w1) and ri(wZ)are real functions that are even with respect to w1 and w2, respectively; if h(nl) and gi(n2) are mirror-image antisymmetric, then Oi = d 2 and 2 can be found in [16].

10.5 LUD REALIZATION OF RECURSIVEFILTERS A s was demonstrated in Sec. 10.3, matrix decompositions such as the LUD

and SVDcan be used to obtain useful realization schemes for nonrecursive

296

Chapter 10

Qb32

Figure 10.6 Direct realization of a 2-D transfer function of order (3, 2).

Realization

297

Figure 10.7 Canonic realization of a 2-D filter with separable denominator.

filters. By representing a recursive filter as an interconnection of nonrecursive filters, the methods of Sec.lO.3 can readily be extended to the realization of recursive filters [5]. By rewriting H(zl, z2) as (10.24) where (10.25) NI N2

D ( Z i l,

l)

=

2 2 bijzFiz,j, i=o j=o

boo = 0

(10.26)

a 2-D recursive filter can be realized usingthe two-block feedback scheme shown in Fig. 10.9a or that in Fig. 10.9b. From Eqs. (10.25) and (10.26), the two blocksof each realizationin Fig. 10.9 can treated be as nonrecursive 2-D filters, and canbe realized by the LUD method. Specifically, the 2-D

298

Chapter 10

Figure 10.8 Canonic realization of a 2-D filter with separable numerator.

polynomials n(z;l, ZF l) and b ( z i l , z, l)can be written in matrix form as N ( Z i ' , ZZ') = z T A ~ and b(Zi1, i p ) = z,TBz, respectively, where A, B E RN'x N 2 . If rank (A) = r;, and rank (B) = r,, then the LU decompositions of A and B give A =

LAUA,

LA E R N l x r a ,

UA E P X N 2

B =

LBUB,

LB E R N l X r b

UB

and 9

E RrbXN2

respectively. From Sec.lO.3, it immediately follows that

Realization

0

299

-

(b)

Figure 10.9 Two feedback schemes for the realization of H(z,, z2).

with

and (10.28) with

Figure 10.9 in conjunction withEqs. (10.27) and (10.28) implies that r,

+

300

Chapter 10

Figure 10.10 Realization of a recursive filter using LUD. r b filters are needed to realize m(zcl, 2 ;') and b ( Z < ' , z ~ ' ) ,and each of these filtersis composed of two cascadedl-D nonrecursive subfiltersas in Fig. 10.4. The complete realization scheme is depictedin Fig. 10.10

10.6 INDIRECT REALIZATION OF RECURSIVE FILTERS A number of different indirect l-D realizations can be obtained by converting analog-filter networks into corresponding digital-filter networks through the application of network-theoretic conceptsin conjunction with some simple transformations. By applying the wave network characterization in conjunction with the bilinear transformationof Eq. (7.2) to equally terminated LC filters, the class of l-D wave digital filters proposed by Fettweis [l71 andlater developed further by Sedlemeyer and Fettweis [l81 can be obtained. Similarly, by transforming an analog network configuration using generalized-immittance converters (GICs) [19], the class of l-D GIC digital filtersreported in [20] can be obtained. These realizations have several attractive properties such as low sensitivity to coefficient quan-

Realization

301

tization and low output roundoff noise; further, they can be designed to be free from limit-cycle oscillations. In Sec. 10.6.1, we showthat the methodology of Fettweis canbe applied for the realization of 2-D circularly symmetric lowpass wave digital filters. Then, in Sec. 10.6.2, we showthat the l-D GIC approach can be extended to include the realization of arbitrary 2-D transfer functions. Finally, the realization of circularly symmetric 2-D GIC digital filters is presented in Sec. 10.6.3.

10.6.1 CircularlySymmetricWaveDigitalFilters In Sec. 7.4, transformations that generate rotation in the amplitude response of an analog l-D filter were usedto solve the approximation problem in the design of 2-D recursive filters having piecewise-constant amplitude responses with circular symmetry. These transformations can also be applied in solving the realization problem for this classof filters. The steps involved are as follows: Obtain an analog equally terminated LC ladder realizationof the lowpasstransferfunction HA1(s) givenby Eq. (7.25)using one of the classical synthesis methods[21-241.

STEP 1:

For each rotation angle identifiedby Eq. (7.26), apply the transformation S = gl(sl, s2) of Eq. (7.12a) to the analog realization obtained in Step 1 to obtain a set of 2-D analog realizations.

STEP 2:

STEP 3:

For each 2-D analog realization obtained in Step 2, apply the methodology in [17]-[l81 to obtain a set of 2-D digital realizations. STEP 4: Cascade

the 2-D digital realizations obtained in Step 3 to yield the required 2-D wave digital filter.

The application of the transformation of Eq. (7.12a) to an impedance sL, yields (10.29) SLOls=g1(s,,s2) = SlLl + S2L2 where

L1 = -Lo sin Pk

and

L2 = Locos Pk

Similarly, for an admittance sC,, we have Scols=gl(s*,m) =

SlCl

+ S2C2

where Cl = -Co sin Pk

and

C2 = COCOS Pk

(10.30)

Chapter 10

302

Therefore, the 2-Danalog filtersrequired in Step 2 can readily be obtained by replacing each inductor in the l - D analog filter by a pair of series inductors and each capacitor by a pair of parallel capacitors, as depicted in Fig. 10.11. Realizationsobtained by applyingthe transformation in Eq. (7.12a) have a nonessential singularityof the second kindon the bicircle U,(see p. 206) and can be unstable. This problem can be overcome by using the transformation S = g&, S,) of Eq. (7.43) instead of that in Eq. (7.12a) (see Sec. 7.4.3). With this transformation, an impedancesL0 becomes

(4

1

c,s’

4 T

(b)

Figure 10.11 (a) 2-D analog network for inductance; (b) 2-D analog network for capacitance.

303

Realization where L1 = LO cos

L,

=

c1 = c/&

cos P k C, = c/Losin P k

Pk,

Lo sin P,

Similarly, an admittance sC, becomes

where CO cos P k , C, = C, sin P k ,

c 3

=

L3 = cos P k L4 = c/L, sin P k

Hence the 2-D analog filters can readily be obtained by replacing each inductor in the l-D analog filter by two parallel resonant circuits in series and each capacitor by two series resonant circuits in parallel as depicted in Fig. 10.12 [25].

10.6.2 Realization of 2-D GIC Digital Filters The concept of the current-conversion GIC (CGIC) has been used extensively in the past for the realization of l-D digital as well as analog filters [19,20,26]. In this section, we show that the CGIC can also be used for the realization of 2-D digital filters. Three general steps are involved as follows: STEP

1: The 2-D counterpart of the l-D CGIC is defined.

STEP 2: A general configuration comprising 2-D CGICs andconductances that realizes arbitrary 2-D continuous transfer functions is obtained. STEP 3:

The 2-D analog configuration obtained in Step 2 is transformed into a corresponding 2-D digital configuration. A 2-D GIC is defined as a two-port analog network that can be represented as shown in Fig. 10.13 where

v, = v,,

ZI =

-h(s,, s*)Z,

(10.33)

The function h(s,, S,) issaid to be the admittance-conversion function (ACF) of the CGIC.A 2-D digital realization of the CGICcan be obtained by applying the wave characterization [17,27]:

A , = V , + Zv/Gv B, = V , - Zv/Gv

(10.34a) (10.34b)

Chapter 10

P

Figure 10.12 (a) 2-D analog network for inductance; (b) 2-D analog network for

capacitance.

for v = 1, 2, to the network of Fig. 10.13. In Eq. (10.34) A,, B,, and G , represent, respectively, the incident and reflected wave quantities and the port conductance of the vth port. By solving Eqs. (10.33)-(10.34) for B , and Bz and then applying the double bilinear transformation of Eq. (7.14), we obtain A2

+

(A, - A,)F(z,,

22)

=AI

+

(A, - AZ)F(Zl,

22)

B, = B2

(10.35a) (10.35b)

where (10.36)

Realization

305

and

qz1, z2) = h(sl , S2)IS,-(l-zr)/(l+zr), r=l,2 Consider now the two special cases

h(s,, ~

(10.37) (10.38a)

2 )= s r

and

h(s,, s2) = vsr

(10.38b)

for r = 1, 2. If we assume that (10.39)

G1 = G2

then from Eqs. (10.36), (10.37), and (10.39) we obtain @l,

z2) = z,

and F(z1, Z J =

-2,

for Eqs. (10.38a) and (10.38b), respectively. The digital realization of the 2D CGICs for the preceding two cases and their symbolic representations are depicted in Fig. 10.14a and b, respectively. These will be referred to as GIC adaptors. Let us consider a 2-Danalog reference network comprising n 2-D CGICs and n + 1 conductances connected as shown in Fig. 10.15. The transfer function realized by the network can be written as (10.40)

Port 1

Figure 10.13 2-D digital realization of the CGIC.

Port 2

Chapter 10

306

CO

M (b) Figure 10.14 GIC adaptor: (a) for h(s,, s2) =

S,;

(b) for h@,, s2) = US,.

307

Realization where

NA(s1, ~

2 )=

kOGo

+ klGlh1 +

*

+ k,G,hlhz

*

h,,

and

DA(s1, ~

2 )=

Go

+ Glh1 +

*

. .+ Gnh1h2

h,,

with

hi = hi(Sl,s2) By letting

G , = b, and k, = a,/b,

for r = 0, 1, . . . , n

(10.41)

in Eq. (10.40), the 2-D continuous transfer function

is realized. A reference network that realizes a 2-D transfer function of a specific order can be obtained by applying the following step-by-step procedure: Choose the ACFs such that NA(sl,s2) and DA(sl,s2) in Eq. (10.40) represent general polynomials of the same order as thetransfer function. This can be achieved by constructing a 2-D polynomial map containing the terms that are present in the polynomial. The polynomial maps corresponding to first-, second-, and third-order transfer functions are given in Fig. 10.16a, b,andc, respectively. Each box in themaprepresents a particular term sisi that is present in the polynomial. The values of the variables i, j are indicated in the maps as shown in Fig. 10.16. The map is constructed in such a way that any two adjacent boxes containing the terms S';@ and sisi, satisfy the condition

STEP 1:

[ p - il

+ 1q - jl

= 1

(10.43)

The element in each box of the map (except the first box) is assumed to represent a product of ACFs in the polynomials NA(sl,s2) and DA(s1,sz). The element in the first box (denoted as box I) is assumed to be sys! = 1. 2: Draw a path in the polynomial map satisfying the following conditions:

STEP

1. 2. 3. 4.

It starts from the first box corresponding to the term S!$. It passes through only adjacent boxes that satisfy Eq. (10.43). It covers all the boxes. No box is covered more than once.

Chapter 10

308

i I

1

I

O

S ’ S?%”-

“””

(4

Figure 10.16 Polynomial maps: (a) first-order transfer function;(b) second-order transfer function; (c) third-order transfer function.

For the polynomial maps given in Fig. 10.16a, b, and c, two possible paths are illustrated in each case. STEP 3:

Obtain the required ACFs as

hl = hII, hz

=

hIII/hn, h, = hw/hIll, . . . , etc.

where hI, hrI,hm, etc., are the elements representedby the boxes covered by the path sequentially. These elementsare related to ACFs h,, h2, . . . , h,, as hI = 1, hlI = hl, hIn = h1h2,hlv = hlh2h,, . . . , etc. Connect n 2-D CGICs as shown in Fig. 10.15 havingthe h values obtained in Step 3 to yield the required reference network. The h values obtained usingthe paths correspondingto the solid lines inthe polynomial maps shown in Fig. 10.16a, b, and c are given in Table 10.1.

STEP 4:

A 2-Dwave digital-filter structure canbederivedfromareference configuration by using the wave characterization of Eq. (10.34) and assuming that the ACFs of the CGICs are asgivenin Eq. (10.38). For example, the digital-filter structure corresponding to Fig. 10.15 is derived as shown in Fig. 10.17 by using the GIC adaptors ,and the known digital equivalents for voltage sources, resistors, and parallel interconnections. The digital structure shown in Fig. 10.17 comprises n GIC adaptors and n - 1parallel adaptors. The assignment ofport conductances isdone such that port 2 of the first n - 2 parallel adaptors is made reflectionfree [18]. In a three-port parallel adaptor, the port conductance corresponding to the reflection-free port is equal to the sum of the other two port conductances. From Eqs. (10.41) and (10.42) and the formula for the multiplier constants given in [18], it can be shown that the values of the multipliers

309

Realization

i

0

1

3

2

in Fig. 10.17 are given by (10.44a)

m,,-1

=

m,,=

(10.44b) (10.44~)

310

Chapter 10

Table 10.1 Values of ACFs Obtained from Fig. 10.16a-c Using Solid-Line Path

Third-order Second-order First-order h(s1, s2) 10.16a) (Fig. 10.16c) (Fig. 10.16b) (Fig. h, h2 h3 h4 h, h6 h, h8 h9 h10 h11 h,, h13 h14 h15

S1

S2

SI

S2

S2

S1

lls,

S1

S1

S1

S2

lls, lls, 1/s2

lls, lls, lls,

S1

S2 $1

S1 S1 S2

lls, us,

us,

and for r = 0, 1, . . . ,n

k, = a,/b,

(10.44d)

The digital transfer function realized by Fig. 10.17 is related to that of the reference configuration of Fig. 10.15 as H&,,

GIC 1

I c

'

22)

PA 1

t : l

BO Ai

= - = 2HA(S1, S2)Isi=(l-zi),(l+zi)

GIC2

PAn-2 PA2

PAn-I GICn-l

G?

1

...

Figure 10.17 2-D digital filter corresponding Fig. 10.15.

for i = 1, 2

GlC n

Realization

311

10.6.3 Realization of 2-D Circularly Symmetric G C Digital Filters The concept of GIC can also be used to realize circularly symmetric lowpass filters [29]. The steps involved are described below. For the sake of simplicity, it is assumed that the 2-D filter to be realized has been designed by using the transformation method described in Sec. 7.4.1. STEP

1: Decompose the l-D continuous lowpass transfer function as

STEP 2: For each rotation angle identified by Eq. (7.26), apply the transformation S = g(sl, S,) where S2)

= ClSl (10.46) + c92

with c, = -sin

Pk,

c, = cos Pk

to obtain a set of 2-D analog transfer functions. STEP 3: For each 2-D analog transfer function obtained in Step 2, apply the methodology in Sec. 10.6.2 to obtain a set of 2-D digital realizations.

Cascade the 2-D digital realizations obtained in Step 3 to obtain the required 2-DGIC digital filter.

STEP 4:

The first-order digital filter corresponding to the jth first-order section in Eq. (10.45) has the transfer function (10.47)

where ND,(Zl, 22) = kdnl(1 DD,(zl, 2,)

21

+ 2 2 - ZlZZ) + kl(2 - m1 - m2)

x (1 + z1 + z2 + zlzJ + k2m2(l + z1 - 2, - tlz2) = 1 + (1 - ml)zl + (1 - m2)z2 + (1 - m, - m2)2122 ko = 1, kl = a,/b,, k2 = 1

and m, =

2Cl

C,

+ b1 +

7

m2 =

2%

Cl

+ bl + c,

312

Chapter 10

0

Ai

(6)

Figure 10.18 (a) 2-D first-orderrotateddigitalfilter; structure.

Figure 10.19 2-D second-order rotated digital filter.

(b) corresponding filter

313

Realization

Two GIC adaptors and one parallel adaptor are required to realize the transfer function inEq. (10.47), as shown in Fig. 10.18a. The corresponding digital filter structure is shown in Fig. 10.18b. The second-order digital filter correspondingto the jth section in Eq. (10.45) can be realized by using six GIC and four parallel adaptors, as shown in Fig. 10.19. The values of multiplier constants m,,. . . , ms of the parallel adaptors are given by

m, =

m,

=

m3 =

e1

e, +

g1c1

e1 + g1c1

e, +

g1c1

e1

e,

m4 =

e,

ms =

e,

+ c:

+ g1c1 +

6

+ g,c, + c: + 2c1c2 2(e, + g&, +

2ClC2)

+ g,c, + c: + 2c,c, + g,c2 + c$ 26

+ g,c, + 6 + 2c,c2+

g,c,

+ c$

and the values of multiplier constants kj are given by

ko

= file,,

kl

=

dllg,, k2 = 1, k3 = 1, k4 = d,/g,, ks = 1.

REFERENCES 1. J. H. McClellan, The design of two-dimensional filters by transformations, Proc. 7th AnnualPrinceton Conf. Information Sciences and Systems, pp. 247-

251, 1973. 2. R. M. Mersereau, W. F. G. Mecklenbrauker, andT. F. Quatieri, Jr., McClellan transformationsfor two-dimensional digital filtering: I-Design,ZEEE Trans. Circuits Syst., vol. CAS-23, pp. 405-414, July 1976. 3. W. F. G . Mecklenbrauker and R. M. Mersereau, McClellan transformations for two-dimensional digital filtering: II-Implementation, ZEEE Trans. Circuits Syst., vol. CAS-23, pp. 414-422, July 1976. 4. J. H. McClellan and D. K. S. Chan, A 2-D FIR filter structure derived from the Chebyshev recursion, ZEEE Trans. Circuits Syst., vol. CAS-24, pp. 372378, July 1977. 5. A. N. Venetsanopoulos and C. L. Nikias, Realization of two-dimensional digital filters by LU decomposition of their transfer function, Proc. ZEEE Znt. Conf. Acoust., Speech, Signal Process., pp. 20.4.1-20.4.4, March 1984.

314

Chapter 10

6. A. N. Venetsanopoulos and B. G. Mertzios, A decomposition theorem and its implicationsto the design and realizationof two-dimensionalfilters, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-33, pp. 1562-1575, Dec. 1985. 7. W.-S. Lu, H.-P. Wang, and A. Antoniou, Design of two-dimensional FIR digital filters by using the singular value decomposition,IEEE Trans. Circuits Syst., vol. CAS-37, pp. 35-46, Jan. 1990. 8. S. Y. Kung, B. C. Levy, M. Morf, andT. Kailath, New results in 2-D systems theory, Part 11: 2-D state space model-Realization and the notions of controllability, observability, and minimality,Proc. IEEE, vol. 65, pp. 945-961, 1977. 9. R. Eising, Realization and stabilization of 2-D systems,IEEE Trans. Automat. Contr., vol. AC-23, pp. 793-799, Oct. 1978. 10. C. Eswaran, T. Venkateswarlu, and A. Antoniou, Realization of multidimensional GIC digital filters, IEEE Trans. Circuits Syst., vol. CAS-37, pp. 685-694, June 1990. 11. A. Antoniou, Realization of digital filters, IEEE Trans. Audio Electroacoust., vol. AU-20, pp. 95-97, March 1972. 12. A. Antoniou, Digital Filters: Analysis and Design, New York: McGraw-Hill, 1979 (2nd ed. in press). 13. G. Strang, Linear Algebraand Its Applications, 2nd ed. ,New York: Academic Press, 1980. 14. E. D. Sontag, On first-order equations for multidimensional filters, ZEEE Trans. Acoust., Speech, Signal Process., vol.ASSP-26,pp.480-482, Oct. 1978. 15. S. H. Zak, E. B. Lee, and W.-S. Lu, Realization of2-D filters and timedelay systems,IEEE Trans. Circuits Syst., vol. CAS-33, pp. 1241-1244, Dec. 1986. 16. T. Venkateswarlu, C. Eswaran, and A. Antoniou, Realization of multidimensional digitaltransfer functions, Multidimensional Systems and Signal Processing, vol. 1, pp. 179-198, June 1990. 17. A. Fettweis, Digitalfilter structures related to classical filter networks, Arch. Elektron. Uebertrag., vol. 25, pp. 79-89, 1971. 18. A. Sedlmeyer and A. Fettweis, Realization of digital filters with true ladder configuration, Int. J. Circuit Theory, vol. (3-18, pp. 314-316, March 1973. 19. A. Antoniou, Novel RC-active-network synthesis using generalized-immittance converters, IEEE Trans. Circuit Theory, vol. CT-17, pp. 212-217, May 1970. of 20. A. Antoniou and M. G. Rezk, Digital-filtersynthesisusingconcept generalized-immittance convertor, IEE J . Electron. Circuits Syst., vol. 1, pp. 207-216, NOV. 1977. 21. J. K. Skwirzynski, Design Theory and Data for Electrical Filters, London: Van Nostrand, 1965. 22. R. Saal, The Design of Filters Using the Catalogue of Normalized Low-Pass Filters, Backnang: Telefunken AG, 1966. 23. A. I. Zverev, Handbook of Filter Synthesis, New York: Wiley, 1967. 24. L.Weinberg, NetworkAnalysis and Synthesis, New York: McGraw-Hill, 1962.

Realization

315

25. G . V. Mendonsa, Design, Realization and Implementation of 2-D Circularly Symmetric Pseudo-RotatedDigital Filters, Ph.D. thesis, Concordia University, 1984. 26. A. Antoniou, Realization of gyrators using operational amplifiers, and their use in RC-active-network synthesis, ZEE Proc., vol. 116,pp. 1838-1850, NOV. 1969. of designing digital filters imitating classical filter 27. A. Fettweis, Some principles structures, IEEE Trans. Circuits Syst., vol.CAS-18,pp.314-316,March 1971. 28. A. Fettweis and K. Meerkotter, On adaptors for wave digital filters, IEEE Trans. Acoust., Speech, Signal Process., vol.ASSP-23,pp.516-525, Dec. 1975. 29. T. Venkateswarlu, C. Eswaran, and A. Antoniou, Design of circularly symmetric two- and three-dimensional digital filters using the concept of the GIC, IEE Proc., Vol. 138, pt.G, pp. 523-529, Aug. 1991.

PROBLEMS 10.1 Verify Eq. (10.5).

10.2 Realize the filter designed in Examble 6.5 of Sec. 6.4.3 by using the

method of McClellan and Chan.

10.3 Obtain theS W - L U D realization of the 2-D filter designed in Problem 9.2. 10.4

Obtain the S W - L U D realization of the 2-Dfilter designed inProblem 9.5b.

10.5

Extend the canonic realization depicted in Fig. 10.7 to the general 2-D transfer function given by

10.6

Extend thecanonic realization shown in Fig. 10.8 to the general 2-D transfer function given by

10.7

Obtain a direct realization for each of the following transfer functions: 1 (a) H(z17") = z1z2+ 0.22, + 0.5'

316

10.8Obtainthe p. 89.

Chapter 10

LUD realization of the filter described in Sec. 4.5.2,

10.9 Obtain theLUD realization of the state-space filter givenin Problem 2.5. 10.10 Obtain a2-D GIC digital-filter structure for the circularly symmetric lowpass filter designed in Problem 7.6. 10.11 Obtain a 2-D GIC digital-filter structure for the filter designed in Problem 7.7.

I1 Finite Wordlength Effects

11.l INTRODUCTION When a 2-D digital filter is implemented in terms of either software on a general-purpose computer or dedicated hardware, numbers representing transfer-function coefficients and signals are stored and manipulated in registers of finite length. Transfer-function coefficients are calculated to a high degree of precision during the solution of the approximation problem, and, consequently, they must be quantized before they can be accommodated in registers of finite length. Similarly, products generated when signals are multiplied by coefficients are always too long to fit in the registers and must again be quantized. The quantization of coefficients and products gives rise to quantization errors that propagatethrough the filter and appear at the output as noise. Product quantization can lead to other problems as well, such as the generation of spurious parasitic oscillations known as limitcycles. Although the effects of coefficient and product quantization are insignificant if a general-purpose computer is to be used for the implementation,owing to thehigh precision of the hardware, particular attention must be paid to these effects when fixed-point arithmetic or specialized hardware with reduced wordlength is to be employed. In such applications, the design process cannot be considered complete until the effects of quantization are studied in detail. 317

318

Chapter 11

This chapter deals with the effects of finite wordlength in 2-D digital filters. In the next section, the quantization errors introduced by the use of finite wordlengths for the representation of coefficients and signals are examined and methods for their computation and characterization [l-31 are considered. In Sec. 11.3, a method for the minimization of output roundoff noise in l-D state-space structures due toMullis and Roberts [4] based on a state-space noise model proposed by Hwang [5] is extended to the case of 2-D digital filters [6]. In Sec. 11.4, two types of parasitic oscillations, namely, quantization and overflow limit cycles, are examined and conditions for their elimination are delineated.

11.2 QUANTIZATION ERRORS AND THEIR COMPUTATION As in l-D digital filters, the use of finite wordlength for the representation of numbers introduces three types of quantization errors, as follows:

1. Coefficient-quantization errors 2. Product-quantization errors 3. Input-quantization errors Coefficient-quantization errors are due to the representation of the filter coefficients by a finite number of digits. Product-quantization errors are due to the quantization of signals after multiplication. Input-quantization errors occur when the signal to be processed is continuous and must be sampled and encoded through an A/D converter prior to processing. Quantization of numbers can be carried out by rounding or truncation, and the magnitude of the error introduced tendsto depend on the type of arithmetic (i.e., fixed point or floating point) and on the number representation (i.e. sign-magnitude, one's complement, and two's complement). Throughout the following analysis, it is assumed that quantization errors are uniformly distributed over the quantization interval and that they are statistically independent from error to error, from sample to sample, and with respect to the input signal (see [ll], Chap. 11, for further details). Under these circumstances, the mean Z and variance g of a quantization error e for fixed-point arithmetic are givenin Table 11.1 [l]. The 2-D transfer function is assumed to be of the form (11.1) where

Finite

Effects

319

Table 11.1 Mean and Variance of Quantization Error e

representation Number

Quantity

Rounding

Truncation

0 EiI12

E012 EgI12

Sign-magnitude One's complement Two's complement Source: Ref. 1.

and

11.2.1 Coefficient Quantization Let Zij and 6, be the quantized values of aij and b , and denote the quantization errors introduced by and

respectively. Also let y(nl, n2) and y(nl, n2) be the output of a 2-D digital filter with and without coefficient quantization, respectively. From Eq. (l.sa), we have

and hence the output error produced is given by

NI NZ

=

u(nl, n2) -

2 j2 biif(nl - i, n2 - j ) =o

i=o

(11.2)

320

Chapter 11

where second-order terms have been neglected in the second equation and NI

N-

(11.3) Combining the 2-D z transforms of Eqs. (11.2) and (11.3) leads to the z-domain description of the output error given by

where

Equation (11.4) can be used to derive formulas for the mean and variance of the output error. Such formulas for the case of fixed-point arithmetic are given in Table 11.2 [l] where 0;and 0;are the variance of the input and output signals, respectively. Table 11.2 Mean and Variance of the Output Error Due to Coefficient Quantization Number representation Sign-magnitude

Quantity

Rounding

Truncation

7

0

0

Of

E,2 ( k o ? + lo;)

E; ( k o t

0 E; -(ka; 12A

+ fop)

E; - (k0;'-+ fu:) 3A

+ fa;)

0 E; -( k o : 12A

7 One's complement

0;

f Two's complement

Source: Ref. 1.

Uj

12A

0 E; -( k u ; 12A

3A

+ lo;)

0

+ zu;)

Finite Wordlength Effects

321

11.2.2 Product Quantization Let x(n,, n2) for n, 2 0, n2 I0 be a quantized input signal. Assuming infinite-precision arithmetic, the output of the filter for n1 2 0, n2 10 can be obtained from Eq. (1.8a) as NI

y(nl,

n2)

=

N?

2 /2= o

NI

aijx(n1

-

i,

n2

-

i=O

N2

C JC= o bijy(n1 - i, n2 - j )

1) -

l=O

-

Denoting a quantized product u - v by (u v ) ~ the , output of a finitewordlength filter y(rz,, n2)for n, ? 0, n2 ? 0 and the corresponding quantization error f ( n , , n2) can be expressed as N:

Ni

y(n1,

n2)

=

CC

i7 n? - j ) l q

[ a l J ~ (-~ ~ l

1=0j = O

NI

N-

where

r=O 1=0

The 2-D z transform of Eq. (11.5) gives

where R(z,, z2)and E(z,, z2)are the z transforms of h(n,, n2) and E ( n l , n2),respectively. Consequently, if h(n,, n2)for n1 2 0, n2 2 0 is the impulse response of the filter represented by l/D(z,, z2),then

Equation (11.6) can be used in conjunction with Table 11.1 to obtain the mean and variance of the output error for the case of a direct realiza-

322

Chapter 11

Table 11.3 Mean and Variance of the Output Error Due to Product Quantization Number representation

Quantity

Rounding

Truncation

0

Source: Ref. 1.

tion of the transfer function (see Sec. 10.4). Formulas for these statistics are given in Table 11.3, where k and I denote the number of coefficients that are different from 0 and 1 in polynomials N(z,, 2,) and D ( z l , z 2 ) , respectively.

11.2.3 Input Signal Quantization Let y(rz,, n2) and 9(nl, n,) be the output of the filter in response to a nonquantized and a quantized input, respectively. It directly follows from Eq. (1.8a) that the output error f(n,, n2) of a 2-D digital filter obeys the recursive equation NI

f(izl,

n2) =

N?

C C a,e(izl

NI

-

i , n2 - j )

-

N?

CC

bIJf(iz,

-

i , 12,

-j)

I=@ J=@

l=o J=@

where

e(i, j )

=

x(i, j )

-

Z(i, j )

The z-domain description of the preceding recursive relation is given by

F(z1, 4

=

H(z1, zz)E(z,, z2)

where E(z,, Z J denotes the 2-D z transform of e(i, j ) and H(z,, z 2 ) is given by Eq. (11.1). The mean and variance of the output error are given in Table 11.4 [l].

Finite Wordlength Effects

323

Table 11.4 Mean and Variance of the Output Error Due to Input Quantization Number representation

Quantity

Rounding

Truncation

Sign-magnitude

f

0

One’s complement

Two’s complement Source: Ref. 1.

11.2.4 Evaluation of C;=, Zr=, h2(i,j ) As can be seen in Tables 11.3 and 11.4, the computation of the output errors due to product and input quantization depends on the quantity Czo h2(i,j ) which is bounded by

Xzo

(11.7) Here we consider the computation for the more general case

(11.8) where H(z,,

2’)

is given by

In what follows, it is assumed that

(11.9) which guarantees the B I B 0 stability of the filter considered.

Chapter 11

324

where N,(zl, z2) and Dl(zl, z2) are polynomials in z1 and z2, and k, and k2 are integers, the integral in Eq. (11.8) becomes

where

The integrand in Jl(zl) can now be decomposed as N Z l , Z,)Nl(Zl,

22)

W Z , , Z 2 ) D l h 22)

Qi(z2) + -2p Q2(z2) 2p = W 2 )

DdZ2)

where Q,(z2),Q2(z2),d(z2) and d1(z2)are polynomials in z2 with coefficients that are polynomials in zl.Note that the stability assumption given by Eq. (11.9) implies that D(z2) # 0

for lzll =1, lz21 < 1

d1(z2)f O

for lzll = 1, lz21 > 1

and

Consequently, for a fixed z1on U,= {zl: lzll = l},

which leads to

By the residue theorem, J,(zl)- is the coefficient of the z;' term in the Laurent expansion of Q2(22)/(D1(22)2~kr). Once Jl(z,) is found, the complex integral in Eq. (11.8) can be computed by calculating the residues of Jl(zl)z:l at all its singularities lying inside the unit disk of the z1plane.

Finite Wordlength Effects

325

Example 11.1 Compute the complex integral J in Eq. (11.8) if

Solution. H(zl, z2) represents a BIB0 unstable filter although its impulse response is square summable (see [12]). Therefore, thepreceding approach is applicable. We can write

W , , 22)H(z;l, 2122

G? - (21 - W 2 - 1) (1 - z1)(1 - 22) 2 ; 2 - 21 - 2 2 22122 - 21 - 22

12zl

and so (21

-

2-21-22

- 1) (1 - z1)(1- 22) 22122-21-22

-1

22

- (1 - 21122 - (1 - 21) (21 - 1122 + (1 - 21) - 2 2 - (2 - zl) (22, - 1)zZ -Z1Z2 - ao(z1)z2 + a2(21)22 + a3(21) 22 - (2 - Zl) (22, - 1)2Z- Z1Z2 +

where

Hence Jl(Zl) =

3(21 a2(21) = 22, - 1

2(22,

-

- 1)' l)(Z, - 2)

Therefore

11.3 STATE-SPACESTRUCTURESWITH MINIMIZED ROUNDOFF NOISE Minimum roundoff noise, subject to dynamic-range constraints based on the L, norm, can be achieved in l-D state-space structures byusing a method proposed by Mullis and Roberts [4], which is based on a statespace noise model developedby Hwang [S]. This method canbe extended to the case of 2-D state-space structures, as will be demonstrated below [61.

Chapter 11

326

11.3.1 Derivation of Noise Model A 2-D digital filter can be represented by the local state-space model given by

= Aq(i, j ) y ( i , j ) = [c, c2]q(i, j )

+ &(it

+ bx(i, j) j)

cq(i, j )

+ aX(i, j )

(11.10a) (11.10b)

, E RNZXN (see 2 Chap. 2). It is assumed that where A, E R N l x N 1A4

which implies the BIB0 stability of the filter (see Chap. 5 ) . If we let

A, = AloAi-l,j + AolAi,j-l

for (i, j ) > (0, 0)

(11.11)

and A - .191. = A.1 , -I . = 0

fori? 1,jz 1

the transfer function of the filter can be expressed as

T =

[" 0

'1

T2

= T, 0 T2

(11.14)

Finite

Effects

327

where TI E RN' x N 1 , T, E R N z x N zand , 0 denotes thedirect sum, tovector q(i, j ) in realization (A, b, c, d ) of Eq. (ll.lO), an equivalent realization (A, b, C, d) where A = T-lAT,

b = T"b,

C = cT,

d=d

(11.15)

can be obtained. Oncematrices A, are defined by analogy with Eq. (11.11) , it is easy to show that A, = T"AijT

for all i, j

Now if finite wordlength is used for the representation of internal signals, the state-space model of the filter becomes

where T1(i + 1, j ) and ~ ~ j( + i ,1) are the random errors generated in the computation of qH(i + 1, j ) and qv(&j + l), respectively, and y(i, j ) is the random error generated in the computation of y(i, j ) . If we define the state-error vector as

and output error (or noise) as

AY& j ) = L(& i ) - Y(i, j ) then Eqs. (11.10) and (11.16) yield

(11.17b)

11.3.2 Output-Noise Power The noise model of Eq. (11.16) can be used to derive an explicit expression for the output-noise power, as will now be demonstrated.

Chapter 11

328

For a fixed point ( i , j )

2

(0, 0), Eq. (11.17) gives

If the quantization of products is carried out by rounding and the quantization errors are assumed to be independent from error to error and from sample to sample, then the variance of each error is E312 where Eo is the quantization step (see Table11.1).From Eq. (11.18), the expected square error is

where

are positive-definite diagonal matrices of dimensions Nl and NZ, respectively, and p and v are the numbers of constants in c and d , respectively, which are neither zero nor one. Therefore, the variance of the output noise Ay(i, j ) can be calculated as

=

12

G

- tr(QW) 12

G (p + v) +-

(11.19)

where

Q

=

Q1 X

0 D

Q2

C

(11.20)

and trp) denotes the trace of a matrix.

Effects

Finite

329

Note that the variance of the output noise given by Eq. (11.19) is dependent on the coordinate system. Specifically, if the transformation in Eq. (11.14) is applied, the output-noise power of realization (A, b , C, a) is given by E(Ajj2) =

G (F + v) G tr(QW) + 5 12

(11.21)

where (11.22) and Q =

Q1

@ Qz.

11-3.3 Dynamic Range Constraints To prevent overflow in a digital filter, signal scaling based on theL2 norm

can be applied, which leads to a set of dynamic range constraints on the local state variables. The local state at (i, j ) due to an input x(& k ) for (0,0) S (I, k ) (i, j ) can be obtained from Eq. (11.12) as

Consequently, if ep is the pth column of the identity matrix of dimension (Nl + N Z ) ,then the pth component of the local state in Eq. (11.23) can be estimated as

r

l2

where

K

=

2

x

1-0 k=O

f(I, k)fT(I, k )

(11.25)

Notice that e,TKep is the pth diagonal element of K and, therefore, if all the diagonal elements of K are equal to one, Eq. (11.24) implies that

Chapter 11

330

the amplitude of each state component is no more than llxll. The dynamic range constraints on the state variables are, therefore, given by

Further, once a similarity transformationT is used to reduce the outputnoise power, the preceding dynamic range constraints should alsobe satisfied by the transformed realization. Since m o o

K

=

C f(i, j ) P ( i , j )

j=o j = o

=

(11.26)

T-~KT-T

the dynamic range constraints on the new variables qH(i,j ) and qv(i, j ) are given by

* 1

*

1

1

(11.27)

1

11.3.4 2-D Second-OrderModes If we denote

and

then Eqs. (11.22) and (11.27) yield KlIW11 = TilKllWllTl

(11.28a)

T;1K22W22T2

(11.28b)

KZ2Wz2

=

In other words, the eigenvalues of CP = KllWll and 9 KZWw, denoted by 4 = { k : 1 I i S Nl} and $ E {Jlj: 1 I j S N Z } ,are invariant under a

.

331

Finite Wordlength Effects similarity transformation. We call sets c$ {&: 1 Ii 1 Ij IN Z }the second-order modes of the filter.

IN , }

and I) = {$:

11.3.5 Minimization of Output Noise For the sake of simplicity, it is assumed that the computation of each component of the new state variables q"(i + 1,j ) and qv(i, j + 1) always involves ( N , + NZ +. 1) multiplications; that is, both Aand b have neither zero nor one entries. This assumption implies that Q in Eq. (11.21) is equal to ( N , + NZ + 1)1 andis independent of the similarity transformation used. Thus Eq. (11.21) now becomes

E(Ay2) =

(NI + NZ 12

+ 1)E; G- + -(p E; + V) 12

(11.29)

where

G = tr(TTWT)

(11.30)

cT, d). is referred to as the unit noise of realization (T"AT, T"b, By using the singular-value decomposition and noting the invariance of the second-order modes of the filter, it can be shown that a transformation T = T, Q T2 can be found that minimizes the unit noise G subject to the constraints in Eq. (11.27) [6]. The resulting state-space realization, namely, {A, b, C, d} {T"AT,T"b, cT, 4, is said to be optimal. The required transformation T can be computed through the following steps:

1: Compute positive-definite matrices W and K using Eqs. (11.20) and (11.25), respectively.

STEP

STEP 2:

Find matrix P = P, 8 P2such that p-lm-T =

['NI

rT

'1

IN2

and compute

3: Find block-orthogonal matrix R = R, 8 R2 such that R ~ W l l R 1= diag {U, uN1}and RTWZ2R2= diag {uN1+,* u ~ ~ + ~ ~ } . STEP

STEP 4:

-

Form matrix A* as

A* = AT Q A;

332

STEP 5:

Chapter 11

Find block-orthogonal matrix S = S1 8 S2 such that

using the algorithm given in the appendix of [5]. STEP 6:

Form

T = PRA*ST This procedure is illustrated by the following example. Example 11.2 Consider a stable state-space digital filter of order (2,2) represented by Eq. (llf.10) where

C

iI -0.2.191 0.01

= [0.2889 -0.0912

Find the similarity transformationthat will lead to an optimal state-space realization. Solution. As the filter is stable, we can use finite sums M I Mz

c

i=o j = o

c c f(i, j)f*(i, MI M2

ACcTcAij

and

j)

i-0 j = O

with sufficiently large M,and M2 to approximate W and K.Taking M1=

Finite Wordlength Effects

333

M, = 240, numerical computation gives W =

and

K

=

[ [

1.1336 -1.0329 i 0.9779 1.7744 -1.0329 0.9652 . -0.9411 -1.6723 --"""""""""""""-t""""""""""""""-. 0.9779 -0.9411 i 87.1245 85.2572 1.7744 -1.6723 85.2572 87.1724

1 I

1

87.1245 85.2572

85.2572 87.1724

1.6398 -1.5391

1.3212 -1.2332

1

i

1.6398 1.3212

-1.5391 -1.2332

1.1336 -1.0329

-1.0329 0.9652

1 1

.---------------------------.:----------------------------l

1

1

The unit noise of this filter after scaling is the sum of products of the corresponding diagonal entries in W and K and is given by 4

Go =

21 wiikii= 365.8049

i=

Following these steps, we obtain T1

=

[

-3.3098 10.44281, -5.328410.6095

T2 =

[

- 0.9493 - 0.0659

and hence the required similarity transformation T = T1 0 T, can be formed. The characterization of the optimal state-space realization can now be obtained as

A

i

0.9645 -0.1190 0.1724 0.9245

-0.4731 i -0.2376

-0.1350 -0.0678

1

= T - ~ A T= .--------------------------i---------------------------0.0453 0.0107 0.0166 0.0220 6 = T"b = [0.11320.0569 S = CT = [-0.4703 2.0493

I

1

1

0.1816 0.8884 i -0.1281 1.0006 -0.1168 0.2985IT -0.2005 -0.05731

The corresponding matrices W and K are 3.3898 -0.0009 -1.2564 0.2644

-0.0009 3.3898 -0.5394 0.0075

- 1.2564 0.2644 -0.5394 0.0075 3.3889 0.0 0.0 3.3910

1

334

Chapter 11

and 1.0 0.4757 0.1724 0.0673 0.4757 1.0 0.2047 0.0968 0.1724 0.2047 1.0 0.4754 0.0673 0.0968 0.4754 1.0

These give the unit noise of realization {A,b, C} as

c

1

4

G=

=

i=l

13.5595

11.4 LIMIT CYCLES Two types of limit cycles may occur in a recursive 2-D filter, namely, quantization and overflow limit cycles. The first type is due to the quantization of products, and the second is due to the finite dynamic range of the filter. In this section, conditions for the absence as well as existence of limit cycles are presented.

11.4.1 Quantization Limit Cycles A 2-D sequence CVi; i

y(i

2

0, j

2

0) is said to be periodic if

+ P,, j + P2) = y ( i , j )

forall i, j

where the integer pair(P1,P2) is called the period of the sequence. A 2-D periodic sequence{ y ( i ,j ) : i 2 0, j 2 0) is said to be separable if y(i

+ P,, j + P,)

= y ( i , j ) = y(i

+ P,, j ) = y ( i , j + P2)

for all i, j

Note that if {y(i, j ) } is periodic andseparable, then it can be generatedby repeating the P , X P2 elements of b(i,j ) } indefinitely in both i and j directions. If either P , or P2 is equal to zero, then the periodic sequence is called degenerate. A 2-D limit cycle is a 2-D periodic sequence which may or may not be separable. Now let us consider a 2-D digital filter with zero input as depicted in Fig. 11.1, where Q, (1 1 I 1 m) are quantizers under fixed-point arithmetic satisfying the sector conditions (11.31)

and

335

Finite

where Q,[y(i, j ) ] is the quantized value of y(i, j ) . Constant kI in (11.31) depends on the type of quantization used. For example, for truncation kl = 1 and for rounding kl = 2. Assume that H(zl, z2) in Fig. 11.l has no singularities on the unit bicircle U2 (see Sec. 5.2, p. 108, for definition), and that the initial conditions for Cy(i,j ) } are given by

y(i, 0)

= 0

for 0 S i IM,

(11.32a)

y(0, j )

= 0

for 0 S j

(11.32b)

and S

M2

where M, and M 2are some positiveintegers. The following theorem gives a sufficient condition for the absence of separable limit cycles of period (M19MZ) P O I . Theorem 11.1 If there exists a diagonal matrixD = diag{d,, . . . , dm} with Re d, > 0 for 1 5 i I m such that for positive integers kl and k2 M(kl,k2) DH[H(ePPkl/MIej2nkzlMz ) - K1 +

[HH(ej;?nkllM~ e~2nkdmz ) 9

KID (11.33) is negative definite where K = diag{k;l, . . . , k; l} and DH denotes the complex conjugate transpose of D, then separable limit cycles of period (Ml, M,) are absent from the filter in Fig. 11.1. -

Corollary 11.1 The filter depicted in Fig. 11.1 is free fromseparable limit cycles if there exists a matrix D = diag{d,, . . . , dm} with Re d, > 0 for 1

U

Figure 11.1 A 2-D filter with m quantizers.

Chapter 11

336 S

i

S

m such that

M(el, 0,) = DHIH(ejel,e’”)

- K]

+ [HH(ejel,e’”)

- KID

(11.34)

is negative definite for 0 S e,

I 27~,0 S e2 S 27~. More feasible conditions for the absence as well as existence of limit cycles can be derived for some low-order filters. First, we consider 2-D filters of order (1, 1) with one quantizer Q characterized by

where

Q(Y> 5 1

for y # 0 (11.36) Y A 2-D filter reaches a zero steady state if there exist integers K1 > 0 and K , > 0 such that O S -

for some i < Kl and j < K, 1. y(i, j ) # 0 for i 2 K , and arbitrary j 2. y(i, j ) # 0 or Y(i,j ) = 0 for j 2 K , and arbitrary i The following theorem provides sufficient conditions for limit cycles in the filter represented by Eq. (11.35) [8].

the absence of

Theorem 11.2 Under zero initial conditions, the filter described by Eq. (11.35) reaches a zero steady state, that is, thefilter is free from limit cycles,

if Ibl < f

and

la1

+ IC\



0

(11.43~) The following theorem provides a sufficient condition for the filter described by Eq. (11.43) to be free of overflow limit cycles [9]. Theorem 11.4 Thefilter described by Eq. (11.43) is asymptotically stable, that is, x(i, j ) = 0

lim i andlor j-m

if there exists a diagonal positive-definite matrix D such that

D - ATDA

=

Q

where Q i~a positive-definite matrix. Note that if @ , 6, 2, represents a realization of a 2-D filter such that (IAll < 1, then

a}

I -

ATA > 0

{A,

a>

Therefore, by Theorem11.4, the realization 6, e, is free of overflow limit cycles. Furthermore, if {A, b, c, d)is a state-space realization of a stable 2-D filter such that A satisfies the 2-D Lyapunov equation [l41 W - ATWA = Q

(11.44)

with W = W, 0 W, > 0 and Q > 0, then by taking T = Wln = W112 @ W112 1 2 as a 2-D similarity transformation, Eq. (11.44) yields 1 - (TAT-l)T(TAT-l) = W"'2QW-1'2 > 0 which implies that ((TAT-'((< 1. We have, in effect, demonstrated the following corollary. Corollary 11.2 If matrix A in a state-space realization of a stable 2 - 0 filter sahfies the Lyapunov equation, namely, Eq. (]l.#), for some W = W1 0

Finite Wordlength Effects

339

W, > 0 and Q > 0, then the realization {TAT-l, Tb, cT-I, d), where T is the symmetric square root of matrix W, is free of overflow limit cycles. The BIBO stability of a 2-D state-space realization does not,in general, imply Eq. (11.44),as was demonstrated in Sec. 5.4. Certain special cases where the BIBO stability of a 2-D filter does imply Eq. (11.44) can be found in [lo] and [14]. One of these cases is described by the following theorem.

Theorem 11.5 I n 2-Dstate-space digital filters of order (l, l ) , BIBO stability implies the absence of overflow limit cycles.

REFERENCES 1. S. H. Mneney and A. N. Venetsanopoulos, Finite register length effects in two-dimensional digital filters, Proc. 22nd Midwest Symp. on Circuits and Systems, pp. 669-676, June 1979. 2. L. M. Roytman and M. N. S. Swamy, Determination of quantization error in two-dimensional digital filters, Proc. IEEE, vol.69, pp. 832-834, July 1981. 3. P. Agathoklis, E. I. Jury, and M. Mansour, Evaluation of quantization error in two-dimensional digital filters, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-28, pp. 273-279, June 1980. 4. C. T. Mullis and R. Roberts, Synthesisof minimum roundoff noise fixed point digital filters, IEEE Trans. Circuits Syst., vol. CAS-23, pp. 551-562, Sept. 1976. 5. S. Y. Hwang, Minimumuncorrelated unit noise instate-spacedigital filtering, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-25, pp. 256-262, June 1976. 6. W.-S. Lu and A. Antoniou, Synthesis of 2-D state-space fixed-point digitalfilter structures with minimum roundoff noise, IEEE Trans. Circuits Syst., vol. CAS-33, pp. 965-973, Oct. 1986. 7. T.-L. Chang, Limitcycles in a two-dimensional first-order digital filter, ZEEE Trans. Circuits Syst., vol. CAS-24, pp. 15-19, Jan. 1977. 8. N. G. El-Agizi and M. M. Fahmy, Sufficient conditions for the nonexistence of limit cycles in two-dimensional digital filters, IEEE Trans. Circuits Syst., vol. CAS-26, pp. 402-406, June 1979. 9. N. G. El-Agizi and M. M. Fahmy, Two-dimensional digital filters with no overflowoscillations, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-27, pp. 465-469, Oct. 1979. 10. P. Agathoklis, E. I. Jury, and M. Mansour, Criteria for the absence of limit cyclesintwo-dimensionaldiscretesystems, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-32, pp. 432-434, April 1984. 11. A. Antoniou, Digital Filters: Analysis and Design, New York: McGraw-Hill, 1979 (2nd ed. in press).

Chapter 11

340

12. D. M. Goodman, Some stability properties of two-dimensional linear shiftinvariant digital filters, IEEE Trans. Circuits Syst., vol. CAS-24, pp. 201208, April 1977. 13. C. W. Barnes and A. T. Fam, Minimum norm recursive digital filters that are free of overflow limit cycles, IEEE Trans. Circuits Syst., vol. CAS-24, pp. 569-574, Oct. 1977. 14.B. D. 0. Anderson, P. Agathoklis, E. I; Jury, and M. Mansour, Stability and the matrix Lyapunov equation for discrete two-dimensional systems, IEEE Trans. Circuits Syst., vol. CAS-33,pp. 261-267, March 1986.

PROBLEMS 11.1 Verify the entries in Table 11.2. 11.2 Verify Eq. (11.5).

11.3 Verify the entries in Table 11.3. 11.4 Find upper bounds of the means and variances of the output error due to product quantization for the transfer function H(zl, z2) = N(zl, z2)lD(z1, z2) where D(z,, z2) is given by (a) D(zl, z2) = 2 - z1 - z2.

(b) D(z,,

22)

=2 -

~

1

-~ ~1 2 -

~ 2 .

11.5 A 2-D digital filter is characterized by the allpass transfer function

(a) Show that ifH(zl, z2) is BIB0 stable, the filter canbe represented in the local state space by Eq. (11.10) with all Ai(1 5 i 5 4) scalar. (b) Verify that if a, = a2 = 0, then H(zl, z2) can be realized by the state-space model in Eq. (11.10) where

11.6 (a) Show that for the state-space model given in Problem 11.5b,

K=-

l

1 - a;

I

and

W = (1 - az)I

(b) Find a realization of H(zl, z2)with a, = a, = 0 that minimizes the output roundoff-noise power subject to L2 dynamic-range constraints.

Finite

341

11.7 Consider a B I B 0 stable state-space 2-D filter of order (1, 1). The

filter is described by Eq. (11.10) where A =

[:: 21,

b =

[:l,

c = [c1 c*],

.d = 0

Assume that matrices K and W have been computed using Eqs. (11.25) and (11.20), respectively, and are given by

Find a realization with minimum output-noise power. 11.8 Prove Theorem 11.4.

11.9 Prove Theorem. 11.5.

12 Implementation

12.1 INTRODUCTION Once a transfer function that satisfies the requirements imposed by the application at hand is obtained, the implementation of the filter must be undertaken. A s was stated in the introduction to Chap. 6, a 2-D digital filter can be implemented in terms of software or hardware. In applications where a record of the data to beprocessed is available, a software implementation may be entirely acceptable, whereas in applications where real-time processing is required or where a massive amount of data needsto be processed, a hardware implementation may be the only available choice. In a software implementation, the difference equation, a signal-flow graph, a digital-filter network, or a state-space representation is converted into a computer program that can be run on a general-purpose computer or workstation or on a general-purpose digital signal processing chip. A software implementation is deemed to be good if it entails low computational complexity, needs a small amount of memory, and is insensitive to quantization effects and aliasing errors. Low computational complexity can be achieved through the application of fast-Fourier transforms[l-41, whereas low sensitivity to coefficient quantization can be achieved by emulating a wave digital-filter structure of the type described in Sec. 10.6.1. On the other hand,if low output roundoff noise is required, a state-space structure of the type described in Sec. 11.3.5 can be used.

342

Implementation

343

In a hardware implementation,the digital filter is implemented in terms of a collection of specialized very-large-scaleintegrated (VLSI) circuit chips. Such an implementation is deemed to be good if it is fast, reliable, and economical. Although the advantages associated with wave digital-filter and state-space structures (e.g., low sensitivity and low roundoff noise) are important, the network complexity of these structures often renders them unsuitable for VLSI implementation. To achieve high speed of operation, the filter structure must allow a high degree of concurrency, that is, a large number of processing elements must operate simultaneously. In such a case, communication among processing elements becomes critical. Since the cost, performance, and speed of a chip depend heavily on the delay and areaof the interconnection network, processing elements should be interconnected by simple, short, and regular communication paths. A class of digital-filter structures that are suitable for VLSI implementation is the class of systolic structures [5-61. These are highly regular structures of simply interconnected processing elements thatpass data from one processing element to the next in a fashion resembling the rhythmical systolic operation of the heart and arteries. This chapter consists of two parts. In the first part, comprising Secs. 12.2 and 12.3, the 2-D discrete Fourier transform (DFI’) is described and methods for its efficient Computation in terms of l-D and 2-D fast-Fourier transforms are considered. These techniques are important tools for the software implementation of nonrecursive filters. In the second part, namely, Sec. 12.4, systolic structures for the implementation of 2-D nonrecursive and recursive filters are examined.

12.22-DDISCRETEFOURIERTRANSFORM In many applications, the 2LD discrete signals under consideration are of finite extent or periodic. In such cases, the 2-D z transform and 2-D Fourier transform described in Chap. 3 are well defined and, consequently, the signal can be represented in terms of the DFI’. In what follows, the D m is defined and its salient properties are described. Its application for the implementation of 2-D nonrecursive digital filters is then considered in some detail.

12.2.1DefinitionandProperties The D l T of a 2-D discrete and periodic signal f p ( n l , n2) with periods Nl and N2 is defined by

344

Chapter 12

We can write

(12.2) and since

for i = 1, 2, Eq. (12.2) implies that

The function at the right-hand side in Eq. (12.3) is called the inverse D l T of Fp(kl,k,) and it can be used to recover the signal fp(nl,n2) from its DE. If f(nl, n2) is a finite-extent, first-quadrant signal with the region of support RNl, = (0 In, 5 Nl - 1 , 0 5 n2 S N2 - l}, a periodic extension f p ( n l , n,) with periods N1 and N2 can be formed as

which has a DFT Fp(nl, n2) given by Eq. (12.1). In what follows, we assume that the signals of interest are periodic. If they are not periodic, they are forced to become periodic through the periodic extension in Eq. (12.4). For the sake of simplicity, subscript p in fp(nl,n,) and Fp(kl, k2) is omitted. The properties of the 2-D DFT are essentially the same as those of its l-D counterpart andcan be summarized in terms of a number of theorems. If f(nl, n2) and g(nl, n2) are periodic signals with periods Nl and N2,

345

Implementation

respectively, such that

m , , n2) * W , , k2) and g@,, n2) * W , , k2) and a, b are arbitrary constants, then the following theorems hold.

Theorem 12.2 Periodicity F(kl, k2)is a periodic function of (kl, k2) with periods N1 and N2; that is, for any integers rl and r, W

1

+

r1N1,

k! + r2N2) = W

, , k2)

Theorem 12.3 Separability Zf Fl(kl) is the Nl-point DFT o f f l ( n l ) and F2(k2)is the N2-pointDFT off2(n2),then f(n1, n2) = fl(nl)f2(n*>*

k2) = ~l(kl)F,(k2)

Theorem 12.4 Parseval’s Theorem Zfg*(nl, n2)is the complex conjugate ofg(n1, n2), then

Theorem 12.5 Duality

F*(%, n2) * ~ l N Z f * ( k lk!) , The circular convolution of f(n,, n2)and g(n,, n2)is defined by

f h , It2) @ g b 1 , n2)

where (ni - mi)Njrepresents an integer pi that satisfies the constraints 0 S pi S Ni - 1 and pi = (ni - mi) + liNi for some integer li. An important property of the circular convolution is given by the following theorem. Theorem 12.6 Circular Convolution

Chapter 12

346

12.2.2 Implementation of Nonrecursive Filters Using the 2-D DFT Consider a 2-D nonrecursive digital filter characterized by the transfer function

The impulse response of the filter h(m,, m2)has a finite region of support in the first quadrant given by Rh = (0 Im, INl - 1 , O Im2 INZ- l}. If x(n,, n2) and y(nl, n2) are the input and output signals of the filter, respectively, then y(nl, n2) can be computed through the convolution N I - l NZ-l

y(nl,

=

c x

m l = 0 m2=O

m2)x(nl

-

n2

-

m2)

If the input signal x(nl, n2) has a finite region of support R, = (0 9 n, I M , - 1 , 0 In2 S M2 - l},then Eq. (12.9) implies that the output signal y(n,, n2) has a region of support R,, = (0 Inl 5 Nl + M , - 1 , 0 5 n2 5 N2 + M2 - l}. In order tofacilitate the application of the D l T for the implementation of the filter, the impulse response, input, and outputmust have a common region of support. Since the regions of support of h(n,, n2) and x(nl, n2) are subsets of that of y(nl, n2), we can use R, = (0 Inl 5 P,, 0 5 n2 I PJ, where P , 1 N , + M , - 1 and P2 2 N2 + M2 - 1, as the common region of support. Augmented versions of x(n,, n2) and h(n,, nz) over region R, can be defined as

(12.10) and

(12.11) From Eqs. (12.9) and (12.10), we can now write P1-l Y(n17

n2>

=

P2-l

c c

m l = 0 m2=0

hu(ml,

m2)xU(nl

-

n2

-

m2)

Implementation

347

and from Theorem 12.6 Y(k17

=

'2)

k2)Xa(k17 k2)

where Y(k,, k,), H,(k,, k,), and X,(k,, k,) are the (P, x P,)-point DFTs of y(n17 n2)7 n2), and x&,, n,), respectively. Once Y(k,, k,) is found, y(nl, n2) for (n17n,) E R, can be computed by using the (P, x P,)-point inverse D l T of Y(k,, k,). The steps involved are as follows: STEP

1: Identify the regions of support of h(n,, n,) and x(n,, n,).

STEP 2:

Identify the region of support R,.

STEP 3:

Augment signals h@,, n,) andx(n,, n,) to h,(nl, n,) andx,(n,, n,) using Eqs. (12.10) and (12.11), respectively. STEP 4:

Compute the (P,

X

P,)-point DFTs for h,(nl, n2) and x&,

nz).

STEP 5:

Compute the (P, X P,)-point inverse DFT of H,(k,, k,)X,(k,, k,) to obtain y(n,, n2).

12.3 COMPUTATION OF 2-D DFT It follows from Eq. (12.1) that computing the DFT of a signal f(n,, n,) over the region Rf = (0 Ik, S NI - 1 , O Ik2 5 N , - l}requires totally N:N$ complex multiplications and N,N2(N1N, - 1) complex additions. More efficient computation can be achieved (1) by decomposing the DFT into a number of l-D DFTs that can be computed using l-D fast-Fourier transforms ( m s ) or (2) by computing the 2-D DFT using 2-D m s .

12.3.1Row-ColumnDecomposition Let

=

-

2-&,, n,)W&?

nz=O

(12.12)

348

Chapter 12

where

Each row of the 2-D signalf(n,, nz), namely,f(n,, n,) for 0 5 n, I Nl 1with n2 fixed, can be regarded asa l-D signal with DFTF(kl, nz). Once E(kl, n,) is found for all n2 in the range 0 5 n, I N2 - 1, a column comprising N2 elements is formed whose DFT is F(kl, k,) as per Eq. (12.12). Therefore, the 2-D DFT off(nl, nz) can be obtained by computing N, l-D N,-point DFTs and Nl l-D N,-point DFTs. If all the l-D DFTs involved are computed directly, then it can easily be shownthat the preceding row-column decomposition method requiresa total of NlN2(N1 + N,) multiplications and aboutthe same numberof additions. For example, the 2-D DFT of a 512 X 512 signal would require approximately 2.685x lo8 multiplications. By comparison, the direct evaluation of the 2-D DFT would require approximately 6.872 X 1O'O multiplications. A more substantial reduction in the amount of computation can be achieved by employing l-D FFTalgorithms [3, Ch.131for the computation of each l-D D m involved. Since the total number of multiplications required in the FFT of an N-point signal is (N/2) log, N, the use of the rowcolumn decomposition approach in conjunction with a l-D FFT would require a total of (N1N2/2)log2(NlNz) multiplications and N1N2 log2(N,Nz) additions. Consequently, the DFT of a 512 X 512 signal would require only 2.36 x lo6multiplications, which representsa 99.65 percent reduction in the number of multiplications relative to direct evaluation.

12.3.2 Vector- Radix FFT An alternative approachto the efficient evaluationof a 2-D DFT is to use a 2-D FFT algorithm. The basic idea in the various l-D FFT algorithms [3, Ch. 131, [4, Ch.81 is that an N-pointDFT can be evaluated by computing two (N/2)-point DFTsor by computing four (N/4)-point DFTs, andso on. The same idea applies to the 2-D case. In what follows, we describe the so-called decimation-in-placeFFT algorithm, which is the 2-D counterpart of the well-known decimation-in-time l-D FFT algorithm. Consider the 2-D DFT of f(n,, n,) given in Eq. (12.1) and assume that Nl = N, = N = 2' A radix-(2

X

2) FFT algorithm can be obtained by writing Eq. (12.1) as

349

Implementation

-

( N E ) 1 ( N E )- 1

2 2

F,,(kl, kz) =

nl=0

f(2n1

n2=0

-

(NE) 1 ( N E )- 1

F,,(kl, k,) =

2 2

nl=0

nz=0

f(2n1, 2n2 + l)WPlW$mz

-

( N E )- 1 ( N E ) 1

F,,(kl, k2) =

2 2

nt=O

n2=0

+ 1, 2n2)WP1W$mz

f(2n1,

+ 1, 2n, + l ) W P * W P

Since F,,, F,,, Fe,, and F, are periodic with periods N/2, it can readily be shown that

Equations (12.14)-(12.17) can be represented by the "radix-(2 X 2) butterfly" flow graph of Fig. 12.1. Since N/2 is a power of 2, F,,, F,,, F,, and F,, can be expressed in terms of (N/4)-point DFTs using the same approach.By repeating this decimation process log, N times, only a number of 2 X 2 DFTs need to beperformed, which do not require complex multiplications. As an example, Fig. 12.2 shows a complete vector-radix FFT for a 4 X 4 array. Note that each cycle of the FFT entails N2/4 butterflies and each butterfly needs three multiplications and eight additions. Consequently, the total number of multiplications required in a radix-(2 X 2) FFT is (3W/4) log, N. The use of the preceding 2-D FFT for thecomputation of the DFT of a 512 x 512 signal would require only 1.77 X lo6 multiplications, which represents a 25 percent reduction in the number of multiplications relative to that required by the row-column decomposition approach.

Chapter 12

350

Figure 12.1 A radix-(2 x 2) butterfly.

12.4 SYSTOLIC IMPLEMENTATION Although a software implementation is entirely satisfactory for many applications, the highest processing rate that can be achieved is limited by the use of general-purpose hardware. Consequently, in applications where a software implementation cannot provide the required processing rate, an implementation in terms of specialized hardware should be considered. Of the numerous structures that have been proposed for the realization of 2-D digital filters (see Chap. lo), systolic structures appear to offer some significant advantages when it comes to VLSI implementation. In this section, some basic systolic structures for the implementation of l-D nonrecursive filters are first described and are thenapplied for the implementation of 2-D digital filters. 12.4.1 Systolic Implementation of l - D Nonrecursive Filters Let

c h,,,zmm N

H(z) =

m=O

(12.18)

be the transfer function of a l-D nonrecursive digital filter. The output of the filter in response to input signal x(n), n = 0, 1, . . . is given by the

Implementation

351

Figure 12.2 A vector-radix FFT for a 4

X

4 2-D array.

convolution N

c

Y(n) = m = O

-

m)

(12.19)

There are a number of realizations for transfer function H(z) that can be transformed directly into corresponding systolic structures. A realization of this type, which can readily be obtained by using the direct realization method [3], is illustrated in Fig. 12.3a. This realization can easily be rearranged into a pipelined structure comprising a number of cascade processing elements (PES) as shown in Fig.12.3b. The unit delay in the rightmost PE and the adder in the leftmost PE are included in order to achieve a regular structure with identical PES. An obvious disadvantage of this structure is that, unlike the N multiplications, which can be performed concurrently, the N additions haveto be carried out sequentially. Therefore, the processing time required is 7 , + where 7 , and T~ denote the time needed to perform one multiplication and one addition, respectively. The processing rate in the preceding structure can be improved by adding a pair of unit delays in each PE, as shown in Fig. 12.4. The addition of

352

Chapter 12

J

r"-"""+""""-

Rocessing element #"-

I

I

I

I

I

I

c""""-l""""-,

" "_

... .

Figure 12.3 (a) A direct realizationof H ( z ) ;(b) corresponding pipelined structure.

the N delay pairs will delay the signals in the topand bottom pathsin Fig. 12.3b by N sampling periods without destroying their relative timing. Consequently, the output will be delayed by N sampling periods. In other words, if y,(n) is the outputof the modified structure and y ( n ) is the output of the original structure, then y,(n) = y ( n - N). It is noted that in the modified structure only one multiplication and one addition are required

Implementation

353

"""""""""""""""""""".

" " " " " " " " " - 4

Figure 12.4 A systolic structure based on the structure of Fig. 12.3(b).

during each digital-filter cycle. Hence the processing time is now independent of the order of the filter and is given by T, + 7,. An alternative direct realization of H(z) is depicted in Fig. 12.5a [3]. This can be rearranged as a pipelined structure, as shown in Fig. 12.5b. A drawback of this structure is that the inputsignal must be communicated to all the PES simultaneously. When the order of the filter is high, the propagation delays due to long signal paths may decrease the processing rate significantly. Global communication can be eliminated by using padding delays, as depicted in Fig. 12.5~. 12.4.2 Systolic Implementation of l-D Recursive Filters Consider now the systolic implementation of l-D recursive digital filters characterized by the transfer function H(z) =

airi

bo = 1

zE0

(12.20)

The output of the filter in response to input signal x(n), n = 0, 1, . . . is given by

y(n) =

N

N ..

i=O

i=l

c a,x(n - i) - 2 b,y(n - i)

(12.21)

Equation (12.21) implies that output y(n) can be obtained by computing two convolutions in which the N most recent input and output values are used. Therefore, systolic structures forthe implementation of H(z) can be obtained by properly combining the structures discussed in Sec. 12.4.1. Figure 12.6a shows one such combination [6]. In this structure, two types of PES are used, as depicted in Fig. 12.6b. The implementation requires totally 2N adders, 2N multipliers, and 2N unit delays, and the minimum

354

Chapter 12

Figure 12.5 (a) A direct realizationof H@);(b) corresponding pipelined structure; (c) systolic structure.

cycle time is 7 , in parallel.

+ 27, as the multiplications in each PE can be performed

12.4.3 Systolic Implementation of 2-D Filters Consider a nonrecursive 2-D filter with a transfer function given by NI-l NZ-l H (2Z21),

=

2 2

n l = 0 nz=0

h(n1, n&;"'z,"2

(12.22)

From Sec. 10.3.1, it follows that the filter can be realized by first realizing r, pairs of l-D nonrecursive transfer functions Li(zl)and Ui(zz)and then connecting pairs of cascaded l-D subfilters in parallel, as depicted in Fig. 10.4. Parameter r, is the rank of the coefficient matrix of the transfer function, and

N,

(12.24) k=i

A systolic implementation of the 2-D filter can be obtained by realizing

the preceding transfer functions in terms of the systolic structures discussed in Sec. 12.4.1. It is noted that the ith pair of the l-D transfer functions given by Eqs. (12.23) and (12.24) can be written as (12.25) (12.26)

355

Implementation

"""_

l"""""""""""""

x(n)

I

I I

I

I

I

I

l

I

I I

I

""""""-r"""""",

I

I

I

I

I I I

I I

---- --------- , I

r""""""""-

T""""""""-c""""""""

I

I

I

I I

I I I

I

I

I

I

I I

I

I

I

I I

... C " " " " " " " " " " " " " " " " " .

I

I

c-"""""""",

l

(C)

respectively and,therefore,the implementation of Li(zl)and requires only Nj - i cells plus i - 1 unit-delay elements for j = 1 and 2, respectively. The systolic implementation obtained above can be used as a building block for the implementation of 2-D recursive filters. The transfer function

can be written as

whereN(z;', z;l) and D(zcl, z ~ ' )are given by Eqs. (10.25) and (10.26), respectively, and hence the recursive filter can be realized using one of the two-block feedback schemes depicted in Fig. 10.9a and b. An alternative approach to the implementation of recursive 2-D filters is to use a systolic implementation for l-D recursive filters as a building

Chapter 12

...

'N- 1

...

"

t

0) Figure 12.6 (a) Systolic implementation for l-D recursive filters; (b) structures of processing elements A and B.

Implementation

357

block. This approach is especially useful if the 2-D filter under consideration is quadrantally symmetric [7], as is demonstrated below. It can be shown that the transfer function of a quadrantally symmetric filter has a separable denominator (see Ref. 6, Chap. 8). If we apply an LU decomposition to its numerator, then it follows from Sec. 10.3.1 that

where r, = rank (A) and am

A = [

a10 aaNN~~Ol

a,,

*

*

a!,

*

*

a0N2 a1,2

uN~N;

Hence the transfer function can be expressed as

where

represent the &h pair of recursive l-D subfilters. Consequently, a systolic implementation of H ( z l , z2) can be constructed by connecting r, pairs of cascaded l-D systolic structures in parallel.

REFERENCES 1. D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing, Englewood Cliffs, N.J.: Prentice-Hall, 1984. 2. J. S. Lim and A. V. Oppenheim (eds.), Advanced Topics in Signal Processing, Englewood Cliffs, N.J.: Prentice-Hall, 1988. 3. A. Antoniou, Digital Filters: Analysis and Design, New York: McGraw-Hill, 1979 (2nd ed. in press). 4. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Englewood Cliffs, N.J.: Prentice-Hall, 1989. 5. H. T. Kung, Why systolic architectures?, ZEEE Computer Mag., vol. 15, pp. 37-46, Jan. 1982. 6. W. Luk and G. Jones, Systolic recursive filters, ZEEE Trans. Circuits Syst., vol. 35, pp. 1067-1068, Aug. 1988. 7 . W.-S. Lu and E. B. Lee, An efficient implementation scheme for quadrantally symmetric 2-D digitalfilters, ZEEE Tran. Circuits Syst., vol. 35, pp. 239-241, Feb. 1983.

358

Chapter 12

PROBLEMS 12.1 Prove Theorems 12.1-12.3. 12.2 Prove Theorems 12.4-12.6. 12.3 Demonstrate the computational

efficiencyof the row-column decomposition for the 2-D DFT by producing a table listing the total numbers of multiplications and additions required for the 2-D DFT of an N X N signal versus N where N varies from 26 to 21°.

12.4 Repeat Problem 12.3 for the vector-radix F R approach. 12.5 A nonrecursive 2-D filter can be implementated by using the SVD-

LUD realization method (see Sec. 10.3.2) in conjunction with l-D DFT. Compare thecomputational efficiencyof this approach to those of the row-column decomposition approach and the vector-radix F R approach. 12.6 Find a systolic implementation for the transfer function H(z,, 2,) = zcCq, where zi = [ l 2 ; ' i = 1, 2, for the followingcases:

- 2r4J

0.0224 0.2186 0.6043 0.3212 0.4143 0.5258 0.5051 0.6137 0.7045 0.6137 0.5051 0.3212 0.4143 0.5258 0.0224 0.2186 0.6043

0.2186 0.0224 0.4143 0.3212

1

.

0.4143 0.3212 0.2186 0.0224

-0.0414 0.0117 0.2217 0.0117 -0.0414 0.2072 0.3114 0.4792 0.3114 0.2072 0.4558 0.6111 0.7367 0.6111 0.4558 0.2072 0.3114 0.4792 0.3114 0.2072 -0.0414 0.0117 0.2217 0.0117 -0.0414

1

.

12.7 Obtain a systolic implementation for each of the following transfer

functions:

-1.6215 1.0 1.0 [l = 0.00895

2,'

z T ~ ] -1.6215 2.6370 1.6213 1.62131.0 1.0 -1.7881 0.8293

1.0

0.8293 -1.7881 3.2064 -1.4927 - 1.4927 0.6982

Implementation

359

12.8 Find a systolic implementation of the state-space filter described by

Eq. (11.10) where

A =

[

1.8890 -0.9122

i i

-1.0

0.0

..................... ....................0.0 0.0

f

0.0

0.0277 -0.0258 i 1.8890 1.0 0.0243 i -0.9122 0.0 -0.0258 b = [0.2191 0.0 1 -0.0289 0.0912IT C = [0.2889 -0.0912 i -0.2191 0.01 d=O

Applications

13.1 INTRODUCTION Two-dimensional digital filters have been used in many different areas in the.past, ranging from the digital processing of satellite photographs, radar maps, and medical x-ray images to the planning of power systems [l-91. In this chapter, some of the applications of linear 2-D digital filters in the areas of image and seismic data processing are briefly reviewed. In Sec. 13.2, two techniques for the enhancement of images are examined. In Sec. 13.3, a well-known constrained least-squares estimation approach [3] to image restoration is described. The approach is formulated in terms of the 2-D FFT and is, therefore, efficient. In Sec. 13.4, the application of 2-D fan filters for the processing of seismic data is discussed.

13.2 APPLICATIONS OF LINEAR 2-D FILTERS TO IMAGE ENHANCEMENT Image enhancement is the process by which an image is manipulated or transformed in order to extract useful information from it, to improve the visual effect of the image, or to change it into some form that is more suitable for further analysis. For example, in robotics a vision system is 360

Applications

361

often required simply because the determination of the geometric shape of the object to be handled by the robot is critical to successful robot motion. In this case,a highpass filter canbe used to strengthenthe contour of the object. In what follows, two image enhancement techniques based on the use of 2-D linear filters are described. Section 13.2.1 deals with a technique for the removal of noise froma digitized imageby using lowpass filters. Section 13.2.2, on the other hand, deals with a technique for the enhancement of edges by using highpass filters [l]. 13.2.1 Noise Removal Transducer and transmission noise in an image tends to be spatially independent, and, therefore, its energy content is usually concentrated at frequencies higher than that of the image. Consequently, processing an image by usinga lowpass filtertends to removea large amountof the noise content without changing the image significantly. The basic requirements imposed on the filter are as follows: 1. A record of the image is usually available. Hence the filter need not be causal. 2. Phase distortion is usually objectionable in image processing (see Ref. [6] of Chap. 6), and hence the filter should have zero or linear phase response. 3. The sum of the impulse-response values should, if possible, be equal to unity to prevent bias in the light intensity during the processing. This requirement ensures that the image background will not be affected by the noise removal process. A 2-D nonrecursive filter of order

(Nl, NZ)where Nl and NZare odd can

be represented by

where C, = (Nl - 1)/2, C, = (NZ- 1)/2 and {x(nl, n,): 0 In1 IM, 1, 0 5 n, S M, - l} represents a discretized version of the image to be processed. The quantity x(nl, n,) is a measure of the light intensity at coordinate (nlTl, n2T,) of the image. The transfer function of the filter is given by

362

Chapter 13

and if =

-Ita)

for -C1S n, S C,,- C ,

S

n2 S C,

.

(13.3)

a zero phase response is achieved (see Sec. 6.3.1). The impulseresponses of three 3 X 3nonrecursivefilters commonly used for noise removal are as follows [l].

that are

h, = {h(n,, n,): nl = -1,O, 1, n2 = 1,0, -l} =

h, = {h(n,, 12,):11, = - 1,0,1, 12, = 1,0, -l} =

These filters satisfy Eq. (13.3) and, therefore, have zero phase response, as required. In addition (13.7) that is, requirement 3 above is satisfied. The amplitude response of the filter represented by Eq. (13.6) is shown in Fig. 13.1. The application of this filter for the removal of noise is illustrated in Fig. 13.2. An alternative approach to image smoothing isto use the DFT in conjunction with analog-filter approximations [2].A normalized Butterworth lowpass filter of order N has the transfer function H(s) =

'1 -

D(s)

where D(s) is the Nth-order stable polynomial given by

D(s)D(1 s ) = 1 + s2N (see Chap. 5 of [lo]). We can write

363

Applications

-a

a

It

Figure 13.1 Amplitude response of filter represented by Eq. (13.6).

and hence a 2-D zero-phase frequency response can be obtained as HL(W17 W2)

= 1

+

1 C [ D ( W , , W2)lWc]"

where

+ (W*T2)2 (13.8) Constants c and W, depend on the desired cutoff frequency. If Tl = T2 = 1 and c = .\/z - 1, then W, is the 3-dB cutoff frequency of the filter and HL(wl, w2) can be expressed as WO,,

02)

=

J ( W I W

(13.9)

In this way, a 2-D lowpass circularly symmetric frequency response is achieved. Figure 13.3 shows a 3-D plot of HL(ol, w2) for the case where N = 4 and W, = 2.0 rad/s. If X(k,, k2) is the discrete Fourier transform

364

Chapter 13

Figure 13.2 (a) Idealimage; (b) image of (a) contaminated by high-frequency noise; (c) the processed image.

(DIT) of the original noise-contaminated image and H&,, k,). = H&&,, kfl,) where f l l = 2dT1 and Cn, = 2n/T2, then the filtered image can be obtained as the inverse DFT of HL(kl, k2)X(k1,k2) as

Figure 13.4 shows the output obtained by processing the image of Fig. 13.2b with N = 4 and W, = 2 rad/s. Note that if the processing is to be carried out by using the Dm, the frequency response of the filter need not correspond to a rational transfer function since it is unnecessary to realize the filter in terms of unit delays, adders, and multipliers. A s a matter of fact, it is not possible to find a rational transfer function whose frequency response is given byEq. (13.9).

13.2.2 EdgeEnhancement Many image-related applications such as pattern recognition and machine vision require images of objects with enhanced edges. In an optical image, the pixel intensity changes rapidly across edges, which means that the frequency content due to edges tends to be concentrated at the high end

Applications

365

366

Chapter 13

-K

K

K

Figure 13.3 Amplitude response of filter represented by Eq. (13.9) (n = 4 and W, = 2 radh).

of the spectrum. As a consequence, edge enhancement can be achieved by using highpass filters. Three 3 x 3 nonrecursive filters that have been used for edge enhancement in the past [l]are characterized by the following impulse responses:

-K]

h, = {h(nl, 112): n1 = -1, 0, 1, n2 = 1, 0, -1) =

h, = =

[-l 0

-15

0 -1 (h(n1, 112):

[

-1 -1 -1

-1 9

=

(13.10) -1, 0, 1, 112 = 1, 0, -l}

-1 -l] -1

(13.11)

367

Applications

Figure 13.4 The processed image.

h3

=

=

(h(n1, n2): n1 = - 1, 0, 1, n2 = 1, 0, -l}

[

1 -2 -2 5 1 -2

-21

(13.12)

Like the lowpass filters of Sec. 13.2.1, the preceding highpass filters have zero phase and satisfy the constraint in Eq. (13.7). Figure 13.5 shows the amplitude response of the filter represented by Eq. (13.10). To demonstrate the effect of a highpass filter on an image, the filter represented by Eq. (13.10) was used to process the image of Fig. 13.6a. The image obtained, depicted in Fig. 13.6b, demonstrates that edges are enhanced. A s in the case of noise elimination, edge enhancement can be accompished by using the DIT in conjunction with analog-filter approximations. A highpass zero-phase frequency response can be obtained from the Butterworth approximation as [2]: 1 w2)

if D ( o , , 02) # 0

=

otherwise

Chapter 13

368

c

-K

Figure 13.5 Amplitude response of highpass filter represented by Eq. (13.10).

where D(w,,02) is given by Eq. (13.8) and c and W, depend on thedesired cutoff frequency. If TI = T2 = 1 and c = .\/z - 1, then W, is the 3dB cutoff frequency and HH(wI,02) can be expressed as (13.13) The frequency response achieved is depicted in Fig. 13.7. As in Sec. 13.2.1, we first compute the DFI? of the image X(kl, k2), then multiply it by the sampled frequency response HH(kl, k2), and then compute the inverse DFT of the product to obtain the processed image y(n17

=

9"[HJf(k19

k2)X(k17

k2>1

Figure 13.8 shows how the edges in the image of Fig. 13.6a are enhanced by using a Butterworth highpass filter with n = 4 and o, = 2 radls. The

369

Applications

(b) Figure 13.6

(a) The-original image; (b) the processed image.

370

Chapter 13

I

-K

Figure 13.7 Highpassfrequencyresponsebased (n = 4 and W, = 2 rad/s).

on Butterworthapproximation

spikes are due to the increased high frequency content which is brought about by the steep transitions at the edges. 13.3 IMAGE RESTORATION

Images are often corrupted by bandwidth reduction during the formation of the image, nonlinearity of the recording medium (e.g., photographic film), and noise introduced during the transmission, recording, measurement, digitization, etc.[4]. The purpose of image restoration is to improve or criterion. In this section,we a recorded image according to some norm

371

Applications

Figure 13.8 The processed version of the image in Fig. 13.6a.

present a constrained least-squares estimation approach to the image restoration problem proposed by Hunt [3].

13.3.1The

ImageRestorationProblem

The basic model that describes the process of image formation can be presented as [4]

where f ( x , y) represents the ideal image, h(x7y; S, t) is the point-spread function of the image formation system, Yp[.] represents the sensor nonlinearity, n(x, y) is a noise term, and g(x, y) is the recorded image. If the image formation system is stationary across the image and the sensor nonlinearity is negligible, then the image model becomes

Chapter 13

372

If the image f(x, y) is defined over the rectangle R, = (0 Ix IA , 0 I y IB} and the point-spread function can be well represented by its values over the rectangle R,, = (0 Ix IC, 0 5 y 5 D},then therecorded image g(x, y) is a finite-extent, first-quadrant 2-Dsignal withthe region of support R , = (0 5 x IA + C, 0 5 y 5 B + D}. A discrete version of the preceding process can be obtained by approximating the integral in Eq. (13.14) by a double summation. In this way, the light intensity of the recorded image at point (nl&, n2Ay) can be evaluated as

+ n(n1, n2)

(13.15) where g(nl, n2) = g(n,Ax, n2Ay), h(n, - m,,n2 - m 3 = h[(n, - m,)Ax, (122 - mJAy] hx Ay, n(n1, nz) n(nl&, nzAy). If A = (M1 - l)&, B = (M2 - l)Ay, C = ( N , - l)&, and D = (NZ - l)Ay, then Eq. (13.14) implies that {g(nl, n2)} and {n(nl, n2)} are matrices of dimension (M, + Nl - 1) X (M2+ N2 - 1). Now define the augmented matrices Ve(m1, m2)) and {he(n,, n2)) as fe(m1, m21

- f(m,, m2) - Io he(nl, nz) =

for 0 5 m, IM, - 1 and 0 Im2 5 M2 - 1 for M, Im, IP , - 1 or M2 5 m, IP2 - 1

Z V ,

for 0 5 n1 I - 1 and 0 In2 IN2 - 1 for Nl In, IP1 - 1 or N2 5 n2 5 P2 - 1

h(n,, nz)

Io

where P , and P2 are integers satisfying the conditions P, 2 M, + NI 1 and P2 2 M2 + N2 - 1. In addition, define three column vectors f,g , and n by lexicographically ordering the matrices {fe(m,,%)), {&l, n2)>, and n2)) as

W,,

f = [fd fel

* * *

fe,Pl-

11T

g = [go g,

.

n

. . . npl-,JT

=

[no n,

* *

g,l-llT

where fei,g,, n, are the ithrow vector of matrices {fe(ml, m2)},{g(nl,n2)}, and {n(nl, n2)},respectively. Under these circumstances, Eq. (13.14) can

373

Applications

be expressed in a vector-matrix form as (13.16)

g=Hf+n

where H is a matrix of dimension P1P2 X P1P2given by H =

[

...

H0

HP,-1

.

H 0

* * *

HP,-2

-

H I

%-l

*

H1

H.]

(13.17)

H0

with each submatrix Hi formed by the jth row of matrix {he(nl, n2)} as a circulant matrix Hi =

[

h e ( j , 0) he(!,

1)

he(j, P2 - 1) he(!,

he(j, p2 - 1) he(j, P ,- 2)

*

1)

:l

*

he(j7

*

h e ( j ,(13.18) 2)

0)

he(;, 0)

In terms of the preceding formulation,the image restoration problem can be stated as: Given an image formation system charactlerized by matrix H and a recorded image represented by vector g, find the column vector that optimally approximates the representation of the original image f under some norm or criterion. There are two problems in image restoration that need to be carefully addressed. The first is associated with the ill-posedness of the image restoration process. As was stated earlier, Eq. (13.16) is a discrete approximation of the integral in Eq. (13.14), which is often called the integral equation of the first kind [ll]. Solving Eq. (13.14) for a given g(x, y) often leads to numerical difficulties because the unboundedness of the inverse of the integral operator associated with Eq. (13.14) tends to magnify the noise componentn(x, y) and, therefore, leads to an unacceptable numerical solution. A s a result, H is always an ill-conditioned matrix; that is, the is very large and, therefore, ratio of the largest to the smallest singular value a small noise component n can lead to large restoration errors. Consequently, physically meaningful constraints needto be imposed on the approximatesolution of Eq. (13.16)toreducenumericalinstability. The second problem is directly related to the computational complexity associated with the solution of Eq. (13.16). The dimension of an image in most cases is at least 512X 512. If the point-spread functionof the degradation system can be represented by a matrix {h(n,, nz)} of dimension 32 x 32, then the dimension of matrix H is 295,936 x 295,936. Obviously, solving such a large linear system of equations is unrealistic for most existing computer systems.

Chapter 13

374

Solutions to thepreceding problems have been obtainedby Hunt in his classic work on this subject [3]. The first problem is solved by using a constrained leastsquares estimation approach. The second problem is solved by noting that matrix H is block-circulant (i.e., H as a block matrix is circulant and each block matrix Hi is also circulant) and then using a frequency-domain approach. In this way, the required computation is carried out by computing the DFTsof several 2-D signals of dimension P , X P2. These solutions are described in some detail below.

13.3.2 The Constrained Least-Squares Estimation Approach If f is the solution to Eq. (13.16), then f satisfies the equation

Ik - Hfl12= Ilnll’

(13.19)

The constrained least-squares estimation approach [3] yields an approximate solution f for Eq. (13.16) that minimizes

llCCll2

(13.20)

Hfl12= Ilnll’

(13.21)

subject to the constraint llg -

where C is a matrix of dimension P , X P2 representing a measure of smoothness. On comparing Eq. (13.21) with Eq. (13.19), we observe that Eq. (13.21) provides a reasonable constraint on the solution f. To identify matrix C in Eq. (13.20), the Laplacian of function f ( x , y ) defined by

= f(n,

+

1, nz)

+ f(n, -

1, n2)

”

m , ,

n2

+

1)

+ m , , 122 - 1) - 4 m * ,n2) (13.22) with f(n,, n2) = f(n,Ax, n2Ay) = f ( x , y ) is employed as a measure of smoothness since A2f responds to transitions in image light intensity. The discrete analogy of the Laplacian described by Eq. (13.22) can be realized byzhe 2-D convolution of f ( x , y with ) signal p ( x , y ) given by

To avoid wraparound error in the discrete convolution, the augmented

375

Applications

matrix pe(nl,n2) is defined as Pdn17 n2)

where P,

2

Cf(rnl,r n 2 ) } .

=

0 5 nl

5 2 and 0 In2 5 2 3 s n 1 5 P 1 - 1 o r O s n 2 s P 2- 1

(p(nl, n2) 0

M 1 + 2, P2 2 M 2 + 2, and M 1 X M 2is the dimension of The discrete Laplacian can now be expressed as

Proceeding as in Sec. 13.3.1, a block-circulant matrix C can be formed as

...

'1

(13.23)

CO

where each submatrix Cj is a P, x P2 circulant matrix generated by the jth row of matrix {pe(nl,n2),0 5 nl 5 P, - 1, 0 5 n2 5 P2 - l), that is,

, Having defined matrix C of Eq. (13.20), we can now form theobjective function

J(f)

= llCfIl2

+ 4llg - Hill2 - llnl12)

where a is the Lagrange multiplier. It is known from calculus that the vector f that minimizes IlCf(12can be obtained by solving

for i, that is,

f

= (HHH

+ yCHC)"HHg

(13.25)

where y = l / a has to be chosen so that givenby Eq. (13.25) satisfies constraint (13.21). To find such a y, define scalar function +(y) as

4f-d

= llg - HfIl2

where f is given by (13.25) so that it depends on y. It can be shown [3] that d+(y)/dy is nonnegative and, therefore, 4(y) is monotonically in-

Chapter 13

376

creasing with respect to y. Since the constraint in Eq. (13.21) can be expressed as

+W

=

llnllZ

the monotonicity property of +(y) leads immediately to the following algorithm for the computation of the optimizing p. STEP

1: Choose an initial value of y that is greater than zero.

STEP 2:

'

Compute using Eq. (13.25) and compute +(y).

If +(y) 5 [Ilnllz - E , lln112 + E ] where E is a prescribed tolerance, then take output f as the solution and stop.

STEP 3:

4: If +(y) < l)n112 - E, increase y using an appropriate algorithm such as the Newton-Raphson algorithm and return to Step 2.

STEP

STEP 5: If +(y) > (In((' + and return to Step2.

E,

decrease y using an appropriate algorithm

13.3.3 Implementation of the Constrained Least-Squares Estimation A s can be seen from Eq. (13.25), the constrained least-square estimation of the image being restored requires theinversion of matrix HTH y C T .

+

The dimension of this matrix is very large, as was demonstrated in Sec. 13.3.1 (see p. 373), and its inversion is, therefore, not practical. The key fact that enabled Hunt [3] to obtain a very efficient implementation of the algorithm is that matrices H and C, defined by Eqs. (13.17)-(13.18) and (13.23)-(13.24), respectively, are block-circulant. Consider first a circulant matrix S of dimension N X N given by

It can readily be shown that SW,, =

XNKWNK

f o r k = 0, 1,

. . . ,N

-1

(13.26)

where

=

SIWNk

(13.28)

and s1denotes thefirst row of matrix S. Equation (13.26) implies that W N k

Applications

377

for k = 0, 1, . . . ,N - 1 are N eigenvectors of S with the corresponding eigenvalues h, for k = 0, 1, . . . , N - 1, that is, S = WNDNW,’

where WN

=

fWNO wNl

[

WN,N-ll

AN0

DN=

0

”,”’ ’ . .

h

(13.29)

]

NN- 1

Note that theeigenvectors W N k specified by Eq. (13.27) are independent of matrix S . In other words, W N k for k = 0 , 1 , . . . ,N - 1 can be regarded as the complete set of eigenvectors for any circulant matrix of dimension N X N. In addition, it can easily be verified that W,’

1N N

= “W

(13.30)

wN

where is the complex conjugate of W N .Furthermore, the eigenvalues h, given by Eq. (13.28) can be written as hNk = ~ ( 0 )+ s(N - 1)ejZTklN + =s(O)

. . . + s(l)ei2~(N-l)k/N

+ s(N - l)e-j24N-lWN +

. . . + s(l)e-jzrk/N

N-l

= S(k)

(13.31)

where S(k) is the kth component of the Dm of the signal {s(l): 0 N - l}.

I 5

I

Similarly, for the block-circular matrix H given by Eq. (13.17), one can find a transformation matrix W and a diagonal matrix D such that H = WD,,W-l

(13.32)

where W,

WO,

W10 W W11 = WP1-1,0 WP1-l,l

-*

-

*

W,,,,

-1

WP,- l,Pl - 1

(13.33)

Chapter 13

378 W,,, = ~ P l ( i m)WP2 , W,, = { W p 2 ( k , I ) ) k = 0,1,. . . ,P2 - 1, I = 0,1,.

(13.34)

. . ,P2 - 1

(13.35)

w P , ( i , m) =

ei2mimIP1

(13.36)

Wp2(k, I) =

ei2TkllP2

(13.37)

and the diagonal elements of matrix D*,namely, the eigenvalues of matrix H, are given by

d,(k, k ) = He(

[61,

k mod P 2 )

for k = 0,1, . . . ,P ,

-1

(13.38)

where He is the 2-D DFT of he(nl,n2) defined in Sec. 13.3.1, [k/P2]denotes the biggest integer not exceeding k/P2, and k mod P2 is the remainder obtained by dividing k by P2. It is observed that the transformation matrix W given by Eqs. (13.33) to (13.37) does notdepend on the entries of matrix H, that is, it can be used as the transformation matrix for diagonalizing any block-circulant matrices with the same block structure as matrix H of Eqs. (13.17) and (13.18). Furthermore, by using Eqs. (13.33) to (13.37), it can be shown that

where

Ur,m . =

1 Wj-,l(i, m)W,'

p1 1 e-j2mim/P~w-l -P2 p,

and

W,'

=

{i

e- j2=klM'2]

k = 0 , 1 , . . . , P2-1,

1 = 0 , 1 , . . . , P,-l

Now let us consider the evaluation of the restored signal 8. Equation (13.25) can be written as

(HHH

+ yCHC)f = HHg

(13.40)

Applications

379

where matrix H can be replaced by WDhW-l and matrix C can be replaced by W0,W-l. Since =

Eh

DC WH = W = P,P,W-' D: =

Eq. (13.40) implies that (&Dh

+ yDJI,)W-'f = DhW-'g

that is, which leads to

where He(kl, k,), Pe(kl, k,), G(kl, k2), and Qk,, k2) are the 2-D DFTs of (he(n1, n2)}7 (~e(n17n2)I' FJI, and {.fe(n>, n2)I. once (E(k1, is found from Eq. (13.41), the restored signal f can then be found by computing the inverse DFT of p(kl, k2).

To illustrate the implementation approach, consider the 64 X 64 ideal image shown in Fig. 13.9a. Figure 13.9b shows an image that was blurred by linear motion and was then contaminated by additive zero-mean white noise with variance U = 0.1. The application of Hunt's algorithm leads to y = 0.36. The restored image obtained by employing the DFT implementation approach is depicted in Fig. 13.9~. 13.4 APPLICATIONS OF LINEAR 2-D FAN FILTERS TO

SEISMIC SIGNAL PROCESSING The structure of subsurface ground formations can be explored by using the so-called seismic reflection method [12]. In this method, a seismic wave generated by an explosion of dynamite near the surface travels through different wave paths to different horizontal interfaces and returns to the surface after being reflected by the interfaces. If a detector is placed on the earth surface near the point of the explosion, the time of arrival of each reflection can be recorded and hence the depth of each reflecting subsurface formation can be determined. If a trace of-theground motion

380

Chapter 13

(a) Figure 13.9 (a) Ideal image; (b) an image blurred by linear motion and contaminated by additive white noise with U = 0.1; (c) the restored image.

versus time is recorded, distinct ground interfaces can be identified from the intensity of the trace. If a linear array comprising NI equally spaced detectors is used, as depicted in Fig. 13.10, the waveforms collected can be used to construct a 3-D plot in which the x , y, and z axes represent distance along the array, time, and ground motion, respectively. In plots of this type, which are often referred to as seismic images, subsurface formations can be identified as well-defined ridges. Seismic waveforms consist of two distinct components: a component due to reflections from the ground and subsurface formations and a surface component often referred to as ground roll. The required information is embedded in the reflected component. The ground roll is believed to consist largely of Rayleigh waves [12], which are known to travel along free surfaces of elastic solids. Its presence in a seismic image is highly undesirable since it distorts the patterns produced by the reflected wave and thus renders the image less discernible. An interesting property of the surface wave is that its velocity is much lower than that of the reflected wave, as can be seen in Fig. 13.11, and as a result its spectrum is concentrated at low frequencies.

Applications

381

Chapter 13

382 source

detector array

Figure 13.10 Measurement of seismic data.

The ground motion can be represented by the 2-D discrete signal v(nl, n2) = v(n,L, n,T) where L is the distance between adjacent detectors and l/Tis thesampling rate used to sample waveforms. An actual seismic image of this type comprising 48 horizontal spatial points and 1000 vertical temporal points is illustrated in Fig. 13.12a. The well-defined ridges inclined at approximately 45" to the n1 axis are due to theground roll.

Figure 13.11 Travel time for reflected and surface waves.

383

Applications

(4

nl

Figure 13.12 (a) Raw seismic image; (b) the processed image (from [6]).

In order to study the effect of the ground roll on a seismic image, assume that

zl + *

fo(y) =

1

l {sin ( k w g ) log k

+ cos ( k o g ) )

is the signal generated by the explosion and neglect any reflections from subsurface formations.The signal will propagate along the surface at some velocity F and will arrive at distance x along the axis of the array x l p seconds later. Consequently, the signal received at point x will be f(x, y, =

zl + *

1

l log k [sin

[

(y - $kwo]

+ COS

[

(y - $ k w o ] ]

Obviously, f(x, y) contains a wide range offrequencies with respect to line X

y=-+c P

Chapter 13

384

Figure 13.12 Continued

of the ( x , y) plane. Consequently, the frequency spectrum of f(x, y) is concentrated on a line (13.42) as illustrated in Fig. 13.13a and b. In practice, the line in Eq. (13.42) forms an angle of3" to 5" [6] and, therefore, the ground roll can largely be eliminated by using a fan bandstop filter with an amplitude response of the type shown in Fig. 13.14. Methods for the design of nonrecursive and recursive fan filters have been described in Chaps. 6, 8, and 9 and by Bruton et al. [6]. The effect of a fan filter with 8, = 10" [6] on the seismic image in Fig. 13.12a is illustrated in Fig. 13.12b.A s can be seen,the groundroll ridges have been removed. "2

=

W 1

385

Applications

(a) 3'

2-

1-

0-

-1 -

-3

-2

-1

0

1

2

3

(b)

Figure 13.13 (a) Amplitude spectrum of signal f(x, y); (b) contour plot of the amplitude spectrum.

386

Chapter 13

Figure 13.14 Amplitude response of an ideal fan filter.

REFERENCES 1. W. K. Pratt, Digital Image Processing, New York: Wiley, 1978. 2. R. C . Gonzalez and P. Wintz, Digital Image Processing, 2nd ed., Reading, Mass.: Addison-Wesley, 1987. 3. B. R.Hunt, Theapplication of constrained leastsquares estimation to image restoration by digital computer, IEEE Trans. Computers, vol. C-22, pp. 805812, Sept. 1973. 4. J. Biemond, R. L. Lagendijk, and R. M. Mersereau, Iterative methods for image deblumng, .Proc. IEEE, vol. 78, pp. 856-883, May 1990. 5. G . Garibotto, 2-D recursive phasefilters for the solution of two-dimensional wave equations, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP27, pp. 367-373, Aug. 1979. 6. L. T. Bruton, N. R. Bartley, and R. A. Stein, Stable 2-D recursive filter design usingequi-terminated lossless N-portstructures, Proc. European Conf. Circuit Theory and Design, pp. 300-307, Aug. 1981. 7. K. L.Peacock, On the practical design of discrete velocity filters for seismic data processing, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP30, pp. 52-60, Feb. 1982. 8. A. N. Venetsanopoulos, A survey of multidimensional digital signal processing and applications,Alta Frequenza, vol. LW, pp. 315-326, Oct. 1987.

Applications

387

9. H. L. Willis and J. V. Aanstoos, Some unique signal processing applications in power system planning,ZEEE Tran. Acoust., Speech, Signal Process., vol. ASSP-27, pp. 685-697, Dec. 1979. 10. A. Antoniou, Digital Filters: Analysis and Design, New York: McGraw-Hill, 1979 (2nd ed. in press). 11. R. Kress, Linear Integral Equations, New York: Springer-Verlag, 1989. 12. M. B. Dobrin, Introduction to Geophysical Prospecting, New York: McGrawHill, 1976.

PROBLEMS 13.1 (a) Using MATLAB generate a 128 X 128 binary image as the orig-

inal image. The 3-D plot of the 2-D array generated should display an image with clear images, which can be used to evaluate edge detection or enhancement algorithms. For example, images of letters are suitable for this purpose. (b) Add a 128 x 128 random noise array to the array obtained in part (a) to form a noisy image. This can be done in MATLAB by using the commands rund('normul') and r = rund(128, 128). The r is a zero-mean normally distributed 2-D array with variance = 1.0. Modify r by multiplying it by a small constant to reduce the variance of the noise to an acceptable level. X 3 to remove the noise of the image generated in Problem 13.lb. (b) Use one of the 3 X 3 lowpass filters described by Eqs. (13.4)(13.6) to filter the same noisy image.

13.2 (a) Use a lowpass nonrecursive filter of order higher than 3 '

.

3 X 3 highpass nonrecursive filters described in Eqs. (13.10)-(13.12) to enhance the edges of the image generated in Problem 13.la. (b) Does the same filter work if only the noisy image obtained in Problem 13.lb is available? (c) Use an appropriate combination of different kinds of filters to enhance the edges of the noisy image.

13.3 (a) Use the

13.4 Apply the image restoration technique proposed by Hunt to remove

the noise from the image obtained in Problem 13.lb.

Index

Adder, 14 Admittance matrix, 245 Aggarwal, J. K.,190 Ahmadi, M., 189 Algorithms: least pth minimax, 239-242 quasi-Newton, 232-236 Allpass transformation, 212 Amplitude response: defined, 89 in nonrecursive filters, 134 octagonal symmetry, 268 quadrantal symmetry, 268 sampled, 251 Amplitude spectrum, 66 Analog-filter transformations, 185-187 Analysis: frequency-domain, 86-89 induction method, 21-25 space-domain, 2, 16, 84-86 stability, 3 frequency domain approach, 107-118

Lyapunov approach, 121-124

388

[Analysis: Continued] state-space domain approach, 118-121

Antoniou, A., 178, 240, 260 Applications: edge enhancement, 364-370 fan filters, 384-386 image enhancement, 360-370 image restoration, 370-379 noise removal, 361-364 processing of seismic signals, 379-386

Approximations: for nonrecursive filters (see Design: nonrecursive filters) for recursive filters (see Design: recursive filters) minimal mean-square-error, 252 the approximation process, 2, 132 Attasi, S., 38 Bandpass filters, 95 circularly symmetric, 218 generalized, 195

Index Bandstop filters, 95 generalized, 194 Baseband, 90 Bessel function, 141, 148 BFGS formula, 233 BIB0 stability, 27, 108 Bilinear transformation, 177-178 Blackman window function, 141 Block-circulant matrix, 375 Bose, N. K., 111 Brierley, S. D., 116 Broyden-Fletcher-Goldfarb-Shanno formula, 233 Bruton, L.T., 245,384 Butterfly, 349 Butterworth highpass filters, edge enhancement, 367-370 noise removal, 362-364 Canonic realization, 295 separable transfer function, 295 Causality, 11 Chan, D. K. S., 163 Characterization: by difference equation, 12-14 in state-space domain, 45 Charalambous, C., 240, 243,266, 269 Chebyshev polynomial, 142 realization, 288 Chiasson, J. N.,116 Circulant matrix, 373 Circular symmetry, 91 Coefficient quantization, 3, 319-320 error, 318 Compensation technique, 257-262 Complex-convolution theorem, 63 Complex-differentiation theorem, 59 Complex-scale-change theorem, 58 Computation of 2-D DFT, 347-350 Computational complexity, 3, 284 Conjugate direction method, 269 Constantinides, A. G . , 189, 218 Constantinides transformations, 218-219 Constrained least-squares estimation, 374-379 implementation, 376-379

389 Continuous signals, 1, 5 Controllability, 50-51 Convergence of Fourier transform, 66 Convolution summation, 25-27 Coprime (see Factor coprime or Zero coprime) Costa and Venetsanopoulos design method, 204-212 Costa, J. M., 204 Cutoff frequencies, 94 Davidon-Fletcher-Powell formula, 233 DeCarlo, R. A., 111 Decomposition: signals, 9 singular-value, 10 Design: approximation step, 132 implementation, 133 nonrecursive filters: bandpass filters, 150 prescribed specifications, 156-159 bandstop filters, 150 prescribed specifications, 159-161 based on the McClellan transformation, 163-173 circularly symmetric, 147-150 prescribed specifications, 150-154 fan, 162-163 highpass filters, 149-150 prescribed specifications, 154-156 lowpass filters, 147-149 prescribed specifications, 151- 154 Remez algorithm, 279 selectivity, 266 using Fourier series (l-D), 138-142 using Fourier series (2-D), 142-143 using minimax method, 267-269 using singular-value

390 [Design: Continued] decomposition method, 271-282 realization, 133 recursive filters: based on two-variable network theory, 244-251 circularly symmetric,204-218 bandpass, 218 bandstop, 218 based on SVD method, 256-257 highpass, 215-218 lowpass (prescribed specifications), 223-226 error compensation, 257-262 lowpass with rectangular passbands (prescribed specifications), 222-223 nonsymmetric half-plane, 249-253 l-D, 253 prescribed specifications, 219-226 quarter-plane, 244-249 sample points, 243 transition band, 258 using least pth method, 231-236 using minimax method, 236-244 using singular-value decomposition method, 251-262 using the method of Costa and Venetsanopoulos, 204-212 using the method of Goodman, 212-213 using the method of Hirano and Aggarwal, 190-204 with half-plane symmetry, 200-204 with quadrantal symmetry, 191-200 study of quantization effects, 133 DFP formula, 233 DFT, 343 Dihedral group, 182 Direct realization, 284 of recursive filters, 294-295

Index Discrete exponential, 21 Discrete Fourier transform, l-D, 348 2-D, 343-345 Discrete image model, 372-373 Discrete signals, 1, 6 sinusoid, 21 Dolph-Chebyshev window function, 142 Double bilinear transformation, 187 Dynamic range constraints, 329-330 Edge enhancement, 364-370 using Butterworth highpass filter, 367-370 using nonrecursive highpassfilters, 366-367 El-Agizi, N. G . , 123 Elementary signals (see 2-D signals) Elements for 2-D digital filters: adder, 14 horizontal unit shifter, 14 multiplier, 14 vertical unit shifter, 14 Elliptical symmetry, 91 Error function, 239 Expected square error (roundoff noise), 328 Factor coprime, 107 Fahmy, M. M., 123 Fan filters, 98-102 application, 384-386 Fast-Fourier transform: l-D, 348 . 2-D, 347-350 Fettweis, A., 300 m, 347 Filters: bandpass, 95 bandstop, 95 fan, 98-102 highpass, 95 lowpass, 94 practical, 102 Finite wordlength effects: limit cycles, 334-339 quantization error, 317

Index Finite-impulse response filters, 2 FIR filters, 2 Fletcher, R., 269 Fornasini and Marchesini model, 38 Fornasini, E., 38 Fourier transform (see 2-D Fourier transform) Frequency response, 89 nonrecursive filters, 135-137 Frequency-convolution theorem, 67 Frequency-domain analysis, 86-89 Frequency-shifting theorem, 67 Fundamental theorem, 3 Generalized bandpass filter, 195 bandstop filter, 194 Generalized-immittance converter, 303 (see also Wave digital filters) Gibbs’ oscillations, 138 Givone and Roesser model: for l-input, l-output filter, 38-45 for m-input, p-output filter, 46 for p-input, p-output filter, 46 Givone, D. D., 38 Globally type-preserving transformation, 189 Goodman design method, 212-213 Goodman, D. M., 109, 212 Gradient vector: defined, 232 for least pth method, 235 for minimax method (recursive filters), 240, 241 Ground roll, 380 Group, 182 Group delay, 90, 134 Hamming window function, 140 Hardware implementation, 343 systolic, 350 Hereditary property, 233 Hessian matrix, 232 Highpass filters, 95 Hirano and Aggarwal design method, 190-204

391 Hirano, K.,190 Horizontal unit shifter, 14 Huang, T.S., 110, 127, 134, 144 Hunt, B. R., 376 Hwang, S. Y . , 325 Idealized filters: bandpass, 95 bandstop, 95 fan, 98-102 highpass, 95 lowpass, 94 IIR filters, 2 Ill-posedness, 373 Image enhancement, 360-370 phase distortion, 361 Image model, 371-373 Image restoration: applications, 370 block-circulant matrix, 375 circulant matrix, 373 constrained least-squares estimation. 374-379 continuous image model, 371-373 discrete image model, 372-373 ill-posedness, 373 Lagrange multiplier, 375 Laplacian, 374 point-spread function, 371 Implementation: as a process, 2, 133 hardware, 343 noncausal filters, 183-185 nonrecursive filters, 346-347 software, 342 systolic structures, 350-357 Impulse response, 23 Impulse sampler, 69 Indirect realization, 284 Induction method, 21-25 Infinite-impulse response filters, 2 Initial-value theorem, 59 Input quantization error, 318 Input signal quantization, 322-323 Interrelations between 2-D, sampled, and discrete signals, 74

392 Inverse Fourier transform, 66 Inverse z transform: 59 Jury, E. I., 111, 219 Jury-Marden stability criterion, 3, 112 table, 112 Justice, J. H., 186 Kaiser window function, 141 Karivaratharajan, P.,213 King, R. A., 189 Lagrange multiplier, 375 Laplacian, 374 Least pth method, 231-236 minimax algorithm, 239-242 Lee, E. B., 116, 122 Limit cycles: degenerate periodic sequence, 334 Lyapunov approach, 337-339 overflow, 337-339 quantization, 334-337 separable, 335 Linear phase filters, 134-135 Linear transformations, 178-185 Linearity: additivity, 10 defined, 10 Fourier transform, 67 homogeneity, 10 phase-response, 4 superposition, 11 z transform, 58 Local minimum, 240 Locally type-preserving transformation, 190 Lossless network, 245 Lowpass filters, 94 Lu, W.-S., 122 LUD realization, nonrecursive filters, 289-292 recursive filters, 295-300 Lyapunov approach to limit cycles, 337-339 stability theory, 121-124 theorem, 121

Index Main lobe, 139 Marchesini, G . , 38 McClellan transformation, 165, 285 McClellan, J. H., 163, 285 Mecklenbrauker, W. F. G . , 166 Mendonsa, G . V.,215 Mersereau, R. M., 151, 166 Minimax method for nonrecursive filters, 267-269 for recursive filters, 236-244 Minimax multipliers, 240 Minimization of output noise, 331-332 Minimum roundoff noise in state-space structure, 325-334 Mitra, S. K.,219 Mullis, C. T.,325 Multidimensional filters, 34-35 Multiplier, 14 Murray, J., 111 Noble, B., 113 Noise model, 326-327 Noise removal, 361-364 using Butterworth lowpass filter, 362-364 using nonrecursive lowpassfilters, 362 Nonessential singularitiesof the second kind, 80 elimination, 213-215 in rotated filters, 209 Nonrecursive filters: amplitude response, 134 causal, 279 characterization, 12-13 design based on the McClellan transformation, 163-173 by using Fourier series, 138-142 design of bandpass filters, 150 prescribed specifications, 156-159 bandstop filters, 150 prescribed specifications, 159-161 bandstop filters, 150 circularly symmetricfilters, 147-150

393

Index [Nonrecursive filters: Continued] prescribed specifications, 150- 154 fan filters, 162-163 highpass filters, 149-150 prescribed specifications, 154-156 lowpass filters, 147-149 prescribed specifications, 151-154 l-D filters, 138-142 2-D filters using Fourier series, 142-143 frequency response, 135-137 group delay, 134 linear phase, 134 nonsymmetric half-plane, 13 phase response, 134 properties, 133-137 symmetry properties, 267-268 systolic implementationof l-D filters, 350-353 2-D filters, 354-355 zero phase (l-D), 278 d2-phase (l-D), 278 Nonsymmetric half-planefilters, 249-251 Normalized frequencies,251 Norms: Frobenius, 252 L,, 252 Nyquist frequencies,90 O’Connor, B. T. , 127 Objective function: defined, 231 for least pth method, 239 for minimax method (nonrecursive filters), 268 for minimax method (recursive filters), 248, 251 Observability, 50-51 Octagonal symmetry, 267 l-D discrete-Fourier transform, 348 l-D fast-Fourier transform, 348 l-D window functions, 139-142

Optimization:

Broyden-Fletcher-Goldfarb-Shanno formula, 233 Davidon-Fletcher-Powell formula, 233 gradient vector, 232 (see also Gradient vector) hereditary property, 233 Hessian matrix, 232 initial point, 233 least pth method, 231-236 local minimum, 240 minimax method for nonrecursive filters, 267-269 for recursive filters, 236-244 minimax multipliers, 240 objective function, 231 (see also Objective function) quasi-Newton algorithms, 232-236 termination tolerance, 233 unconstrained, 231, 238 using conjugate direction method, 269 Output noise: minimization, 331-332 power, 327-329 unit noise, 333 Overflow, 329 limit cycles, 337-339 Parseval’s formula for continuous signals, 67 for discrete signals,63 for 2-D DFT, 345 Pass filters: quadrant, 200 two-quadrant, 202 Passband, 94 Passband boundary, 94 Pendergrass, N. A., 219 Periodicity of 2-D signals, 7 Phase distortion, 4, 361 Phase response, 89 nonrecursive filters, 134 Phase spectrum, 66 Phase-response linearity, 4

394 Point-spread function, 371 Poisson’s summation formula, 67 Polynomial map, 307 Powell, M.J. D., 269 Practical filters, 102 Prescribed specifications in nonrecursive filters, 150-161 in recursive filters, 219-226 Processing element, 351 Processing time, 351,353 Product quantization, 321-322 error, 318

Index

[Realization: Continued] direct, 29, 284 GIC, 303-313 indirect, 284 LUD (nonrecursive filters), 289-292 LUD (recursive filters), 295-300 parallel, 30, 197 state-space, 52 SVD-LUD, 292-294 the realization process, 2, 133 wave, 301-313 Rectangular symmetry,91 Rectangular window function, 140 Recursive filters: Quadrant pass filters, 200 canonic realization,295 Quadrantal symmetry, 92,268 characterization, 13-14 Quantization: circularly symmetric, 204-218 coefficient, 3, 319-320 design of circularly symmetric effects, 133 bandpass, 218 error: circularly symmetric bandstop, coefficient, 318 218 computation, 318-325 circularly symmetric highpass, input signal, 318 215-218 mean, 318-323 circularly symmetric lowpass product, 318 (prescribed specifications), variance, 318-323 223-226 input signal, 322 lowpass filters with rectangular limit cycles, 334-337 passbands (prescribed product, 321-322 specifications), 222-223 signal, 3 design using the method of Costa 2-D signals, 7 and Venetsanopoulos, Quarter-plane filters, 244-249 204-212 Quasi-Newton algorithms, 232-236 the method of Goodman, Quatieri, Jr., T. F., 166 212-213 the method of Hirano and Ramachandran, V., 213 Aggarwal, 190-204 Ramamoorthy, P. A., 245 direct realization, 294-295 Real-convolution theorem, 59 implementation of noncausal filters, Realization: 183-185 analog reference network,305-308 LUD realization, 295-300 based on McClellan transformation, systolic implementationof l-D 285-288 filters, 353-354 canonic, 295 2-D filters, 355-357 cascade, 30, 196 with half-plane symmetry, 200-204 Chebyshev, 288 with quadrantal symmetry, 191-200 circularly symmetric highpass filters, zero-phase, 211-212 215-218

Index Reddy, H. C.,213 Region of support: for 2-D signals, 7 nonsymmetric half-plane, 13 Remez algorithm, 279 Representation of 2-D digital filters: by difference equations, 12-14 by flow graphs, 14-16 by networks, 14-16 Residue theorem, 62 Resultant matrix, 116 Ripple ratio, 139-140 Roberts, R., 325 Roesser, R. P.,38 Rotated filters, 187 stability, 206-210 zero-phase, 211-212 Rotation angle, 186, 205 Roundoff noise: dynamic range constraints, 329-330 expected square error, 328 model, 326-327 output-noise power, 327-329 second-order modes, 330-331 Row-column decomposition(FFT) , 347-348 Rudin, W., 111 Saeks, R., 111 Sampled amplitude response, 251 Sampling process, 68-70 Sampling theorem, 70-74 Schur-Cohn Hermitian matrix, 113 matrix, 112 stability criterion, 113 Second-order modes, 330-331 Sedlmeyer, A., 300 Seismic image, 380 ground roll, 380 Seismic signal processing, 379-386 Selectivity, 266 Separability: 2-D signals, 9 z transform, 60 Shanks’ theorem, 108 Shanks, J. L., 82, 186

395 Shift invariance, 11 Side lobes, 139 Signal quantization, 3 Signals (see 2-D signals) Siljak, D. D., 114 Silverman, H., 110 Similarity transformation, 326 Singular values, 252 Singular-value decomposition: correction section, 255 defined, 10,252 design of circularly symmetric filters, 256-257 nonrecursive filters, 271-282 recursive filters, 251-262 main section, 255 sampled amplitude response, 251 singular values, 10, 252 symmetry properties, 273-274 Sinusoidal response, 86-89 Software implementation, 342 Space-convolution theorem, 67 Space-domain analysis: defined, 2, 16 impulse response, 23 state-space, 48 using the z transform, 84-86 Space-scaling theorem, 67 Space-shifting theorem, 67 Spatial frequency, 2 Speake, T. C., 151 Stability: analysis in frequency domain, 107-118 in state-space domain, 118-121 bounded-input bounded-output, 27, 108 defined, 27 in 2-D filters, 3 Jury-Marden criterion, 3,112 table, 112 low-order filters, 124-127 Lyapunov, 121-124 necessary and sufficient conditions, 27-29, 108 of rotated filters, 206-210

396 [Stability: Continued] properties, 107-111 Schur-Cohn matrix, 112 Shanks’ theorem, 82, 108 theorems, 107-111 z-domain conditions, 82 State variables: defined, 40 horizontal, 42 vertical, 42 State-space methods: boundary value, 49 characterization, 45 controllability, 50-51 minimum roundoff noise, 325-334 models, 38 observability, 50-51 realization, 38 space-domain analysis, 48 stability, 118-121 state variables, 40 Stopband, 94 Strictly Hurwitz polynomial,245 Strintzis, M. G . , 110 Structures: cascade, 30, 196 direct, 29 parallel, 30, 197 separable, 32 SW-LUD realization, 292-294 Swamy, M. N. S., 213 Symmetry: amplitude response, 90 circular, 91 elliptical, 91 Fourier transform, 67 octagonal, 267 properties of nonrecursive filters, 267-268 properties of singular-value decomposition, 273-274 quadrantal, 92, 268 rectangular, 91 Systolic implementation: l-D nonrecursive filters, 350-353 l-D recursive filters, 353-354

Index [Systolic implementation: Continued] processing element, 351 processing time, 351 2-D filters, 354-357 Termination tolerance, 233 Theorems: circular convolution of 2-D DFT, 345 complex convolution, 63 complex differentiation, 59 complex scale change, 58 convergence of Fourier transform, 66 duality of 2-D D m , 345 frequency convolution, 67 frequency shifting, 67 Fundamental, 3 initial-value, 59 linearity of Fourier transform, 67 linearity of 2-D DFT, 345 linearity of z transform, 58 Lyapunov, 121 Parseval’s formula for continuous signals, 67 for discrete signals, 63 for 2-D DFT, 345 periodicity of 2-D DFT, 345 Poisson’s summation formula, 67 real convolution, 59 residue, 62 Shanks’ theorem, 82 space convolution, 67 space scaling, 67 space shifting, 67 stability, 107-111 symmetry, 67 translation, 58 Transfer function: all-pole, 248 defined, 77 derivation from difference equation, 78 from digital-filter network, 80 from state-space characterization, 81

Index [Transfer function: Continued] for nonsymmetric half-plane nonrecursive filters, 79 recursive filters, 79 separable denominator, 236,252 Transformations: allpass, 212 bilinear, 177- 178 Constantinides, 218-219 double bilinear transformation, 187 for analog filters, 185-187 globally type-preserving, 189 linear, 178-185 locally type-preserving, 190 McClellan, 165, 285 similarity, 326 Transition band, 258 Translation theorem, 58 Treitel, S., 186 2-D digital filters: applications, 1 causality, 11 characterization, 12-14 design (see Design) elements, 14 finite-impulse response, 2 infinite-impulse response, 2 linearity, 10 multiple-input multiple-output, 32-34 nonrecursive, 2, 12-13 nonsymmetric half-plane, 13 nonsymmetric half-plane, 249-251 order, 13 overflow, 329 realization, 29-32 recursive, 2, 13-14 nonsymmetric half-plane, 14 representation, 14-16 shift invariance, 11 space-domain analysis, 16, 21-25 stability, 27-29 system representation, 1, 10 systolic implementation, 354-357 wave, 301-313

397

2-D discrete Fourier transform: circular convolution, 345 computation, 347-350 defined, 343 duality, 345 implementation of nonrecursive filters, 346-347 linearity, 345 Parseval’s formula, 345 periodicity, 345 2-D discrete transfer function (see Transfer function) 2-D fast-Fourier transform, 347-350 butterfly, 349 row-column decompositionmethod, 347-348 vector-radix method, 348-350 2-D Fourier transform: amplitude spectrum, 66 convergence, 66 defined, 66 frequency convolution, 67 frequency shifting, 67 inverse, 66 linearity, 67 notation, 66 Parseval’s formula, 67 phase spectrum, 66 Poisson’s summation formula, 67 space convolution, 67 space scaling, 67 space shifting, 67 symmetry, 67 2-D signals: continuous impulse, 69 continuous, 1, 5 decomposition, 9 singular-value, 10 discrete, 1, 6 elementary signals, 16-21 discrete exponential, 21 discrete sinusoid, 21 unit impulse, 17 unit pulse, 21 unit step, 17 unit-line impulse, 18

398 [2-D signals: Continued] unit-line pulse, 2 1 finite extent, 7 interrelations, 74 mixed, 6 notation, 6 periodicity, 7 quantization, 7 region of support, 7 first-quadrant, 7 right half-plane, 7 sampled, 68 separability, 9 2-D window functions, 143-147 2-D z transform: complex convolution, 63 complex differentiation, 59 complex scale change, 58 defined, 56 initial-value theorem, 59 inverse z transform, 59 linearity, 58 Parseval's formula, 63 real convolution, 59 region of convergence, 56 separability, 60 theorems, 58-59 translation, 68 Two-quadrant pass filter, 202 Unit bicircle, 70, 88 impulse, 17 noise, 333 pulse, 21 step, 17 Unit-line impulse, 18 pulse, 21 Vector-radix FFT method, 348-350 Venetsanopoulos, A. N.,204

Index Vertical unit shifter, 14 von Hann window function, 140

Walker, R. J., 116 Wave digital filters: analog reference network, 305-308 admittance-conversion function, 303 based on equally terminated LC filters, 301-303 GIC, 303-313 current-conversion GIC, 303 GIC, 303 adaptor, 305 incident wave, 304 polynomial map, 307 port conductance, 304 reflected wave, 304 Window functions: Blackman, 141 Chebyshev polynomial, 142 design of nonrecursive filters using, 139-140 Dolph-Chebyshev, 142 Hamming, 140 Kaiser, 141 main lobe, 139 l-D, 139-142 rectangular, 140 ripple ratio, 139- 140 side lobes, 139 2-D, 143-147 von Hann, 140

Zero coprime, 107 Zero-phase rotated filters, 211-212 z transform (see 2-D z transform)