FAST TRANSFORMS A l g o r i t h m s , A n a l y s e s , Applications
Douglas
F. Elliott
Electronics Research Center Rockwell International A n a h e i m , California
K. Ramamohan
Rao
Department of Electrical Engineering The University of Texas at A r l i n g t o n A r l i n g t o n , Texas
A C A D E M I C PRESS, INC. (Harcourt Brace Jovanovich, Publishers)
Orlando
San Diego
Toronto
Montreal
San Francisco Sydney
Tokyo
New York Sao Paulo
London
COPYRIGHT © 1 9 8 2 , BY ACADEMIC PRESS, INC. ALL RIGHTS R E S E R V E D . N O P A R T O F T H I S P U B L I C A T I O N M A Y B E R E P R O D U C E D OR T R A N S M I T T E D I N A N Y F O R M OR B Y A N Y M E A N S , E L E C T R O N I C OR M E C H A N I C A L , I N C L U D I N G P H O T O C O P Y , RECORDING, OR A N Y I N F O R M A T I O N STORAGE A N D RETRIEVAL S Y S T E M , W I T H O U T PERMISSION I N WRITING F R O M THE PUBLISHER.
A C A D E M I C
PRESS,
Orlando, Florida 32887
United
Kingdom
Edition
INC.
published
by
A C A D E M I C PRESS, INC. ( L O N D O N ) 24/28 Oval R o a d , L o n d o n N W 1 7DX
LTD.
Library of Congress Cataloging in Publication Data Elliott, Douglas F. Fast transforms: algorithms, analyses, applications. Includes bibliographical references and index. 1. Fourier transformations—Data processing. 2. Algorithms. I. Rao, K. Ramamohan (Kamisetty Ramamohan) II. Title. III. Series QA403.5.E4 515.7'23 79-8852 ISBN 0-12-237080-5 AACR2
AMS (MOS) 1 9 8 0 Subject Classifications: 6 8 C 2 5 , 4 2 C 2 0 , 6 8 C 0 5 , 4 2 C 1 0
P R I N T E D I N T H E U N I T E D S T A T E S O F AMERICA 83 84 85
9 8 7 6 5 4 3 2
To Caroiyn and Karuna
CONTENTS
Preface Acknowledgments List of Acronyms Notation
Chapter
1
1.0 1.1 1.2 1.3 1.4
Chapter 2 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
xiii xv xvii xix
Introduction Transform D o m a i n Representations F a s t Transform Algorithms F a s t Transform Analyses F a s t Transform Applications Organization of the B o o k
1 2 3 4 4
Fourier Series and the Fourier Transform Introduction Fourier Series with Real Coefficients F o u r i e r Series with Complex Coefficients E x i s t e n c e of F o u r i e r Series T h e F o u r i e r Transform S o m e F o u r i e r Transforms and Transform Pairs Applications of Convolution Table of F o u r i e r Transform Properties Summary Problems
6 6 8 9 10 12 18 23 25 25 vii
vlii
CONTENTS
Chapter 3 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12
Chapter 4 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9
Chapter 5 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8
Discrete Fourier Transforms Introduction D F T Derivation Periodic P r o p e r t y of the D F T Folding P r o p e r t y for Discrete Time Systems with Real Inputs Aliased Signals Generating kn Tables for the D F T D F T Matrix Representation D F T Inversion—the IDFT T h e D F T and I D F T — U n i t a r y Matrices Factorization of W Shorthand Notation Table of D F T Properties Summary Problems
,
E
33 34 36 37 38 39 41 43 44 46 47 49 52 53
Fast Fourier Transform Algorithms Introduction Power-of-2 F F T Algorithms Matrix Representation of a Power-of-2 F F T Bit R e v e r s a l to Obtain F r e q u e n c y O r d e r e d Outputs Arithmetic Operations for a Power-of-2 F F T Digit R e v e r s a l for Mixed Radix Transforms M o r e F F T s b y M e a n s of Matrix T r a n s p o s e M o r e F F T s b y M e a n s of Matrix I n v e r s i o n — t h e I F F T Still M o r e F F T s by M e a n s of F a c t o r e d Identity Matrix Summary Problems
58 59 63 70 71 72 81 84 88 90 90
FFT Algorithms That Reduce Multiplications Introduction Results from N u m b e r T h e o r y Properties of Polynomials Convolution Evaluation Circular Convolution Evaluation of Circular Convolution through the C R T C o m p u t a t i o n of Small N D F T Algorithms Matrix Representation of Small N D F T s K r o n e c k e r Product E x p a n s i o n s
99 100 108 115 119 121 122 131 132
fx
CONTENTS
5.9 5.10 5.11 5.12 5.13 5.14 5.15
Chapter 6 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9
Chapter
7
7.3 7.4 7.5 7.6 7.7 7.8 7.9
Chapter 8
136 138 139 145 154 162 168 169
D F T Filter Shapes and Shaping Introduction D F T Filter R e s p o n s e I m p a c t of the D F T Filter R e s p o n s e Changing the D F T Filter Shape Triangular Weighting H a n n i n g Weighting and H a n n i n g W i n d o w Proportional Filters S u m m a r y of Weightings and W i n d o w s Shaped Filter Performance Summary Problems -
7.0 7.1 7.2
8.0 8.1 8.2
T h e G o o d F F T Algorithm T h e Winograd Fourier Transform Algorithm Multidimensional Processing Multidimensional Convolution by Polynomial Transforms Still M o r e F F T s by M e a n s of Polynomial Transforms C o m p a r i s o n of Algorithms Summary Problems
178 179 188 191 196 202 205 212 232 241 242
Spectral Analysis Using the FFT Introduction Analog and Digital S y s t e m s for Spectral Analysis Complex D e m o d u l a t i o n and M o r e Efficient U s e of the F F T Spectral Relationships Digital Filter Mechanizations Simplifications of F I R Filters D e m o d u l a t o r Mechanizations O c t a v e Spectral Analysis Dynamic Range Summary Problems
252 253 256 260 263 268 271 272 281 289 290
Walsh-Hadamard Transforms Introduction Rademacher Functions Properties of Walsh F u n c t i o n s
301 302 303
CONTENTS
X
8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10
Chapter
9
9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13
Chapter
10
10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12
Walsh or S e q u e n c y Ordered Transform ( W H T ) H a d a m a r d or N a t u r a l O r d e r e d Transform ( W H T ) Paley or Dyadic Ordered Transform ( W H T ) C a l - S a l O r d e r e d Transform ( W H T ) W H T Generation Using Bilinear F o r m s , Shift Invariant P o w e r Spectra Multidimensional W H T Summary Problems W
P
CS
h
310 313 317 318 321 322 327 329 329
The Generalized Transform Introduction Generalized Transform Definition E x p o n e n t Generation Basis F u n c t i o n F r e q u e n c y A v e r a g e Value of the Basis F u n c t i o n s Orthonormality of the Basis F u n c t i o n s Linearity Property of the Continuous Transform Inversion of the Continuous Transform Shifting T h e o r e m for the Continuous Transform Generalized Convolution Limiting Transform Discrete Transforms Circular Shift Invariant P o w e r S p e c t r a Summary Problems
334 335 338 340 341 343 344 344 345 347 347 348 353 353 353
Discrete Orthogonal Transforms Introduction Classification of Discrete Orthogonal Transforms M o r e Generalized Transforms Generalized P o w e r Spectra Generalized P h a s e or Position S p e c t r a Modified Generalized Discrete Transform ( M G T ) P o w e r Spectra T h e Optimal Transform: K a r h u n e n - L o e v e Discrete Cosine Transform Slant Transform H a a r Transform Rationalized H a a r Transform Rapid Transform r
362 364 365 370 373 374 378 382 386 393 399 403 405
CONTENTS •
10.13
Chapter
11
11.0 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13
•
Summary Problems
Xi
•
410 411
Number Theoretic Transforms Introduction N u m b e r Theoretic Transforms M o d u l o Arithmetic D F T Structure F e r m a t N u m b e r Transform M e r s e n n e N u m b e r Transform R a d e r Transform C o m p l e x F e r m a t N u m b e r Transform C o m p l e x M e r s e n n e N u m b e r Transform P s e u d o - F e r m a t N u m b e r Transform C o m p l e x P s e u d o - F e r m a t N u m b e r Transform P s e u d o - M e r s e n n e N u m b e r Transform Relative Evaluation of the N T T Summary Problems
417 417 418 420 422 425 426 427 429 430 432 434 436 440 442
Appendix Direct or K r o n e c k e r P r o d u c t of Matrices Bit Reversal Circular or Periodic Shift Dyadic Translation or Dyadic Shift modulo or m o d Gray C o d e Correlation Convolution Special Matrices Dyadic or Paley Ordering of a S e q u e n c e
444 445 446 446 447 447 448 450 451 453
References
454
Index
483
PREFACE
Fast transforms are playing an increasingly important role in applied engineering practices. N o t only do they provide spectral analysis in speech, sonar, radar, and vibration detection, but also they provide b a n d w i d t h reduction in video transmission and signal filtering. F a s t transforms are used directly to filter signals in the frequency domain and indirectly to design digital filters for time domain processing. T h e y are also u s e d for convolution evaluation and signal decomposition. P e r h a p s the r e a d e r can anticipate other applications, and as time p a s s e s the list of applications will doubtlessly grow. . At the p r e s e n t time to the a u t h o r s ' knowledge there is no single b o o k that discusses the m a n y fast transforms and their u s e s . T h e p u r p o s e of this b o o k is to provide a single source that covers fast transform algorithms, analyses, and applications. It is the result of collaboration b y an a u t h o r in the aerospace industry with a n o t h e r in the university c o m m u n i t y . T h e authors h o p e that the collaboration has resulted in a suitable mix of theoretical development and practical u s e s of fast transforms. This b o o k has g r o w n from notes used by the authors to instruct fast transform classes. O n e class w a s sponsored by the Training D e p a r t m e n t of Rockwell International, and another w a s sponsored b y the D e p a r t m e n t of Electrical Engineering of T h e University of T e x a s at Arlington. S o m e of the material w a s also u s e d in a short course sponsored b y the University of Southern California. T h e authors are indebted to their students for motivating the writing of this b o o k and for suggestions to i m p r o v e it. The development in this b o o k is at a level suitable for a d v a n c e d undergraduate or beginning graduate students and for practicing engineers and scientists. It is a s s u m e d that the r e a d e r has a knowledge of linear system theory and the applied m a t h e m a t i c s that is part of a standard u n d e r g r a d u a t e engineering curriculum. T h e emphasis in this b o o k is on material not directly covered in other b o o k s at the time it w a s written. T h u s r e a d e r s will find practical a p p r o a c h e s not c o v e r e d elsewhere for the design and d e v e l o p m e n t of spectral analysis s y s t e m s .
xiv
PREFACE
The long list of references at the end of the b o o k attests to the volume of literature on fast transforms and related digital signal processing. Since it is impractical to c o v e r all of the information available, the authors h a v e tried to list as m a n y relevant references as possible u n d e r some of the topics discussed only briefly. T h e authors h o p e this will serve as a guide to those seeking additional material. Digital c o m p u t e r p r o g r a m s for evaluation of the transforms are not listed, as these are readily available in the literature. Problems h a v e b e e n used to convey information b y m e a n s of the format: If A is t r u e , use B to show C. This format gives useful information both in the premise and in the conclusion. T h e format also gives an a p p r o a c h to the solution of the problem.
ACKNOWLEDGMENTS
It is a pleasure to acknowledge helpful discussions with our colleagues w h o contributed to our understanding of fast transforms. In particular, fruitful discussions w e r e held with T h o m a s A. Becker, William S. Burdic, TienLin Chang, R o b e r t J. D o y l e , Lloyd O. K r a u s e , David A . O r t o n , David L . H e n c h , Stanley A. W h i t e , and L e e S. Y o u n g of Rockwell International; Fredric J. Harris of San Diego State University and the N a v a l O c e a n Systems Center; I. Luis A y a l a of Vitro T e c , M o n t e r r e y , M e x i c o ; and Patrick Yip of M c M a s t e r University. It is also a pleasure to a c k n o w l e d g e support from T h o m a s A. B e c k e r , M a u r o J. Dentino, J. David Hirstein, T h o m a s H . M o o r e , Visvaldis A . Vitols, and Stanley A. White of Rockwell International and Floyd L . C a s h , Charles W. Jiles, J o h n W. R o u s e , Jr., and A n d r e w E . Salis of T h e University of T e x a s at Arlington. Portions of the manuscript w e r e reviewed by a n u m b e r of people w h o pointed out corrections or suggested clarifications. T h e s e p e o p l e include T h o m a s A. B e c k e r , William S. Burdic, Tien-Lin C h a n g , Paul J. Cuenin, David L . H e n c h , L l o y d O. K r a u s e , J a m e s B . L a r s o n , L e s t e r Mintzer, Thomas H . M o o r e , David A. Orton, Ralph E . Smith, Jeffrey P . S t r a u s s , and Stanley A. White of Rockwell International; H e n r y J. N u s s b a u m e r of the Ecole Polytechnique F e d e r a t e de L a u s a n n e ; R a m e s h C. Agarwal of the Indian Institute of Technology; M i n s o o Suk of the K o r e a A d v a n c e d Institute of Science; Patrick Yip of M c M a s t e r University; R i c h a r d W . H a m m i n g of the N a v a l P o s t g r a d u a t e School; G. Clifford Carter and Albert H . Nuttall of the N a v a l U n d e r w a t e r S y s t e m s Center; C. Sidney B u r r u s of Rice University; Fredric J. Harris of San Diego State University and the N a v a l O c e a n Systems Center; Samuel D . Stearns of Sandia L a b o r a t o r i e s ; Philip A. Hallenborg of N o r t h r u p Corporation; I. Luis Ayala of Vitro T e c ; George Szentirmai of C G I S , Palo Alto, California; and Roger Lighty of the Jet Propulsion L a b o r a t o r y . The authors wish to t h a n k several h a r d w o r k i n g people w h o contributed to the manuscript typing. T h e bulk of the manuscript w a s t y p e d b y M r s . XV
xvl
ACKNOWLEDGMENTS
Ruth E . Flanagan, M r s . V e r n a E . J o n e s , and M r s . Azalee T a t u m . T h e authors especially appreciate their patience and willingness to help far beyond the call of duty. T h e e n c o u r a g e m e n t and understanding of our families during the preparation of this b o o k is gratefully acknowledged. T h e time and effort spent on writing m u s t certainly h a v e b e e n reflected in neglect of our families, w h o m w e thank for their forbearance.
LIST
ADC AGC BCM BIFORE BPF BR BRO CBT CCP CFNT CHT CMNT CMPY CNTT CPFNT CPMNT CRT DAC DCT DDT DFT DIF DIT DM DST ENBR ENBW EPE FDCT FFT FGT FIR FNT
OF A C R O N Y M S
Analog-to-digital converter A u t o m a t i c gain control Block circulant matrix Binary Fourier representation Bandpass filter Bit reversal Bit-reversed order Complex B I F O R E transform Circular convolution property Complex F e r m a t n u m b e r transform Complex H a a r transform Complex Mersenne n u m b e r transform Complex multiplications Complex n u m b e r theoretic transform Complex p s e u d o - F e r m a t n u m b e r transform Complex pseudo-Mersenne n u m ber transform Chinese remainder theorem Digital-to-analog converter Discrete cosine transform Discrete D transform Discrete Fourier transform Decimation in frequency Decimation in time Dyadic matrix Discrete sine transform Effective noise b a n d w i d t h ratio Equivalent noise b a n d w i d t h Energy packing efficiency F a s t discrete cosine transform Fast Fourier transform F a s t generalized transform Finite impulse response F e r m a t n u m b e r transform
FOM FWT GCBC GCD GT (GT) r
HHT (HHT),. HT IDCT IDFT IF IFFT IFNT IGT (IGT)
r
IIR IMNT KLT LPF lsb lsd MBT MCBT MGT (MGT) MIR MNT MPY msb msd mse
r
Figure of merit Fast Walsh transform G r a y code to binary conversion Greatest c o m m o n divisor Generalized discrete transform rth-order generalized discrete transform H a d a m a r d - H a a r transform rth-order H a d a m a r d - H a a r transform H a a r transform Inverse discrete cosine transform Inverse discrete Fourier transform Intermediate frequency Inverse fast Fourier transform Inverse F e r m a t n u m b e r transform Inverse generalized transform rth-order inverse generalized transform Infinite impulse response Inverse Mersenne n u m b e r transform K a r h u n e n - L o e v e transform L o w pass filter Least significant bit Least significant digit Modified B I F O R E transform Modified complex B I F O R E transform Modified generalized transform rth-order modified generalized discrete transform Mixed radix integer representation Mersenne n u m b e r transform Multiplications M o s t significant bit M o s t significant digit M e a n - s q u a r e error xvii
LIST OF ACRONYMS
xviii MWHT (MWHT) NPSD NTT PSD RF RHT RHHT (RHHT) RMFFT RMS RNS RT
r
h
Modified W a l s h - H a d a m a r d transform H a d a m a r d ordered modified W a l s h - H a d a m a r d transform Noise power spectral density N u m b e r theoretic transform Power spectral density R a d i o frequency Rationalized H a a r transform Rationalized H a d a m a r d - H a a r transform rth-order rationalized H a d a m a r d H a a r transform Reduced multiplications fast Fourier transform R o o t m e a n square Residue n u m b e r system R a p i d transform
SHT (SHT), SIR SNR ST WFTA WHT (WHT)
CS
(WHT)
h
(WHT)
p
(WHT)
W
zps
S l a n t - H a a r transform rth-order s l a n t - H a a r transform Second integer representation Signal-to-noise ratio Slant transform. W i n o g r a d Fourier transform algorithm W a l s h - H a d a m a r d transform Cal-sal ordered W a l s h - H a d a m a r d transform H a d a m a r d ordered W a l s h H a d a m a r d transform Paley ordered W a l s h - H a d a m a r d transform Walsh ordered W a l s h - H a d a m a r d transform Zero crossings per second
NOTATION
Meaning
Symbol A,
B,...
A® A A'
B
T
1
Symbol
Matrices are designated by capital letters T h e K r o n e c k e r product of A and B (see Appendix) The transpose of matrix A T h e inverse of matrix A D C T matrix of size ( 2 x 2 ) Periodic D F T filter frequency response, which for P = 1 s is given by L
D(f)
L
[# (L)] S
Meaning W a l s h - H a d a m a r d matrix of size ( 2 x 2 ) . T h e subscript s can be w, h, p , or cs, denoting Walsh, H a d a m a r d , Paley or cal-sal ordering, respectively. H a a r matrix of size ( 2 x 2 ) r t h order ( H H T ) matrix of size ( 2 x 2 ) Opposite diagonal matrix, e.g., L
[Ha(L)] [Hh (L)] r
L
I
m
D'(f)
exp[-77i/(l - 1/AO] [sin(7r/)]/(7c/) D'(f)
DFT[jc(w)]
\P {L)-\ E r
Nonperiodic frequency response of D F T with weighted input (windowed output) T h e discrete Fourier transform of the sequence MO), x ( l ) , ...,x(N1)} 7'th matrix factor of [G>(L)] Expectation operator 7'th matrix factor of \_M (L)~\ rth F e r m a t number, F = ( 2 + 1), t = 0, 1 , 2 , . . . ( G T ) matrix of size ( 2 x 2 ) M W H T matrix of size (2 x 2 )
7^
m
7^
w
t
2t
L
r
iH {Ly] mh
L
L
L
C o l u m n s of I are shifted circularly t o the right by m places C o l u m n s of I are shifted circularly to the left by m places C o l u m n s of I are shifted dyadically by / places Identity matrix of size (R x R) T h e imaginary part of the quantity in the square brackets T h e inverse discrete Fourier transform of the sequence {X(0),X(l),...,X(N-l)} K L T matrix of size ( 2 x 2 ) Integer such t h a t N = cc Mersenne number, M = 2 — 1, where P i s a prime n u m b e r N
1% I Im[ ] R
IDFT[X(/c)]
r
Ft
L
~ 0 0 0 f 0 0 1 0 0 1 0 0 1 0 0 0
N/JNsm(nf/N)
Periodic frequency response of D F T with weighted input (windowed output) Nonperiodic D F T filter frequency response which for P = 1 s is given by
L
r
L
sin(7t/) D(f)
L
[K(L)] L M P
N
N
L
L
L
P
P
xix
NOTATION
XX
(MGT),. matrix of size (2 x 2 ) Transform dimension Multiplicative inverse of the integer TV such that TV x TV" = 1 (modulo M) 1. Period of periodic time function in seconds 2. I n Chapter 11, prime number Diagonal matrix whose diagonal elements are negative integer powers of 2 ( W H T ) circular shiftinvariant power spectral point /th power spectral point of (GT), rath sequency power spectrum 1. R a t i o of the filter center frequency a n d the filter b a n d w i d t h (Chapter 6) 2. Least significant bit value (Chapter 7) The real part of the quantity in the square brackets R a t e distortion R H T matrix of size ( 2 x 2 ) ST matrix of size ( 2 x 2 ) L
TV N'
1
Symbol
Meaning
Symbol
X(f)
o r XJif)
L
\X(f)\
2
1
P
X(k)
\X(k)\
2
X
c
h
^(/)
Q
Re[ ] *(D) [Rh(L)]
L
L
L
[S (t) evaluated at t = nT M a g n i t u d e of (•) Integerize by truncation (or rounding) Smallest integer ^ (•), e.g., [3.51 = 4, r— 2.51 = - 2 Largest integer ^ (•), e.g., L3.5J = 3 , L - 2 . 5 J = - 3 Signed digit addition performed digit by digit modulo a
BAD
r
a
2
<M") KOI
IKOII
r(-)i L(-)J
X
2
k
CHAPTER
1
INTRODUCTION
1.0
Transform Domain Representations
Many signals can be expressed as a series that is a linear combination of orthogonal basis functions. The basis functions are precisely defined (mathematically) waveforms, such as sinusoids. The constant coefficients in the series expansion are computed using integral equations. Let the basis functions be specified in terms of an independent variable t and be represented as cj) (t) for k = . . . , — 1, 0, 1, 2 , . . . . Let x(t) be the signal and X(k) be the kth coefficient. Then the signal x(t) can be decomposed in terms of the basis functions (j> {t) as k
k
X
x{t)=
(1.1)
XiftMt)
If (1.1) describes x(f) for all values of t, it also describes x(t) for specific values of t. Suppose these values are nT where T is fixed and « = . . . , — 1, 0, 1, 2 , . . . . Define x(n) and cf) (n) as x(t) and (f) (t), respectively, evaluated at t = nT. Then (1.1) becomes k
k
(1.2) N o w suppose that only N of the coefficients in (1.1) are nonzero, and let those nonzero coefficients be X(0), X(l), X{2),X(N1). Then (1.2) reduces to
x(n) = ^X{k)Un)
(1-3)
N
Let
o(N-l)
h(N-l)
4> -i(0) N
(1.4)
•••
0 _ (iV-l)J N
t
1
1
2
INTRODUCTION
and let X be the vector defined by X = [X(0), X ( l ) X(2), , . . , X(7V - 1 ) ]
(1.5)
T
3
where the superscript T denotes the transpose. Then (1.3) can be written as a matrix-vector equation that specifies N variables x(0), x ( l ) , . . , , x(N - 1): x = X
(1.6)
x = [x(0), x(l), x(2), ...,x(N-
1)]
(1.7)
T
The TV coefficients in (1.5) scale the values of
)-£
where e is an arbitrarily small interval. The Fourier transform ofthe periodic function is thus an infinite series of delta functions spaced 1 /P H z apart whose strengths are the Fourier series coefficients. Transforming (2.14) yields £
X(k)e
i2nk,lf
=
X
(2.57)
X(k)d(f-k/P)
we again see that the Fourier transform of the Since X(k) = a — jsign(k)b , Fourier series gives spectral lines at / = 0, + l/P, ± 2/P,..± k/P,.... Figure 2.1 represents the Fourier transform coefficients if the vectors representing X(k) are considered to be delta functions whose strengths are a /2 and b /2. k
k
k
TRANSFORM OF A SINGLE SIDEBAND MODULATED FUNCTION
Single
k
sideband
modulation of a time signal is used extensively in spectral analysis and communications. It accomplishes a frequency shift which preserves a signal's
2
16
FOURIER SERIES AND THE FOURIER TRANSFORM
spectrum without duplicating the spectrum. This makes single sideband modulation more efficient than double sideband modulation which duplicates the spectrum. Single sideband modulation is accomplished by multiplying a signal x(t) by the modulation factor exp( ±7*271/00- The Fourier transform of a modulated function is
^[x(t)e- ]
x(t)e- e- ^dt
=
j2nfot
j2nfot
j2n
+ f )t] dt = X(f + f )
x(t)exp[-j2n(f
0
(2.58)
0
The inverse transform of X(f + f ) is found by a change of variables. Iff is fixed and z =f + / , then dz = df a n d 0
0
0
X(f + fo)e * df j2
=
ft
X(z)e ej2nzt
dz
j2nfot
(2.59)
The factor exp(— j2nf t) does not vary with z and m a y be factored out of the integral, giving the Fourier transform pair 0
x
(
t
)
e
- J 2 n f o t ^
X
(
f
+
f
o
)
(
2
6
Q
)
Equation (2.60) specifies a frequency shifted spectrum and the single sideband modulation property is frequently called the frequency shift property. T h e shift for positive f is to the left. F o r example, consider the spectral value X(f ) occurring at f in the original spectrum. The value X(f ) is found at / = 0 in the shifted function X(f + / ). 0
0
0
0
0
TRANSFORM OF A TIME SHIFTED FUNCTION
Suppose
we
have
the
Fourier
transform pair x(t) X(f) and want the Fourier transform of the time shifted function x(t — T). W e find this transform by the change of variables z = t — T, which gives
#x(£
The factor e resulting in
-
T) =
j 2 n f x
x(t -
T)e~ dt j27lft
j2nfT
The factor e~
j2nfx
j2nfz
dz
(2.61)
does not vary with z a n d can be taken out of the integral, &x{t - T ) = e- X(f)
j2llfx
x(z)e- e~
(2.62)
is a phase shift that couples power between the real a n d
17
2.5 SOME FOURIER TRANSFORMS AND TRANSFORM PAIRS
imaginary parts of the Fourier transform spectrum. The result is the transform pair x{t ± T) ++e X(f)
(2.63)
±j2nfv
TRANSFORM OF A CONVOLUTION Convolution is one of the most useful properties of the Fourier transform in the analysis of systems incorporating an F F T . If x(t) and y(t) are two time functions, their time domain convolution is represented symbolically as x{t) * y(t) and is defined as x(t)*y(t)
x(u)y(t — u) du
=
(2.64)
The Fourier transform of (2.64) gives &[x(t)*y(t)]
J2nft
=
x(u)y(t — u) du dt
(2.65)
For most time functions the order of integration in (2.65) may be interchanged, giving x(u)
u)e- dtdu
y{t -
j2nft
(2.66)
Letting z = t — u gives
#-[x(0*X01 =
x(u)
y(z)e-
j2nf{z
+
dz
u)
du
(2-67)
N o t e that u does not vary in the integration in brackets and may be factored out to give &[x(t)*y(t)]
=
y{z)e- dz
x(u)
j2nfz
-j2nfu
(2.68)
The term in square brackets is the Fourier transform of y(t), which we denote Y(f) = &Y(t). W e now have
x(u)e- Y(f)du j2nfu
=
Y(f)
x(u)e- du= J2nfu
Y(f)X(f)
(2.69)
18
2
FOURIER SERIES AND THE FOURIER TRANSFORM
The inverse transform also exists for most applications. Furthermore, we can interchange x(t) and y(t) in (2.69) and get the same answer. This establishes the Fourier transform time domain convolution pair x(t)*y(t)~X(f)Y(f)
(2.70)
If X(f) and Y(f) are two frequency domain functions, then frequency domain convolution is represented by X(f) * Y(f) and is defined as
X(f)*Y(f)
X(u)Y(f-
=
u)du
(2.71)
The inverse Fourier transform of X(f) * Y(f) is similar to the Fourier transform of x(t)*y(t) outlined in (2.64)-(2.70). The transform pairs for time domain and frequency domain convolution are summarized as follows: x(t)*y(t)~X(f)Y(f)
(2.72)
x(t)y(t)^X(f)*Y(f) 2.6
(2.73)
Applications of Convolution
The Fourier transform of a time domain convolution and the inverse transform of a frequency domain convolution are particularly useful for system analysis. The transfer function property illustrates the application of time domain convolution, and analysis of a function with unknown period illustrates the application of frequency domain convolution. TRANSFER FUNCTIONS Determining transfer functions is an important application of the convolution property. Let a linear time invariant system have an input time function x(t) with transform X(f), as shown in Fig. 2.4. Let the system response to a delta function be the output y{t). Let the transform of y(t) be Y(J). System
Input
X(f)
Fig. 2.4
Output
0(f)
Y(f)
Relationships between system transfer functions.
Y(f) is called a transfer function when used to describe a time invariant linear system. We shall demonstrate that the output time function o(t) has Fourier transform 0(f) given by 0(f)
=
X(f)Y(f)
(2.74)
2.6
19
APPLICATIONS OF CONVOLUTION
as shown in Fig. 2 A Equation (2.74) gives the output time function o(t) as o(t) = 0
N e x t show t h a t if a ^ 0 oo
f
a
1
J a + (2nf) 2
-df=2
1
and """
2
a
limT a
a 2
0
f oo
for
(.0
for
J
+ (2TI/)
2
J
/'= 0 0
— oo
T h u s establish u(t)^S(f)
+ l/U2nf)
(P2.19-1)
Likewise establish $5(t)-l/(j2nt)~u(f) 20
Differentiation
21
Integration
dx{t)/dt^2njfX{f)
(P2.19-2)
and
-j2ntx(t)^dX(f)/df
Use time d o m a i n convolution a n d (P2.19-1) to show t h a t
+
*V)
J
j2nf
2
29
PROBLEMS
Use (2.73) and (P2.19-2) to show that x(t)
x(0)d(t)
—j2nt
f
2
22
Horizontal
Axis Sign Change
Prove x( — t)
