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First Published 1974 Pentech Press Limited: London
8 John Street, wei N 2HY
ISBN 07273 1801 2
Printed in England by The Whitefriars Press Ltd., London and Tonbridge
PREFACE
This book was w r i t t e n t o h e l p practISIng enginee r s, and s t u d e n t s o f r a d a r t h e o ry, t o u n d e r s t a n d h ow unce r t a i n t y fun c t i o n t ec hn i q u e s c an be used t o analy se the p e r fo rmanc e o f rada r system s . It is comm o n fo r mode r n systems t o employ sophist ic a t ed m o dul a t i o n, a n d t h e i r chara c t e ristics are n o t always apparen t from a n i n t u i t i ve t r e a t me n t . Engi ne e r s must h ave c o n fidence i n m a t h em a t ic al t e c h n iques (and kn owledge o f any lim i t a t ions) if they are to apply t h em t o new p rob l e m s . It is the a u t ho r ' s opinion that t h i s c o n fidence is best i n s p i re d b y s t ressing b a s i c principles and by giving t h e proofs o f all mathem a t i c al resu l t s, the reby establ ishing t h e c o n d i t i o n s nece ssa ry for t h e i r vali d i t y . This philosophy has b e e n adopt e d i n w r i t i ng t h e book. To facilit a t e a fi rst reading, t h e c o n c l u sions a n d discussion usually precede the de t ailed m a t h emat i c s; t h i s a l so m akes it easy to u se the book for reference purpose s . The book h as b e e n d i v i d e d i n t o t wo part s . Chapt e rs 1 t o 5 cover t h e r a d a r t h e o r y and i nclude nume rous r e fe r e nces t o Ch aptErs 6 and 7 which p rovide a firm fo u n d a t i o n of t h e unde rlying t ra nsfo r m theory . It is sugge s t e d t hat the reade r may l i k e to c o n c e n t r a t e i n i t i ally on Cha p t e r 1, fol lowe d by S e c t ions 4.1, 4.2, 4.6 and Chap t e r 5. This -procedure will acq u a in t h i m w i t h t he significance of the u n c e rt a i n t y fun c t io n a n d h o w i t c an be u s e d t o divi d e r a d a r receivers i n t o t w o cl asses-m a t c h ed fil t e r receive rs and Four i e r t r a nsform receivers . The th e ory is c onsolidat e d in Chapt e r 5 b y ap ply ing the re sult s t o two practical m o d u l ati o n wave forms; the reader should then be in a po s i tion to understand the more advanced resu l t s given i n the established
literature . The second part of the b oo k provides the m athem a ti cal b a c k gro u n d to Chapt ers 1 to 5 . Chapter 6 de als with t h e rel a t i o n s hip be tween Lapl ace and Fourier transforms . The discrete Fourier transform (OFT) is also discussed and it is shown how two OFT theorems lead to the fast Fourier transform ( F FT) process. Chapter 7 covers the Hilbert transform and complex analytic signals. The author has always found 'engineering' treatments of this lat ter sub ject t o be particularly unconvincing-it is hoped that the reader will find the present treatment t o b e much more satisfactory. The author has fou nd Part 2 to be a useful reference source when working in other fields of signal processing. The summary of the main notation in Appendix 5 is also useful in this respect.
Preface The fi rst d raft of this book was written as an inte rnal report for the Wells laboratories of EMI Elect ronics L t d . The mate rial has also been used as the b asi s of a series of aft e r-hours lectures t o the au thor's coll eagues. The au thor would like to th ank EMI Electronics L t d. for permission t o publ ish and a l so his coll eagues for m a ny helpful di scussion s. G. J. A. Bi rd
CONTENTS
Preface
PART l. RA D A R THEORY
I.
THE RADAR UNCERTAINTY FUNCTION
1.1 1.2 1.3
1.4
1.5
2.
T HE
Limitations
and
characteristics
of
the
uncertainty
function The Lm.s. error criterion
3
5
The area of uncertainty of a received signal Resolution and precision The relationship between the
MATHEMATICAL
T,
wand
x,
TREATMENT
6
y planes
OF
7
THE
UNCERTAINTY FUNCTION
2.1 2.2
The mathematical properties of
2.3
The effect of repetition upon the uncertainty function
13
2.4
The effect of linear FM upon the uncertainty function
19
Proofs of the properties of
20
2.5 3.
4.
Derivation of the uncertainty function
X(T,
w)
X(T, w)
9 II
WORKED UNCERTAINTY FUNCTION EXAMPLES 3.1
The rectangular pulse with constant carrier frequency
30
3.2
The rectangular pulse with linear FM
34
3.3
The rectangular pulse with triangular FM
37
SIGNAL PROCESSING METHODS 4.1
The matched filter receiver
41
4.2
The Fourier transform receiver
46
4.3
The effect of finite processing time
51
4.4
The matched filter concept
59
The
62
4.5
mathematical treatment of the matched filter
receiver 4.6
Practical Fourier transform calculators
64
4.7
The mathematical treatment of the Fourier transform
65
receiver
5.
SOME EXAMPLES OF SIGNAL PROCESSING METHODS 5 .1
The rectangular pulse with constant carrier frequency
68
5 .2
The rectangular pulse with sawtooth FM
76
Contents PA R T 2. M ATH E M A TIC A L BA CKGR O U N D
6.
LAPLACE AND FOURIER TRANSFORMS 6.1
The 2-sided Laplace transform
6.2
The significance of the strip of convergence
6.3
The I-sided Laplace transform
6.4
The Fourier transform
6.5
The physical interpretation of Laplace and Fourier transforms
6.6
Fourier transform symmetries
6.7
The Fourier transform of a conjugate function
90
6.8
Limiting cases of the Fourier transform
6.9
The delta function
92 92
94
6.10 Fourier transform notation 6.II Using the p-multiplied Laplace transform notation 6.12 The discrete Fourier transform (OFT)
6.13 Evaluation of the Fourier transform by means of the DFT 6.14 T he fast Fourier transform (FFT) process 6.15 Proofs of the properties of the OFT and FFT
7.
85 86 87 88 89
99 99
100 103 106 108
HILBERT TRANSFORMS AND COMPLEX ANALYTIC
SIGNALS 7.1
Summary of the main results of Chapter 7
114
7.2
The Hilbert transform
116
7.3
Envelope and phase functions
7.4
The
complex
122
7.5
The results of multiplying real or complex analytic
125
7.6
Filtering either real or complex analytic signals
129
7.7
The com pression of a Doppler shifted pulse
131
exponential
approximation
121 to
the
analytic signal signals
Appendix I Complex conjugate terminology
137
Appendix:2
138
Parseval's theorem
Appendix 3
Fresnel integrals and their relationship to linear FM
Appendix 4 A short table of Hilbert transforms
141
Appendix 5
A summary of the main notation
141
138
References
145
Index
146
Chapter 1.
THE RADAR UNCERTAINTY FUNCTION
The unce rtainty functio n , int ro duced by Woodwa rd [ I ] , is a function of two variables represen t ing signal de l ay and Doppler shift. I t may be regarded as a 'figu re of m e rit' by which various waveforms and processing methods can be compare d . This chapter gives a qualitative treatment of the uncertainty func tion and its propertie s .
1.1 LIM ITATIONS A N D CHARA CTERISTICS OF THE U N C ERTAINTY FU N CTION
The return sign al from a radar target is a mo dified version of the t ransmitted sign al . The modifications are due to the paramet e rs of the target which can , in p ri nciple , b e deduced by comparing the re turned signal with the transmit ted signa l . The unc e rtainty function resu l t s from a co nside rat ion of t h e signal retu rned from a single point targe t . An example of such a t a rget would be a small insect flying at constant veloc ity t oward an 'upward looking' radar. The radar retu rn is assumed to differ from the transmi t ted sign al in only two way s , namely: (I) A time del ay (x), p roportional to the radial range of the targe t . ( 2) A constant f reque ncy shift (y) of the whole signal spectru m , p roportional to t h e t arget radial velocity . T h e variable y is c alled the Doppler offset; it is posit ive for targe t s travelling towards the radar. Strictly speaki ng , the assumption of a constant Doppler offse t is an app roximation which is only valid when the target velocity is small com p ared with the veloCity of propagation of the transmit ted signal . The true effect is a distortion of the signal spect ru m , due to an offset which varies with frequency. I t is normally realistic t o assume a constant Doppler offset in the case of elect ro-m agnetic propagat ion in air , but caution must b e exe rcised i n such applications a s sonar [ 2 ] . With the ab ove assumpt ions, the target m ay b e regar de d , at any one time, as a point in the x, y p l ane . Use of the uncertainty fu nction allows one t o define an a rea of unce rtainty in the x , y p l ane , inside
The radar uncertainty function
4
convenient mathematical p rope rt ies. F o r the present d i scussion, 8, will be d e fined th rough the followi ng relationship:
J U{t-x, wo+y)-f{t-x+r, wo +y +w)}2 d t
00
8,2 =
(1.1)
I t is shown in S e c t i o n 2.1 t h a t th e above d e finition leads t o
00
8,2 = 2 wh e r e
r L/(I)]2 df-Ix(r, w)1 cos [ (wo+y+ w)r- Arg{x(r, w)}] x(r,w)= (a(t)a*(r+T)e - i w tdt _'00
(1.2)
( 1.3)
and
(1) a(t) is
a comple x baseband funct ion which is
transmitter modulation, as discussedin
(2) r, ware di
target co-ordinates (see Fig.
(:i)
,1IJ(r)12 df
=
derived
from the
Section 2.1.
1.2).
F i( .e.
In
to as 'the ambiguity function'. Also, x(r, w) and Ix(r, wW c an be found called by either n a me . A full mathematical treatme nt o f x(r, w) i s given i n Chapter 2. Only two the established literature it is s o me time s referred
0), its value at t h e origin : at wh ich po int it is equal to 21-.'. independent of signale ner gy, F. (2) x(r, w) normally changeswith r muc h more slowly than does cosl(wo+.\' +w)r I h ecause Wo is a hi ghcarrier ( I ) x(r, w)is never greater than X(O.
Equation
�
21:'
1.2 =
I
can be re-written in the more cllnvenient form:
-
I 21:'
1).:(1, w)1 cosl(wu +\' +w)r
.
Arg{x(r, w )}]
( 1 .4 )
A graphical interpretation of Equa tion show s a typ ical variation of g 2/21:' along a line in the r, w plane pa rall el to the r a x i s . 1-.'trepresents before the s i gnal can b e said o t different . As Wo is a high carrier frequency, 8,2/21:' is a rapidly fl uc t ua ti n g function of r, a nd always lies between the solid lines in Fig. 1.1. Si n c e pr ec i sion with ambiguities s e p arat e d by distances of th6 01 d e r of the tr ansmi t ted Signal wavelength
5
The radar uncertainty function
2
1
1
2E l x (1,W)1
-- - ------
1
�
--- - ------- -
' 2E I x (1,W)1
--- -
-----
./ Normalised �
threshold
- - -- - ---------
I n s u fficie n t diffe rence
o
1, 1----.-
Fig. 1. 1. A graphical interpretation of Equation 1.4
(m i n i mum value) cu rve i s chose n as a test o f whe t h e r t h e t h re sh o l d h a s been exceeded. T h u s , t h e t a rget range co-o r d i n a t e c a n b e said t o l i e i n t h e range s 0 t o 7), o r 72 to 73. T h e e xam p l e u s e d h a s b o th range-u nc e r t a i n t y and am b i gu i t y . Figu re 1.1 sh ows t h a t ach i evable prec ision c a n be inc reased b y red uc ing th e ra t i o c�/E . S i n c e E t i s u l t im a t e l y set by system n oise , t h i s i m p l ie s a n increase i n t h e t ransm i t t e d e n e rgy , E. 1.3
THE A R E A OF UNCE R T A INTY OF A R ECEIV ED S I G NA L
Figure 1.1 shows that Equ at ion 1.4 can be re-w r i t t e n to read: A rec eiv ed si gnal m ay be l oc a t e d wh ere
IX(7, w)1 � U'
I -E(/2E �
(1.5)
E( in Equat ion 1.5 represe n t s the (arb i tra ry) t h re sh o l d l eve l which t h e mean sq u a re e rror must exc eed b e fore i t i s p o s s i b l e t o m easu re t h e paramet e rs o f t h e ret urned signal. Si nce system n oise has not been considered , t h e i mpl i cat ion is that Et excee ds the syst e m noise level by a s ign ificant amount. Table
1:."/1:.'(
Contour of
define� the
(ratio)
I\(T, wll/21:.· has log,o"
which
of uncertainty
2.5 dB
0.4 dB
10
20
IX(T. w)I/2E
area
6 dB
2
100
1.1
0.04 dB
been
converted to dB
by
the formula:
6
The radar uncertainty function
Equa t i o n 1.5 c a n b e u se d t o c a l cu l a t e c o n t o u r l e v e l s o f I X(T , w) I /2£ i n side wh ich t h e t a rget para m e t e rs of a signal o f a given e n e rgy l i e . T h e a re a e nc l o se d b y such a co n t ou r i s t e rm e d t h e 'a r e a o f u n c e r t a i n t y ' . Exa m p l e s for various signal-t o-t h resho l d e n e rgy ratios a re given i n Tab l e 1.1.
1.4
R ES O L UTION AND P R ECIS ION
Re solut i o n i s use d h e re t o i n d icat e th e abi l i t y t o separa t e t h e re t u rn s from sev e r a l target s (as o p p osed t o 'p rec i si o n' w h i c h w a s used above t o i n d i c a t e t h e p e rfo r m a nce w i t h respect t o a s i ngl e t a rge t). I f the requi re d p re c i s i o n has bee n d e fi n e d a s a n area i n t h e T , w pl ane, and t h a t area i s e ncl ose d wi t h i n a m uch l a rge r a rea o f u n c e r t a i n t y (fo r a given sign al-t o-th reshol d rat io) i t c a n b e s t a t e d t h a t t h e giv e n signal d oe s no t mee t t h e spec ificat i o n. N o amo u n t of c leve r sign a l processing w i l l al t e r t h i s fact (assu m i ng, o f cour se, t h a t t h e signa l-e n e rgy t o t h resho l d-level ca n n o t b e i nc rease d ) . Th e unc e rt a i n t y fu n c t ion c a n n o t b e used d i r ec t l y t o m ake a sim i l a r stat e m e n t rega r d i ng reso lut i o n . If t h e un ce rt a i n t y areas co r respon d i ng t o t wo t a rge t s are p l o t t e d i n t h e x, v pl ane a n d t he y ove rl ap, reso lut i o n wi l l b e di ffic u l t o r impo ssib l e. If t h e t wo signal s were muc h h i gh e r t h an t h e rec e ive r t h resh o l d level t h e co r respo n d ing unce rt a i n t y a r e as wou l d b e s m a l l a n d m igh t no t overl ap. However, o n e sign al co u l d s t i l l b e m u c h st r o nge r t h an t h e o t h e r a n d i t s response s i d e l o b e s coul d compl e t e l y swa m p t h e weake r sign a l . One coul d o n l y b e su re t hat t h i s woul d n o t h ap p e n i f t h e Ixl = 0 con t ours sur ro u n d i ng each target co-o r d i nat e di d n o t i n t e rsec t. Th i s l a t ter c r i t e r i o n is not very use ful fo r the t y p e s of signal wh ich are usual l y s t ud i ed by means of the unce r t a i n t y fu nct i o n , and does n o t h e l p i n t h e case o f a sign al i n cl u t t e r. I n tui t ively, o n e m ight t h i n k t h a t if t h e a r e a o f u nc e r t a i n t y o f t he weakest signal al one was m a rk e d o ff i n t h e x, y plane t h e t est fo r resolut i o n i n t h e p r esence of a sign al 1/ dB s t ronge r w o u l d b e wh e t h e r t he -1/ dB c o n t our o f Ixl/2c·, su rrou n di n g t h e st ro nge r s ig n a l co-o r d i nates, e nc l o sed t h e a rea o f un c e rt a i n t y o f t h e w e ake r sign al . Al t h o ugh t h is argu m e n t has val i d i t y i n t h e c ase o f rece ivers w h i c h gi ve an out put wave fo r m in t h e shape of a cut t h rough X(T, w), it d o e s n o t a p p l y t o cases w h e re th is i s not so. I n pa r t i c u l a r , s u c h a l t e rn a t ive cases may we l l l e a d t o b e t t e r reso lut i o n. A t rue assessm e n t o f t h e p reci sion and reso l u t io n p r op e r t i e s o f a give n rad a r sy s t e m can o n l y be m a d e b y st u dy i ng t h e shape o f t h e p rocessed o u t p u t p u l se w h i c h i s u se d t o d e t e r m i n e t h e val u e o f t h e delay o r D o p pl e r p a ra m e t e r b e i ng me asu re d . I f a single figure i s
7
The radar uncertainty function
required as a me asu re o f p rec ision o r reso l u t io n , it can be ob tained b y measu ring t h e width of t h i s output pulse a t som e arb i t r a ry l evel. In this b ook t he 'precisio n ' o f a system will be defi n e d as the width of its outp u t pulse measu re d be tween its half-am pl i tu de (i.e . -6 dB) point s . Table 1 . 1 shows t h at this implies a sign al-t o-th reshold energy ratio of unity . The 'n dB reso lut ion' of a system will be d e fined as the width of its ou t p u t p ul se me asured between its widest -n dB p o in t s .
The ab ove definitions apply to an actual syst e m , the inherent p recision o f a transmitted signal will be defined as the -6dB width o f t h e a p p ropriate c u t t h rough IX(7, w)l. 7 and ware rel at ed t o range and veloc i t y by the formulae
2r 7= -
( 1 .6 )
c
2wov w= --
( 1 .7)
c
where r and v are range and ve locity (wi th respect t o the ta rget ) , c is the velocity of p rop agation and Wo is t he carrie r frequency . Thus i f, fo r a given system, the d e l ay p recision is give n by 71, the corre spo nding range precision w i ll b e (C71 /2). The velocity precision, correspo n d i ng t o a Dop p l e r p recision of Wi> will be (cwI/2wo) a n d it shou l d be no t ed t hat this c an be improved by inc reasi ng Wo.
1.5 THE REL A TIONSHIP BETWEEN THE
7, w
AND x, y P L ANES
The co-ordi nat es x, y define a point with respect to the receive r , while the co-ordinates 7, W define the same point with respect t o th e t arget , Fig . 1.2. Y
Y,
/compari50n point
r�---:L _____
, I I I I I I I I I-----I....-.I ____
__
Target
Fig. 1.2. Relationship between x, y and
i,
W
8
The radar uncertainty function Y
W
y
,
--D \...W
-------+--�---.x x,
Fig. 1 . 3. T ran slation of X(T, w) to x, y plane
Since i t fol l o w s t h at
x = (Xl -7), X(7, w) = X [-(x-xd, (y- ydl
The above relat i o n ship is illust r a t e d by Fig. 1.3.
Chapter 2 THE MATHEMATICAL TREATMENT OF THE UNCERTAINTY FUNCTION
This chapter is used to show how the r.m.s. error criterion leads to the function
x(r,w)= f a(t)a*(t+r)e-jwtdt. 00
_ 00
Also, some mathematical properties of x(r,
2.1
w)
are studied.
DERIVATION OF THE UNCERTAINTY FUNCTION
It is assumed that a real signal, f(t), is transmitted such that
faCt) = aCt)ejwot is the corresp onding complex analytic signal (Chapter
(2. 1 ) 7).
Equation
implies that the real transmitted signal is given by
j(t) = Re{fa(t)}
2. 1
(2.2)
Defining
aCt) = la(t)1
ejp(t)
(2.3)
leads to
f(t) = la(t)1 cos[wot+(t)]
(2.4)
A subtle point worth n oting is that with the above definition of f(t),
la(t)1
and
(t)
are not arbitrary functions. Let f(t) be written:
f(t) = Re { b(t) e jwo t }
where
bet) = Ib(t)1
that is
where
Ib(t)l, OCt)
ej(J(t)
f(t) = Ib(t)l cos [wot + O (t)]
(2.5)
are defined as the real, arbitrary, functions applied t o
the amplitude and phase modulation terminals, respectively, o f the transmitter.
9
10
The mathematical treatment of the uncertainty function Unless the carrier frequ e ncy W(j i s sufficiently high, it does n ot
follow that
Equ ation
aCt) is equ al to bet).
2.5
Rather,
u si n g the re lationship :
aCt)
has to be calculated from
aCt) e j w o t = fa Ct) = f(t)+ittt)
(2.6)
A ful l discussion o f this point is given in Section 7 .4 . T� e dist in ction
b e twe e n the complex an aly t i c signal an d the exp on e n t ial approxima tion h a s t o be m ade in t he case o f son a r. Some impo rtant d i ffe rences b e tw e en radar an d sonar are de t a ile d b y Krame r [2] . Fo r t u n a t ely , for p ractical rad a r applicat i on s, i t is n o t ne cessary t o use Equ a t i on 2.6 as aCt) c a n u su ally b e conside red equal to b(t ). The r.m .s. e r ro r c rite rion has b e e n discuss e d in S ect i o n 1 .2. The r.m. s . diffe re n ce , &, between tw o re al sign a l s f(t) and get), is defin e d t h rough t h e rela t ionship
J (f(t)-g(t)J2 dt
00
&2 =
(2.7)
An i m p o r tan t p rope r ty o f complex a nalyt i c signals (p rove d i n Se c t i on 7.2.5) i s that Eq u at i o n 2.7 can b e repl ace d by
&2
J lf�(t) - ga(tW dt 00
=
�
(2.8)
Using the r ela t ion sh i p s
f�(t) = f(t)+ jf(t) &(t) = g(t)+ i(t)
Equat i o n 2.8 c a n b e expanded t o give 28,2 =
To p r ove Equat ion 1.2 it i s necessary t o p u t
j{t)=f[t-x, wo+y]
g(t) = f[t-x+ T, Wo+l' + w] Hence
j�(t) = aCt-x)
ej(wo+y)( t -x)
gael) = aU -x + T)
ej(wo + y+w)(t - X+T)
(2.9)
The mathematical treatment of the uncertain ty function Substitution of the above in Equ ation
give s
2.9
J la(t-xW dt + J la(t-x + TW dt -2 J Re{a(t-x)a*(t-x + T) e - jwt e jw x e - j(wo+y+w)T} d t 00
11
00
2&2 =
00
(2. 1 0)
B y u sing the relationship between the spec tra of real and comple x analytic signals (Section 7.1), an d b y the app l ication of Parseval's the orem (Appendix 2), it ca n be shown that
J laCt-x W dt = J la(t 00
00
_00
_00
J [fa (t ) 12 dt = 2 J Lf(t)]2 dt 00
-
x
+ T ) 12 dt =
_00
Al s o , by changing the dum m y var ia b l e Equa tion 2. 1 0 b e come s
{
2 Re e j WX e - j(w o +Y + W)T
Thus, w ith the definition
00
_I a
t to (t + x), the last integral i n
(t) a * ( t + T) e -j(t+x)w dt
f a( t) a *(t + T ) e
}
00
X(T, w ) =
-
(2. 1 1 )
j w t dt
Equ a tion 2 1 0 m ay b e w ri tte n in the form .
J [f(t)]2 dt - Re { e j(w o 00
&2 = 2
-
+ Y + W)T
X( T ,
w)}
(2 . 1 2 )
Equ ation 2. 1 2 i s also equ ivalen t to
J Lf(t)]2 dt-lx(T, w)1 cos[( wo + y + w ) T-Arg X( T , w)] 00
&2 = 2
(2. 1 3)
Th e di s c u ssi on in Se ct ion 1.2 uses Equat i on 2. 1 3 t o estab l ish th e physi cal s i gn ifican ce of X(T, w). 2.2 THE M A THEM A TI C A L PR OPE R TI ES OF
X(T,
w) i s defined by
X(T , w )
J a( t)a * ( l + T ) e -j w' dr 00
X( T , W)=
(2.14 )
The most important properties of X ( T , w) are stated below; proofs a re
gie v n St utt
[3, 4] and Siebert [5].
X( T . w ) = ,�{a(t) a*( r + T )}
(:2.15)
12
The mathematical treatment of the uncertainty [unction
�
XCi, w) = 2
_
r 00
A *Ux)A [j(w + x)] e-jxT dx
(r.16)
X*(i, w) = e-jwT X( -I, - w)
(2. 1 7)
IX(i, w)1 = IX(-i, -w)1
(2. 1 8)
I X(i, w)1 can be regarded a s a 3 -dimensi on a l solid , placed u po n the i , w plane. Tw o-dimensional fun c tions can b e ob t ained by t aking cuts t h rough IX (i, W)I. The symmetry indicate d in Equa tion 2. 1 8 means t hat t he resu l ting fun ctions will be eve n , if the cu t s pass through the origin o f t he i, w plane.
O
f la(t)12 dt � IX(i, w)1 00
X ( , 0) =
(2 . 1 9)
_00
E quation 2 . 1 9 means tha t IX(i, w)1 can never be greater t han i t s value at t he origin . Note t h a t sin ce aCt) w a s de rived from faCt)
f
00
J [f( t)p dt; 00
X(O, 0) is equal to twice t h e ene rgy of f(t). ]2 = 2� fT'X(i, wW di d w [X(O, 0) _00
i.e .
la ( t ) 1 2 dt = 2
( 2. 20)
_00
Equ ation 2 . 2 0 exp resse s the fa ct that the total volu me of t he I X Ci , W)12 sol i d is constan t , rega rdle ss of the for m of aCt). This means that any steps take n to c o n ce n t r a t e W)12 i n a n a rrow spike at t h e origin m u st also resu l t in a l arge spread of low e r leve l s o f t he fun ction. A change i n t h e form of a t) gives the resul t s set o u t in Table 2. 1
IXCi,
C
Table 2.1 a(t)
AUw)
X(T, w)
t
aCt) ejb 2
cJbw2 A Uw)
a(cxt)
e-jbT2 X[T, w + 2bT]
ej bw2 x[T-2bw, w] 1 - X[CH, w/a] lal
I;j 1
A Uaw)
X[T/a, aw]
The significance of Equation 2.21 is discussed in Section 2.4.
Rep e t i t i on o f a b asic w avefo r m , i.e. c hanging aCt) t o n-J i
L
=O
Cia ( t- ik)
(2.21 ) (2.22) (2.23 ) (2.24)
The mathematical treatment of the uncertainty function
13
leads to an uncertainty function given by
n-I n-I-m m C:C,+me-jwlk L e-jw kX(T+mk.w) L m=1 1=0 n-I-m n-I + L X(T- mk. w) L C,C:+m e-jw1k
where C,
m =0
is a real or complex m u ltiplier (commonly unity) a nd 11 the
number of p uls es
S e ct ion 2.3 .
.
The significance of Equation 2.15 is discussed in
The composite function aCt) + bet),
leads to
X(T, w) = Xaa(T, w) + Xbb(T, w) + Xab( T, W)
+
where
XuvCT. w) = or,
by
(:�.25)
1=0
inspec t ion
XLiI.( T.
I
u(t)l'*(t + T)
w) = ;;; f II( t)l' * (1
ejwT Xa*b( - T, -w) e-jwr dt
+
(2.2h) (2.27) (2.2R)
T)} dt
2.3 THE EFFECT OF REPETITION UPON THE UNCERTAINTY FUNCTION
A general by
case o f repetition of a basic w3veform. a(t). 11 times i� given
h(t)
=
1/
L
1
i 0
C/a(t
Ik)
By letting C, take on comple x values. i.c.
C/
= a
/ 1.'1>1
ucceeding pulses can hc givcn differcnt amplitlldcs and initi:t1 ph :l\c d. T hu s , in a given portion o f the T, w p l ane, only two displa ce d Xb fun ctions w ill be involve d in the calculation o f X(T, w). The a b ove is illustrated by Fig. 2.5 which show s t h e regions of influ ence of t he individual b i t fu nctions on t h e X func t ion of a fou r-b it w o rd. W
t I IT+ 2d)--'I-I'-X IT) .... I.O-- X IT- 2 d)----.., I I I I I--- x 11 +3d) . : . X I'"(+d ) --r--- x IT -d) ----X-I.I-.- IT 3d) ------.1
Fig. 2.5. R egi ons in w h ich the displaced bit x' s ex ist for a 4·bit w ord wit h b it dur at ion �d
It should be parti cu l a r l y noted th at along t he lines parall e l to the w axis where T = ±md{m = 0, 1, . . . (L - 1 )}, Xb(T - md, w) i s the only non-zero Xb fu n ction.
The mathematical treatment of the uncertainty function
19
The function represented b y .s# (m, w ) is determined by the code and is indepen dent of the bit shape. For the special case of Ci real ( e. g. 00 , 1800 phase coding) .s#(m, w ) is equal to d*(m, - w ) . A fu rther discussion of the effects of signal repetition and coding is given in Section 4. 3 .
2.4 T HE EFFECT OF L I N EA R FM UPO N T HE U N C ER T A I N TY FUN CT ION
If linear f.m. is added to a c o mp lex modulation functi on a C t), t he resul t is t o give a new m odulation function of t he form aC t) eib f , i.e. an extra qua drati c phase te rm. Table 2.1 show s that this results in the uncertain ty function changing from X(T , w) t o
b e-i r2 X[T, w+2bT]. The above result means tha t the modulus of the new uncertainty fun ction is a she ared version of the modulu s of the original uncertainty func tion . A characteristic which formerly occurred at the co-ordinate s TI, WI is tr ansferred t o w+2bT:=Wt
That i s , to the new co-ordi na tes Til (WI - 2bTI)' The effect is illustrated in F i g . 2.6. When evaluating formulae numerically, some confusion can resu l t over t he choice o f t he parame ter b. I t i s convenie n t , in algeb raic manipulation , to use t he si ngle le tter b; however , when using the results it is m ore useful t o t hin k in terms of the physical 'fre quency sweep' . Interpreting instan t an e ous f re quency a s w· :=
.
I
dcp dt
-
the freque ncy swe ep , in t ime t, cause d by the qu adratic phase term w i ll be 2bt. Thus, for a n enve lope of durati on d, t he quadratic ph ase term will lead to a t o t a l fre quency sweep 6
= 2 bd.
(2.29)
I t follows t hat , when using results base d upon t he linear f.m. formul ae , b should be give n the va lu e b=
6
2d.
(2.30)
20
The mathematical treatmen t of the uncertainty function w
--�------+---�r--1
w
Ix [1,
W +
]I
2b1
Fig. 2. 6. T h e sh earing effect of linear FM u pon t h e ar ea of uncertainty
A term often used in the litera t u re is ' disp e rsion factor'. Using the ab ove notati on , the dispersi on fa ctor is equal t o the dimen sio nless p roduct 6 d. F o r the comm on pra c t i ca l sit u ation o f a fixed envelope d u ration , an increase in the dispe rsio n fa cto r means an in crease in the total frequency swe ep 6 .
2.S PROOFS OF THE PROPERTIES O F
x (r , w )
Th is se ction contains the pr oofs o f the mathematic al properties o f x (r , w) w hich were sta t e d i n Section 2 . 2 . 2.S. 1 Proof of Eq uation 2. 1 S
Equation 2 . 15 foll ow s, b y i nspe ct i o n , fro m t he definitions o f x(r , w ) a n d the F ou ri e r t ra n sfo r m .
The mathematical treatment of the uncertainty function
21
2.5.2 Proof o f Equation 2.1 6
From the results of Sec ti on 6.7
ff{a*(t)} =A*(-jw). Hence
ff{a*(t+r)} = eiWT A*(-jw). Exp anding Equation 2. 15 b y me ans o f the convolu tion t heorem gives 00
1 x(r, w) = 21T
f .
eJXTA*(-jx)A [jew-x)] dx.
_ 00
Changing t he dummy variable to
f
-x
give s
00
1 21T
x(r,w)= whi ch is Equation 2 . 16.
A*Ux)A[j(w+x)] e-iXTdx,
2.5.3 Proof of E quation 2. 1 7
Since E qu ation 2. 14 leads to
[x y z] * = x*y*z*,
J a*(t)a(t+r) e iWT dt 00
x*(r, w) =
_00
Chan ging the dummy va riable to (t - r) gives
J a*(t- r)a(t) e iW (T-T) dt 00
x*(r, w) =
_00
which be comes
J a(t)a*(t-r) e iWT dt. 00
x*(r, w) = e - i uJ T
Th e ab ove integral is equal, by inspection, to x( -r ,-w), t hus proving e quation 2. 1 7 .
2.5.4 Proof of Equation 2.18
Fr om Equation 2 . 1 7
Ix*(r, w)1
=
Ix(-r,-w)l.
The mathematical treatmen t of the uncertain ty function
22
Also , since IX *( 7, w)l = IX(7, w)l , Equ a t i on 2.18 follow s .
2.5.5 Proof of Equation 2.19
The e quality in Equat i on 2.19 foll ow s b y i n spec t i on , reme m b e r i ng that
�12 =xx*.
T he i n e q uali t y i n E qu a t i on 2.19 can be ob t ai n e d from t he Sc hw a rz inequal i t y [6]. Sin ce
I
�
I_L f(x)g(x )
dx
2
�_
l
00
lf(xW dx
_[ (x 00
[g
)12 dx
i t fol lows t h a t
The a p p l ic a t i on of Pa rseva l's t he o rem ( Appe n d i x 2) sh ow s t h a t t he tw o r ight-hand i n t egrals a re equal. Since i t has a l r eady been sh own t h a t
J la(f)12 dt = X(O, 0) 00
it fol l ows tha t
which p r ove s E q u a t ion 2.19.
2.5.6 Proof of Equation 2.20
The p roof of E q u a t i o n 2.20 is ob t ai n ed b y r e p l acing I X( 7 , wW by X(7 , w) X *( 7 , w). X(7, w) is expan ded by means o f Eq ua t i o n 2.14, and X *(7 , w) by Equation 2.16. T he result is
JJ IX(7, wW d7 dw 00
fffa(t)a*(t 00
=
+
7) e -jwt df
fA 00
21n
Ux)A * [j( w +x)] e j XT dx d7 d w .
The mathematical treatment of the uncertain ty function The right-han d si de m ay be re-a rranged to give
I 2rr
IIa(t)A(jX) I 00
00
I
00
a*(t+T) e
jXT d
T
_00
A*U(w+x)] e - j
23
w t dw dt dx
Sin ce the inte grals over T and w, i n the a bove expressio n , contain all the T and w fu n ctions they may be evaluate d separately. Now
J a*(t +T) ejxT dT J a*(T) ej (T - t)x dT 00
00
=
=
Also
A
e -jxt *Ux)
_00
J A*U(w+x)] e - jw t dw J A*Uw) e-j(w-x)tdw 00
_
00
=
00
=
2rr e jx
ta*(t).
The ab ove results follow from the de finitions of the F ourier transform and its inve rse . Substitution in the expressio n for
JJ IX(T, WWdTdw 00
gives
H IX(T, wW dTdw J a(t)a*( t) dt J A(jx)A*Qx)dx 00
00
=
00
Parse val's the ore m sh ows the la st integral to be equal to 2rr times the middle integral. It has already be e n show n that
f a(t)a*(t) dt 00
=
_00
Hence
II
f
00
_00
la(tW dt
=
X(O, 0)
00
2Irr
IX( T, wW dTdw = [x(O,O)f
which is Equati on 2.20.
2.S. 7 Proof of Equations 2.2 1 and 2.22
De fining
f b(t)b*(t+T) e -j wt dt 00
Xl(T, w)
=
_ 00
The mathematical treatment of the uncertain ty function
24
and subs t i t uti ng
gi ves
J a(t ) ej b r a * (t + r) e -j ( f + T )2 b e -j w f d t ""
x \ (r, w) =
J a(t ) a * ( t + r ) e - jbT2 e - j (w + 2 bT )t dt. 00
= Hence
p r o v i ng E q u a t ion 2 . 2 1. S i m i l ar l y t a k i n g
gives
f
00
=
2rr
T - 2 b W )X dx A * U X ) A [j ( W + X ) ] ejb w 2 e -j( .
H e n ce
p r o v i n g E q u a t i on 2. 2 2 . 2 . 5 . 8 Proof of Equ a t ion s 2.23 an d 2 . 2 4 If
b e l ) = a( M )
J a ( al )a * l ( l + r )a j e - j w t dt 00
x \ ( r, w ) = t hen
J
00
XI ( r, w) =
lal
a( l ) a * ( l + ar ) e - j w f/Ci. d t
The mathematical trea tment of the uncertain ty function
25
which is Equ ation 2 . 23 . Also, i f BGw) =AGaw) t hen
The mathematical treatment of the uncertainty function
Since
j
J
00
��
a(t + x) a * (t + y ) e - j w t dt =
i t follows t h a t
J
-�
a(t ) a * (t -x + y ) e - j w ( t - x) dt
00
X I ( T , W) =
b (t) b *(t + T) e - j w t dt + n + e - i w k c;, Q' X ( T + n k , w)
= Co Cl)X( T , w)
+ Co crX ( T - k, w) + e - j w k C I Cl' X [ T , w ] . . .
+ n + e - j w k Cn C fx [ T + (n - l )k , w ]
+ Co c,iX( T - nk , w) + e - j w k CI G X [ T - (n - l )k . w ] + + e - j w n k Cn C* n X(T W) Hen ce X I ( T , w) = e - j w n k X(T + nk , w ) { c;, Co' } + . . . I
,
+ e - i wk X( T + k , wH cl Co' + C2 ci e - j w k . . . + Cn G I e - j w (n - l )k � n C: c;, ci . . . cl k w + w k e j Co T, X( w e j + + } H C�
+ X ( T - k , w H Co C; + C I G e - j w k . . . + c;, _ I c: e - j w (n - I ) k } . . . + X( T - l I k , w H Co G }
which leads
to
X J T, w ) =
n
L
m =1
+
e - j w m k X( T + mk , w ) n
L
m =0
X( T - m k , w)
n -m
L
;= 0
n -m
L
;= 0
G + m Ge - j w ;k
C G+ m e - j w ;k ;
so that the new 'n' i s e q u a l t o the n u m b e r o f Re p l a c i n g n b y n p u lses give s n-I n- I-m m w w = X I ( T, ) L e j k X( T + mk , w) L C; + m G e - j w;k m=1 1=0 +
11 - 1
L
m=O
X( T - mk, w)
n - I-m
L
;=0
c;c7+m e - j w;k
The math ema tical treatmen t of the uncertain ty function
27
2. 5 . 1 0 Special cases o f Equation 2 . 25
Thre e special case s will n ow be considere d .
(I) G at e d sou r ce .
f(t) = H ence
n-I
L
;= 0
la(t - ik ) 1 cos [ wo t + ¢ ( t - ik ) ]
fa ( t) =
n-I
L
;= 0
a(t - ik ) e i w o t
which c orresponds t o C; = 1 . T h u s t h e coeffic i e n t o f X ( T + mk, w ) is a ge ometric p rogression given b y e -i w m k p + e - i w k + . . . + e -i W k (n - l - m ) }
{
-i W k (n - m ) . - e - J wm k e k e -i w _ l
_ I}
Similarly the coefficien t of X (T - m k , w) is
By defining e =
�k
{
e -i W �(n - m ) - I e -J W k _ 1
}
and u si ng the identity
( I _ e - 2 i x ) = 2j e - i x sin (x)
the above coefficients can be written as follows : X ( T + m k , w ) � e - j(n - I + m w X( T - mk , w) � e j (n - l - m W
{ {
sin [(n - m)8 ] . e SIn
sin [ (� - m )8 ] SI n e
} }
which is expressed compactly by saying t h a t the c o e fficient o f X(T - m k , w ) is W l e -j (n - - m
{
Sin [ (n - I m l ) e ] . SIn e
}
fo r 0 � I m l � n - 1 . The expression for X lT , w) then b e c omes : (n l ) -j w sin [ (n - l m l )8 ] i e (n - l - m X [ T _ m k, W ] . SIn e m = - (n - l )
{
(2) Pulsed source . n
{
-I
f(t) = L
;= 0
}
l a(t - ik) 1 cos [ wo (t - ik ) + ¢(t - ik ) ]
}
28
The mathematical treatment of the uncertainty function Hence
faCt) =
n
-I
L
;= 0
which corresponds t o C;
e - i wo ki a (t _ ik )ei wo t
= e- iwoki. Thus
C·I+ m C'" = \:""_i wo km I
Since the above products are not fu nctions o f i, the coefficients of x(r + m k, w) and x(r - m k, w ) are once more geomet ric prog resssions . The same reasoning as in the gat e d case lea ds to the result
x 1 (r, w) =
i
n
m-- n-
(
l
I)
{eiWokm
e -i (n - l _ m )8
{
�
Sin [ ( . - l m l ) e ] ,.10 e
}
x [ r _ mk' W ]
}
(3) N o n - c o h e re n t sou r c e . This i s t he case for Ci = e jq)i w he r e ¢i t akes on r a n dom values be twe en 0 and 21T r a d i a n s . The co-effi cien t of x(r - m k. w) fo r m = 0 i s a ge ome t ric p r ogression n-I
L
i= 0
Ci c1 e -i wik
=
e-i wik i {
n-I
L
i= 0
si n [ n e n ( = e - n - I )8 si n e
f
wh e re e = wk/ 2 . The c o e ffic ients o f X ( T ± mk, w) for m ± 0 are not geom e t ri c p ro g ressi o n s and c a n n ot be exp ressed i n a c l ose d fo r m .
2.5. 1 1 Proof of Equation 2.26
Eq u a t i o n 2 . 2 6 can be prov e d b y usi n g t h e fu n c t i o n aCt) + b et) in the d e fi n i t i on of X (T , w):
r [ a ( t ) +b(t ) ] [a *(t +r)+b * (t + T)] e 00
X( T , w) =
_00
M u l t ip ly i n g ou t gives
-i wt dt
X(T , w) = Xaa ( T , w)+Xbb ( T ,w)+Xab ( T , w)+Xba ( T, w) w h e re
_L u(t) v*(t+T) e 00
Xu v ( T , w) =
- iw
t
dt
The mathematical treatment of the uncertainty function
29
I t follows that
Xba (r , W) = (b(t)a*(t +r) e -j wt dt 00 Changing the dummy variable to (t - r) gives Xba (r , w) = ej w T (a*(t)b(tr) e-jwt dt 00 _
_
Hence
Substitution of the above in the expression for Equ ation 2.26.
x(r, w)
gives
Chapter 3 WORKED UNCERTAINTY F UNCTION EXAMPLES
This chap ter is use d t o illustrate the m e thod of c alculating and representing u n ce rtainty fun ctions .
3.1
THE RECTANG U LAR PULSE WITH CONSTANT CARRIER FREQ UENCY
F o r t h e rectangular pulse w ith constant carrier frequency t h e b aseband m o du lating fu nction is given by
a(t) = H en ce
a * ( t + 7) =
-t
Thus , for 0 < 7 < d
-t+d t --
n-- 1
a(t ) a * ( t + 7 ) =
o
- "C + d
t .....
An d for -d < 7 < 0
a(t ) a * ( t + 7) = F i nally , for 17 1 > d
n-- 1 -t d t .....
a(t ) a * (t + 7) = 0
F r o m Equ ation 2 . 1 5 H e nce for 0 < 7 < d
X ( 7 , w) = .�{a( t)a * (t + 7) } X( 7 , w ) =
l _ e -j (d - T ) W
jw
----:---
30
Worked uncertainty function examples .
31
� sin l � (d - T)]
(3 . 1 )
or
X(T , W) = e -j( w /2 ) (d - T) For -d < T < 0
. r
1 - e - j (d + T) W . --X( T , w) = el W T -JW
Hence
X (T , w) = e - j ( w /2 ) (d - T)
�
sin
[�
Equations 3 . 1 a n d 3 .2 c a n be comb i ne d to give
X ( T , w)
=
e - j ( w /2 ) (d - T)
X ( T , w) = 0,
ITI
�
l�
sin
]
]
(d - I T I ) ,
>d
]
(3 . 2 )
(d + T )
}
(3.3)
T h e cut a l o n g t he T axis o f t h e uncer t a i n t y fu nct i o n i s given b y IX( T , 0)1 , w here I X( T , 0 ) 1 = d - I T I , ( 3 .4) X (T , 0 ) = 0 , W
The cut a l o ng the Ix(O , w) l , w here
}
axis of t h e uncer t a i nt y fu nc t i o n i s given b y
I X( 0 , w ) 1 = d
'1
l
si n ( uxl / 2 ) ( uxl/2 )
I
(3.5 )
Eq ua t i ons 3 .4 a n d 3 . 5 a re i l l ustrated by F i gs 3 . 1 a nd 3 . 2 . T he fu l l 3-dim ensi onal fu n ct i on o f Equa t i on 3 .3 is ill u s t r a t e d b y t h e c om p u t e r p l o t of F ig. 3 . 3 ; F ig . 3 .4 i s a com p u t e r p l o t sh ow i n g t h e a re a o f u n ce r t ai n t y fo rme d by the - 6 dB c on t ou r of I X (T , w) l . -- -
-1
--- d
o
L/cJ
Fig. 3. 1 . I X ( T , 0 ) 1 for a recla l l � lar p u lse wilh CO I I S la l l 1 carrier jrcqu el/ c \ '
Worked u ncertain ty function examples
32
-4
-2
o
w d/ 2n
2
4
Fig. 3.2. I x (O , w ) 1 for a r ectangular pulse with con st a nt carrier fr equ en cy
T
Fig. 3. 3. Th e un certain ty fun ctio n of a r ect angular pulse wit h const ant carrier fr equency (tr uncat ed at half height )
U si ng F i g . 3 .4 t o ge th e r w i t h the q u a n t i t a t i ve d e fi n i t i on s o f pre cisi o n an d r e s o l u t i on (Sec t i o n 1 .4 ) gives t h e i nh e ren t delay a n d D op pl er p re c I s I ons a s T
-
d
wd
=
I
1 .2 2 rr =
(3 .6 ) (3 .7)
33
Worked uncertainty function examples W dj2 Tt 0.8 0.6 0.4 0. 2 - 0.6
- 0.4
-0.2
0. 2
0.4
0.6
l:jd
-0.2 - 0. 4 - 0.6 - 0.8 Fig. 3.4. Th e - 6 dB area of un cert ainty of a rectangular pulse with cons tant carrier frequency
while the correspon din g figures for resolution a re
J = 2 (for complete resolut ion)
c.::: = 6 3 7 (fo r 60 dB resolution)
(3.8)
( 3 .9)
The relationship between T / LV a nd range/velocity is given in Section 1 .4 (Equations 1 .6 and 1 . 7) ; making the appropriate substitutions leads to the followin g relationships for inheren t range and velocity p recision
r = 0 . 5 cd
(3 . 1 0)
= 0.6
(3 . 1 1 )
v
Note th at pulse .
fo d
c fo d
is equal to the numb er of RF cycl e s contained in the
Worked un certain ty jill1ction examples
34
3 . 2 . T H E R ECT ANG U LAR PU L S E W I T H L IN EA R FM
For t h e r e c t a n gu l a r p u l se w i t h l in e a r F M t h e b ase b a n d m od ul a t i n g
func t i on i s gi ve n
by :
I t fo l l ows , fr o m E q u a t i on 2 . 2 1 t h a t , i f X tCT , w ) a p p l i e s t o t h e p u l se w i t h ou t FM , t h e r e q u i r e d fun c t i on is e- j bT 2
X I f T , w + 2b T ]
Hen c e from Se c t i o n 3. 1 ( Eq ua t i on 3. 3 )
X ( T , w ) = e _ jb; 2
e- i (W +2 bT)(d - T)/ 2
sin
X( T , w)
=
0,
(
2 + W �T
__ _ _
[ � w
2b T
)
]
}
(d - I T I ) ,
(3. 1 2)
Th e c u t s a l o n g t h e u n c e r t a i n t y fu n c t i on axes are giv e n b y
I X( T , 0) 1
=
I X ( T , 0) 1
d
=
0,
I ��� Si n [ b
I
- ITI )]
I
,
/ 2) sin( wd I X (0 , W ) 1 = d (wd/ 2 )
ITI < d
ITI > d
I
'I
J
( 3. 1 3 )
(3. 1 4)
Usi n g t h e re la t i on ship b = 6. /2d (Equ a t i o n 2 . 30) , I X (T , O ) I c a n b e w ri t t en i n t h e fo rm
IX( T, O ) I = 0 ,
IJI < I �I
> 1
}
(3 . 1 5 )
Th e a rgum e n t o f t h e s in e fu n c t i o n i s illu s t rate d by F ig . 3.5 . N u l l s o c c u r i n IX (T , 0)1 w h e n e v e r t h e g ra p h o f F ig . 3 . 5 pa sses t h r o ugh i n t egral m u l t i ples o f n . It c a n b e see n that whenever t h e d ispersion fac t o r d6. is large t here w i ll be a l a rge n u mb e r of n ulls-w i t h c o r re s p o n d i n g p e aks b e tw e e n t h e m . For a l arge d i s pe r s i on fac t o r the l as t p eak w i ll occ u r i n t h e vi c i n i t y o f I T lid = 1 a n d w ill have a n a p p r ox im a t e value o f 2d/(d6. ) . Thus for a d isp e rsi on fa c t o r o f 2000 t h e last peak i n I X( T , 0) 1 w i ll be a b o u t 60 dB d own o n t he m'a i n lobe.
..
Worked uncertain ty function examples
(n+1)
11:
I
11:
o
--
- - -t- - I I I I I I
I
I
I --- - - -t- - - -tI - - -tI I I I I I I I I I I I I I I I
n 11:
2 11:
35
I I I -+----1---+-+ ---+I I I I I I II 1 I I
1 I 1 1 t 1I I I 1 1 I I I 1 1 -
-
- - - -
I I
:
L
- -
I I
+ I
- -
1
+
I
- - -
i
:
1
I I I
I
I I I -+ I
I
-
I
T
I
-
I
: :I :
I
I
I
I � I --
Fig. 3. 5. Plo t of
T: (1 - III I )
A fu r t h e r poi n t t o n o t e is t h a t for small values o f 1 7 1/d , i . e .
� d
=
1 76 1 & 0 1 d6 ""'"
I X(7 , 0) 1 c a n be represe n t e d b y I X(7, 0 ) 1
�
d
I
!
sin(76 / 2 ) , (76 / 2)
1 7 6 1 � 0. 1 d6
(3 . 1 6)
Since t he sin(x )/x fun c tion has i t s n ul l s when x is equal to i ntegral mul t iples of 1T , it follows that it is convenie nt to l abel t he 7 a nd w axes fo r t his fu n c t i on in t erms of 76 21T
and
wd
21T
respec tive l y . I t follows from Equ ations 3. 1 4 a n d 3 . 1 6 t ha t t h e Doppler p recision a n d resolution a re given b y : wd
2 1T
= 1 . 2 (prec ision)
'::: = 6 3 7 (60 dB resolution)
(3. 1 7) (3 . 1 8)
36
Worked uncertainty function examples
Fig. d�
=
40
3. 6.
Th e uncertainty fu nction of a rectangular pulse with linear FM and
dB area of uncertain ty of a rectangular pulse with linear FM
37
Worked uncertainty function examples For large values o f dfl the d elay precision will b e given b y
;�
=
( 3 . 1 9)
1 .2
The complete u n ce rtainty fu nction is ill ustrat e d b y F igs 3 .6 a n d 3.7 . No te t hat there i s strong delay-Doppler c oupling, the consequences of t his a re discusse d i n Sec tions 4. 1 , 4 .2 a n d 5 .2.
3. 3 T H E RECT A NG U L A R PU L S E W IT H TRIA NG U L AR FM
The baseband modulating fu nction to be studied in t his se ction is given by
c ( t ) = aCt) + b et )
where
(3 . 20a)
aCt ) =
[ rL -d 1] e- jbr
( 3 . 2 0b )
bet) =
[ rL1] d
( 3 . 2 0c )
0
l-
0
l-
H e nce the in stantaneous fre quen cy is given by
w( t) =
S/JU2bd -d o
d
The above de finitions maint ain t he relation sh i p b. = 2 bd ; i n t his case b. is t he t otal frequency change (rad/s) during either t he down sw eep or the u p swe e p . T h e results of Section 2 .2 ( Equations 2 .2 6- 2 . 2 8 ) show t h a t the com p lete uncertainty fun ction is the sum of the unce r tainty functions of a(t) a n d b et) an d o f the 'cross un certainty fu nctio ns' , i .e .
x(r , w ) = Xaa ( r , w ) + X bb (r , w ) + Xa b (r, w ) + e j w T X:b ( -r , - w)
(3 . 2 1 )
whe re 00
X uv( r , w ) = f u(t) v *(t + r) e -j w t dt _ 00
If
a{ t ) =
[ D1 J l-
e -j
bt2
(3 . 22 )
38
Worked uncertainty function examples
it foll ows that aCt ) defi ned by E quatio n 3 .2 0(b) is given by
aCt ) = cx( t + d )
(3 2 3 )
3 . 2 3 all ow :xaa ( T , w ) and X bJ.T , w ) Section 3 .2 . The final expressi ons a re
Equations 3.20(c) and from the results of
:xaa (T , w ) = ex p j
[� ] [� (: ) ,
:xaa ( T , w) = 0 , and
X bb ( T , w ) =
(T + d) - b Td
exp
j
W 2b T
sin
W 2b T
[� - ] ; [ (} )
]
W
2b T
w 2b T
to b e found
(d - I TI ) , IT I < d
}
(3 .24)
I TI > d
(T d ) - b d T sin
.
�(2t+d)bd
(d
- ]. IT!)
ITI < d I TI > d
}
(3 . 2 5 )
I t is shown i n Section 3 .3 . 2 that t h e c ross term NJ b (T , w ) is given by
Xa b ( T , W ) = O ,
T 2d w here C(u) , S(u ) are F r e sn el integrals (see A p p e n d ix 3 ) arid
UI = U2 =
J( ; ) [ 1 � l J ( : ) [ 1 .- � ] b 2
b 2
+
2 T
2 T .
0
0 , Section 7.6 (Equation 7 42 ) leads to .
CaUw) = F;Uw) Note particularly that F:(jw) i s complex.
(4 .20)
not equal to FaC -jw), since faCt) is
Signal processing methods
64
Using Section 6.7 (Equation 6 . 1 7 ) with Equation 4 .20 gives
&it) = r:c -t)
(4 .2 1 )
It follow s from Equation 4.2 1 that the outpu t from a matched filter receiver can be obtaine d from Equation 4. 1 7 by replacing b(z) by a*(-z ), hence
f a(z)a* (z+x - t)ej (wo - w . - w v + Y)z dz 00
hi t) = !ej (t - x )w .
(4.22)
The integral in Equ ation 4.22 will be re cognised (Equ a tion 2 . 1 4) as [-(t - x), -Cwo - W I - Wv +y)] . Hence , the outpu t R F envelope is given by X
(4 .23 ) Equat ion 4 .2 3 (Equat ion 2 . 1 8 ).
follows
from
4.22
by
virtue
o f Section 2 .2
4. 6 PRA CTIC A L FOU RIER TR A N SFO R M C A L C UL A TO R S
The calcu lator u se d i n t h e F ou rier t ran sform receive r i s requi re d t o fmd
r1{ t) e - jwt dt
b
a t a selected nu mb e r of valu es of w . The param e t e rs n an d k represe nt t h e number o f processed pu lses and the pulse repe tition frequency p e ri o d , resp ec t ively. Once one bat ch of pulses h as been p rocessed it is required that the in tegrat ors be re se t , ready to pr ocess the ne x t b at ch . The direct m e t h o d o f im plementin g the calcu lat o r w ould b e to expand the a b ove integral to give
J
nk
f( t ) cos( w t) d t j -
j
nk
1{ t ) sin( wt ) dt
Since it is t he modulus (or t he squ are of t he modulus) of t he a bove sum wh ich is require d , a direct imp lemen tation me thod w ould be by the system illustrate d in Fig. 4 . 1 7 . I f p recision requiremen ts dictated calculat ion a t m frequencies it would either be n e ce ssary t o use m parallel channels (ea c h with a different value o f w) , for a fast read ou t , or t 9 t ake m times as long by using one chan nel m time s . T h e system of F ig . 4 . 1 7 could be replace d by any o f the well know n syste ms u se d fo r spec t rum analysis. Owin g t o the rapid a dvance s in in tegrated circuit technique s a part icula rly a t t ractive m e thod m ight be to sam p le the outpu t signal from the re ceiver mu ltiplie r and then fee d
65
Signal processing methods ,...---- x
Square
cos ( w t )
Os c i l l a t o r
In put -
Output
90
·
P ha s e - s h i f t
s i n ( wt )
L..---
x
Square
Fig. 4. 1 7. D irect implementation of F. T. calcu lator
t he samples t o a digital system for calc ulation . A parallel method of calculat ion , making the m ost of finite logi c speeds , w oul d be one of the FFT algorithm s discu sse d in Section 6 . 1 4 . The classic sp ect rum analy sis sche me , using a narrow band I F amp lifier i n conjun ction with a sw ept local oscillator i s esse n t iall y a series scheme sin ce the oscillato r must sw eep slow ly fo r a ccurate results. The equ ivalent parallel scheme would employ a bank of narrow band fil ters each tuned t o a different frequency . The filter bandwi d ths should be approximately equal to l / ( n k) H z .
4 . 7 THE
M ATHE M ATI C A L
TREA T M E N T
OF
THE
TRA N S FO R M REC EI V ER
With reference t o F ig. 4.7 the t ran smitted signal i s de fined as
f i t, wo ) = [a C t) [ cos [ wo t + ¢( t ) ] = Re{a(t) ei wo t} The inputs t o t he main m ixe r a re
Je t - x,
Wo
+ y)
and
j{t - tv, Wo - WIJ
F O U R I ER
66
Signal processing methods
Hence the
product Jt t - x ,
Wo
+ y) f(t - t",
Wo - WL) is equal t o
k7(t - x) 1 k7(t - tv) 1 cos [ (w o +y) (t - x) + ¢(t - x) ]
cos [( Wo - wd {t - tv)+¢(t - t v)]
= t la{t - x) 1 kz(t - tv) 1
{cos [ (y + wL)t + ¢(t - x) - ¢( t - tv) +(w o - wdtv -{Wo+y)x ]
+ cos[(2wo - WL +y )t+¢(t - x) + ¢(t - tv) -Cwo - WL)tv -{ WO+y)x] ) The mixer outpu t i s assumed t o be the first (low fre quency) tenn i . e . x o + y)x L L a la( t - x) l Ia( t - tv)1 e W ( t - ) e - j ¢ ( t - tv) ej (y + w )t � (wo - W )tv e j ( w + +a la( t - x ) 1 kz( t - tv) I e -j ct>(t - x) � ¢ ( t - tv) e -j (y W L ) t e -j (w o - W L )tv ej (w o + y )x
w hich is e q ual t o
)x M a( t - x ) a * ( t - t v) eJ (y + w Ot eJ (w o - wO tv e - j (w o + y + a *( t - x ) a( t - t v) e -j(y + w O t e -j ( wo - WL ) t v ej (w o + Y)X }
(4 . 24)
But 00
J a( t - x) a * ( t - tv) e - j w t d t
.?" { a( t - x) a * ( I - tv) } =
00
J a( l ) a *( 1 + x - tv) e -j(t + x ) w d t
=
_
00
( 4 . 2 5)
= e - jwx X Ix - tv, w ]
Si m i l arly
( 4 . 26 ) wh e r e
J a( t ) a * ( t + T) e - j w t dt 00
X( T , w) = _
00
Using Equations 4 . 2 5 a n d 4 . 2 6 t h e Fourier t ransform o f 4 .2 4 can b e w rit t en ! { e j ( w o - w L ) t v e - j ( w + w o - wOx X [ x - tv , W - Y - W L] +
For
a
e - j ( w + W o + y ) t v e j (wo + y )x X [ tv - x , w + y + wd }
real b a n dpass fun c t ion h ( t ) 9"{h ( t) } = HUw) = IliU w ) I ej8 (j w )
(4 . 2 7)
67
Signal processing methods Hence
h(t)
=
1 2 rr
I
00
I
00
HUw) e jw t dw
_ 00
=
0
2 IHUw) I cos[wt +euw)] df
If h(t) were applied to a physical 'Fou rier transform calcu lator' (e .g. a spectrum analy ser) the outpu t wavefo rm w ould represent the amplitude distribution of the various cosine components, measu red in V/Hz , i.e . 2 IHG w)l. It can be seen , from the ab ove remarks , t hat the outpu t from the F ou rier transform receiver w ill be given by twic e the mo dulus of the exp ression 4.2 7 , when evalu ated for w > O. If W L i s sufficiently high, the contribution from X [ tv - x, w + y + WL ] will be negligible leading to an ou tput waveform given by
4 . 7 . 1 T h e effec t o f weigh ting
The e ffect of t ime weighting in a F ou rie r t ransform rece ive r w ill now be compared with that o f fre quency weightin g in a matched fi lter receive r . To investigate the e ffec t o f mUltiplying the signal at the in pu t o f the Fourie r transform calc ulator (Fig. 4 . 7 ) by the func tion met - tv) , i t is necessary to put m et - tv) as a mul tipl i e r before expre ssion 4 . 2 4 . The resu ltan t exp ression for the Fourier t ransform modulus can be written as
II_Ia(t, - x - z)a*( - z) m( -z )
e
-
j ( W L + y - w ), dz
1
( 4 . 28)
The e ffect of frequency weighting in a matched filte r receive r can be investigated b y assuming that t h e comp lex analytic impulse resp onse corresponding to the receiver filter is a * ( -t)e(t) e j w " rather than a*(-t) ej w1 t. Replacing bet - x -. z ) in Section 4 .5 (Equa tion 4 . 1 5 ) by a *(z + x - t)e(t - x - z ) and subsequen t simplification gives the outpu t RF enve lope as
!/ _f a(t - x - z)a*( - z )c(z ) e -j (wo - wl + y - w v)z / dz
Expressions 4 . 2 8 and 4 . 29 are discussed in Sec t i on 4 . 2 . 1 .
(4 . 29)
Chapter 5 SOME E XAMPLE S O F SIGNAL P ROCE S SI NG M ETHOD S
Exampl es are give n in this chap t e r of the methods used to pro cess two t ypes o f modulated signa l . The resu lting p recision and re solution are d isc ussed in t e rms of the quantitative d e finitions g iven i n Section 1 04 .
5 . 1 THE
RECTA N G U L AR P U LSE
W ITH CONSTA N T
C A R R IER
FREQ U E N C Y
A re c t a ng u l a r p u l s e w i t h consta n t carrier frequency c a n be represe nted by
fi t ) =
r -0-1 cos( wot)
(5 . 1 )
Hence , fo r W o sufficiently h igh ( 5 .2)
I
I
The appro p riate unce rt a i n t y fu nct ion was evaluat e d in Sectio n 3. wh e re it was show n ( Equa t io n s 3 . 1 0 and 3 . I ) that t h e inherent range and ve l o c i t y precision s are r = 0 . 5 cd ( 5 .3 ) v
= 0 .6
c
f� d
( S A)
This pe rfo r m a nce can be ach ieved by using a n unweighted matched fi I t e r or Fourier t r ansform receive r.
5. 1 . 1
The Fourier transform receiver
The F o u ri e r t ransfo rm receive r of Fig. 4 . 7 is o ft e n used with this wave form in prac t i cal radar sy stems. So metimes the delay tv is se t at a con stant value and the system used t o me asu re the velocities o f t a rget s 68
Some examples of signal processing methods
69
in a fixed range bracke t . The Fourier transform calculator is often implemented as a bank of narrowband filters each tuned to a different centre frequency . To design such a system, given a specificat ion of the desired range and velocity precision, one should choose the pulse length from Equation 5 . 3 and the carrier frequency from Equation 5 .4 . This procedure will allow the fastest readout of velocity information- the required Fourier transform integration time being equ al to the pulse duration d. In the case o f a filter bank the filter bandwidth (in Hz) should be equal to abo ut l /d. Equation 3 . 5 gives _
I X( O , w) l - d
I
sin( wei12)
(weiI2)
I
( 5 . 5)
To avoid loss of p recisio n d ue to the inte rference of the two unce rtainty func t ions correspond ing to a given target (see F ig. 4.8) it is necessary to choose an offset frequency WL such that I x(O, w) 1 is ve ry small for w equal t o 2 wL ' The expression in Equation 5 . 5 will be more than 6 0 d B dow n for wei > 2000, thus a sui t able choice for WL (and hence the I F frequency) would be � 1 000 ( 5 .6 )t wL :::- dThe pulse repetition frequency period k woul d b e c ho sen high enough to eliminat e seco nd t ime round errors. Al tho ugh it is t heoret ically po ssible to carry out the above p rocedure, in m any cases the required carrier frequency will be too high fo r p ractical implementatio n . Comb ining Equat ions 5 .3 and 5 .4 give s the required carrier frequency as 0.3 fo = vr
c2
( 5 . 7)
Taking an uppe r prac tical limit of 1 00 GHz for fo (in t e rms of lase rs the upper limit could b e as high as 1 06 G Hz), gives vr
( 5 8)
� 0. 1
.
The units of v and r are mile/s and miles respectivel y . By w a y of example, a p recision specificat ion of 1 mile a nd 3 60 mph would lead to d = 1 0 . 8 J1s , t
fo = 1 00 GHz,
I t is usual to em ploy a lower value o f I F
h � 1 5 MHz (w V
than that indicated by
Equation 5 . 6 . This is permissible beca use the (sin x )/x expression of Eq u a t ion 5 . 5
applies to a pu lse with z ero rise and fall times.
Some examples of igllal pr ocessing metJlU ds
70
b d l f i l t e r i n t h e fi l t e r b a n k w u u l d r e q u i re a b a n d w i d t h o f a bo u t 1 00 k i l l . I f t he
r eq u i r e d c a r r i e r fr e q u e n c y c a n n o t b e a c h i e v e d a c u m b i n e d
r a n g e a n d v e l o c i t y p r e c i s i o n s p e c i fi c a t i o n c a n b e s a t i s f i e d b y u s i n g t h e
i m p r o ve m e n t
in
Doppl e r
p r ec i sio n
d u e t o r e pe t i t i o n . T o e x p l o i t t h e
p r e c i s i o n i m p ro v e m e n t i t i s o n l y n e c e ssa ry t o i n c r e a se t h e c a l c u l a t o r i n t eg r a t i o n t i m e t o 1 1 k . w h i c h g i ve s a n i n t e g r a t i o n o v e r
II
p u l se s . T h e
fi l t e r b a n k b a n d w i d t h s s h o u l d b e r ed u c e d t o a p p ro x i m a t e l y I
1( lIk ) H z .
T h e I l e w e x p r e s s i o l l fo r v e l o c i t y p r e c i s i o n ( r e p l a c i n g E q u a t i o n 5 .4 ) i s v =
0.6
j'
c
II � I
k
0 11 '
( 5 .9 )
e q u a t i o n 5 .9
i n d i c a t e s t h a t v e l ( ) c i t y p r e c I si o n c a n be b y l e ng t h e n i n g t h e i n t e g r a t i o n t i m e it s h o u l d b e n o t e d t h a t t h e r e ad o u t t i m e be co m e s p ro g ressive l y l o n g e r . h e n ce c h a n g e s i n v e l o c i t y w i l l t ake l o n g e r t o be c o me a p p a r e n t . A l t hough
i m p r o v e d i n d e fi n i t e l y
[O, w� /
E nv e l o p e of X
� ./ r
...
-
" , "
.,., /
_ _
-
./
, I ' I
" " , , , , , , , , I
:
, ,
,
, , , , , , '
.:�=:� .:: -=- r= -
E n velope of _ _ _
.
1' , . • "
" I' " .'
" " "
x [o. w - yJ
---:: ----....--.. - - - - / ,
� -=
"
;,
"
" " , , , , , ,
' ' ' ' ' '
-y
o
y
1
'2 PRF
w -
Fig. 5. 1 . Th e sli m of '« (0, w - y) a nd x((), the wciRh til/g fll nc t io n du e to r ep etit io n
w
, , , , , , , , , ,
\
+ y ) when bo th are multiplied by
I f o n e is rel y i ng u p o n t h e Doppl e r ' ba r' e ffe c t t o give
v eloc i t y
t h e syst em c a n be s i m p l i fied b y o m i t t i n g t he o ffse t a h o mo d y n e receive r- t h e p r i c e p a i d i s a l o s s o f k n ow l e d ge o f t h e sign o f v e l o c i t y c h a nges. R e fe re n c e t o F ig , 5 . 1 w i l l show I h a t . for t h i s spec ial case . o m i ss i o n o f t h e o ffse t o sc i l l a t o r d o e s n o t l e a d t o a l o s s o f p r e c i s i o n a s l o ng a s t he ma x i m u m Dopple r fre q ue n cy i s less t h a n h a l f t h e p u l se re p e t i t io n fr e q ue n c y . precision
osci l l a t o r - l e a d i ng t o
Some examples of signal processing methods
71
5.1.2 The matched filter receiver
It will be seen from the above discussion that the implementation of a Fourier transform receiver for a constant carrier rectangular pulse is relatively straightforward. The physically realisable form of the matched filter receiver would need a filter with an impulse response . given by a suitably delayed version off(-t). Reference to Equation 5.1 shows that, for this particular waveform, such an impulse response would be given by the expression for
f(t) itself. The corresponding filter
transfer function would be given by
HG*[j( -w -wo)] +G [j(w-wo)]}
(5.1 Oa)
where (5.1 Ob) The matched filter would actually be implemented at a suitably high IF frequency wI-rather than at wo-and used in the system of Fig. 4.1. It turns out that it is not particularly easy to construct such a filter (a method is given by Skolnik
[14] ) so most practical systems would use a 4.1 scheme.
non-matched filter in the Fig.
As an example of a non-matched system the performance using a single tuned circuit type of filter will now be considered. It is assumed that the complex analytic form of the filter transfer function is (S.11a) where
BUw)
=
a . a+Jw
(S.11b)
The corresponding baseband impulse response is
b(t)
=
a e -at u(t)
(5.12)
It is shown below that use of the above filter leads to a receiver
output envelope of the same shape as a cut (parallel to the r axis) through the following 3-dimensional shape
0,
11jI(r, w)i
rd
72
Some examples of signal processing methods
To evaluate the above expression it is necessary to decide upon a value for the product ad. It is commo n practice to make the 3 dB bandwidth of the filter (in Hz) equal to the reciprocal of the pulse
width. The 3 dB bandwidth of the RF filter is equal to 2a rad/s, thus = 1T.
the above philosophy implies ad
An alternative approach would be to use a filter having the same noise bandwidth as the matched filter. It is shown below that this =
implies ad
2.
Figures 5.2 and 5.3 show the effects of the above choices of filter bandwidth upon the delay and Doppler performance; matched filter responses are also shown for comparison. The results are summarised in Table 5.1 in which the delay figures are values of figures are values of wd/21T.
T /d
and the Doppler
It can be seen that either choice of bandwidth leads to results which are not too different from those obtained with a matched filter. Neither
o
--------.--_
/
"
- 20
/
I
/
/'
-
"- "-
-"
Single tuned circuit
----- Matched filter
'\
I
dB
-40
-60 �------�--� o
o
-20
------ - --,-_ _
I , I
/
I
"" I
'"
"- "-
/
""
\
\ \
\
\
dBl -40
-60 �------�--� o 2 3 4
'tId
(b)
(h)
Fig 5.2. Delay performance (a) ad = rr (i.e. IF bandwidth lid Hz) =
\ /" .... , \,' \ /'-, \: \\ '/ \\ '/._\\ \1 I , \ II I, II II \
-20 dB
-40
-60
I I I
:
I
II
II II \I
U •
I
H
: I
:
I I
/-\\
I
\ \
II II \I
\
"
I
/' ....,
I I II 1/
i
I
1/
I,,'
\
\
I
/
-
\
"
\
I
\/ II II
\ , I I
., II 1/
,
/-,
\ \
\
\ \
I
I
: I , : �I �I : I I I I I I �---L----�--L---�--� o 2 4 6 eWd/2n (b)
Fig. 5.3. Doppler performance (a) ad (b) ad rr (i.e. IF bandwidth l id Hz) =
=
2 (ie.
IF bandwidth
=
2/rrd Hz
=
Table
5.1
Matched filter Delay precision dB delay resolution Doppler precision Comparative SiN (dB)
60
1
2
1.2
o
ad
=
2
1.05 4.5
1.4 -1.26
ad
=
rr
1
3.2
1.6
-2.34
the matched filter nor the single tuned circuit will give good Doppler resolution, since both IX(T, w)1 and IW(T, w)1 fall off as l/wfor large values of w.
5.1.3 Derivation of results
The equations leading to Figs 5.2 and 5.3 will now be derived.
]4
Some examples of signal processing methods
Single pole filter The expression for 11/1(7, w)1 can be derived by using Equation 5. 12 with Section 4.5 (Equation 4. 15 ); this gives the output RF envelope as
Remembering that u(t - x assuming that (wo-
=
For 0 < (t
-
=
=
-
z) is equal to zero for z> (t - x) and
l
�
!!..- e-d
-f
Laplace and Fourier transforms
93
The first expression in the right-hand limit will be recognised as 1TO sgn(t) =
0,
t = 0
-I,
t(-o
{
I _r2
}
2rr(l-2rcost+r2)
(6.29)
The most useful characteristic of the delta function is its sifting property,
f h(x)o(t -x) dx = h(t) 00
(6.30)
Equation 6.30 follows from Equation 6.2 7, remembering that oCt - x) is zero except at the point x = t. An alternative representation of the delta function can be obtained by considering its Fourier transform.
J oCt) 00
.7{0(t)} =
e-jwt
dt = 1
(6.31)
The result of Equation 6.31 follows from Equation 6.30 by noting that e-jwt =
1,
for t
=
0
Equation 6.31 can also be written in the form of an inverse Fourier transform giving
1 oCt) = _
2rr
f
00
ej wt
dw
(6.32)
Lap/ace and Fourier transforms
96
Equation 6 .32 may be used to obtain another useful result-the interpretation of o(a t + b).
o(at +b) = +-�rr f
00
ej(at+b)W dw
Changing the dummy variable to wla gives
o(ar+b) = �1rra f
ej(t+bja)w
o(ar+ b) = -=-� 2rra f
ei(t+b ja)w dw,
a>O
00
dw.
_ 00
Hence
a c. Equation 6.34 may be written in the form
jR (t) where
II (t) f2(t)
= =
= r
(6 .35)
II ( ) +/ir) - jet)
f(t) + e -Ckf{r k) + e -2ckf(t _.
-
2k) + ...
f(t) + e -Ckj(t + k) + e -2Ckf(t + 2k) + .. .
Laplace and Fourier transforms
97
It will be noted that the factor e-Inlck ensures that the 'pulses' decrease in amplitude each side of the point t = O. This effect can be made negligible by allowing c to tend to zero. From Equation 6.35
+e-cke-pk +e-2ck e-2pk +... ]
F 1 (p) = F(p)[ 1
(6.36 ) (6.37)
Equations 6.36 and 6.37 are geometric progressions which summed to infinity give
(6.38)
F2(P) =
F(p)
I -e -ck epk'
le-ckepkl 1
(6.39)
-c and Re(p)< c, respectively. Hence
FR(P) = F(P) where Re(p) =
[ 1 -e-;k e-pk + 1 �Ck pk -1] e
-e
(6.40)
-c
< Re(p) < c. Since the strip of convergence includes O,fR(t) has a Fourier transform given by
FRUw) = rC'U ' W)
[ 1 -e -ck1 e-J"
1
+
-e e
"
-
1]
(6.41)
Equation 6.4 1 simplifies to
FRUW) = FUw)
e-ck
[1_2e-lc�c:�2:: +e-2Ck] + 0, r 1 - O.
(6.42)
21TFUw) L 8(wk-21Tn)
(6.43)
Putting = r it is seen that for c -+ and 6.29 are compared, it follows that
c-+O {FRUW)} Lt
=
"
-+
If Equations 6.42
n =--
The Fourier transform inversion integral gives
(6 .44) Substituting Equation 6.43 in 6.44 and using Equation 6.33 gives
c ��
0
{fRet)}
=
-
t f FUw) ejwr f
-
-
)
8 (w - 21Tn dw k \ n=--
(6 .45)
iwo t}
e
.
The
'
{f(��)}
expression
-
formed
from
J .
I
J
'
Fourier
transform
FUw) eJwt dw
F(�j27Tt)
2n
1
W
()
T
J
F,(x)G,(w� x)dx
J
w F,(w) ei t dw F,(�27Tt)
27T
..'F -'{F, (w)G, (w)}
27
F, (w �wo)
e
� iWT F (w) ,
F
!at 'a
1
( J
F2(f) F2 (�t)
J
F 2 (x)G2 (f� x) dx
i e 27T!t df
§. -, {F 2 (f)G 2 (f)}
00
��)
�i27T!T
F2 (f)
() f F2 a
F2 f �
e
!at
7T t
in this way does not necessarily represent the Laplace transform
FUx)G[j(w�x)J dx
.'7 -'{FUw)GUw)}
27T
'"
F[j(w � wo)J
e�iWT FUw)
.
( )
1 [F W Ja j(i
the
throughout the strip of convergence.
t
.�-'{f(f)} = '7
f(t)
f(x)g(t -x)dx
00
J
.�{f(t)g(t)}
) .� (rU
.� {f(t� T)}
.� {f(at)}
F2* (�f)
F* ( �jw)
.'F {ru>} F,*( �w)
F2 �j
F, (�jp)t
F(p)t
£ {f(t)}
( f)
F2(f)
F,(w)
FUw)
Result
.�{f(t)}
Property
Table 6.1
� ""
....
�
�;::s
�.
� l:::
� � I::l ;::s I::l...
E)
t--o
-§
\0 00
Laplace and Fourier transforms
99
Applying the result of Equation 6.30 to 6.45 leads to the Fourier series Lt {fRet)} = 1 k c-+o
I: [j 2rrn ]ej(21Tn1k)t
n=-oo
F
_
k
(6 .46 )
6.10 FOURIER TRANSFORM NOTATION
The expression defined as the Fourier transform in Section 6 .4 can be called FQw), F(w) or F(f) depending upon the author. Although the differences are mathematically trivial they can lead to confusion when theorems are quoted. For the convenience of the reader, various results are quoted in Table 6 .1 in the three 'languages'. DEFINITION g;- {lU)}
= FUw) = F I( w) = F2(f) =
where
6.11 USING
f fit) e -jwt dt 00
w = 2rrf.
THE
P-MULTlPLIED
LAPLACE
TRANSFORM
NOTATION
Van der Pol and Bremmer [8] have given extensive lists of Laplace transform theorems and results under the respective headings o f 'grammar' and 'dictionary'. A s the information is given i n the p-multiplied form it is useful to be able to translate it into the standard notation. A typical 'dictionary' entry is valid for 0 < Re(p) < 00
u ( t) --+ 1,
To convert this to the standard notation, simply divide the given Laplace transform by p. Thus
£{u(t)}
=
1
-
p
,
valid for 0 < Re(p) 2B, i.e. the normal sampling condition. The implications of (2), from the radar point of view, can be obtained from Sections 4.2 and 4.3 by noting that the signal processing time is (N - I )T seconds. If the above conditions are satisfied then FGw) can be evaluated for N values of w by using Equations 6.67 and 6.68 in conjunction with Fig. 6.2.
104
Laplace and Fourier transforms
o f(Q)
o
f---(b)
Fig. fl. 2. (a) the form of FUw) in Equation 6.64; (b) the form ofrFs(jw) in Equation 6. fl4
In a similar man n e r , the inverse Fo u r i e r t r a n s form of FUw) is represented by the seque nce {ail if the sequence {An} is fo rmed from samples of FUw) t aken � Hz apart . The necessary c o n d i t ions a re
( I ) I / � must be grea t er than t he t o t al d u r a t i o n o f f(t). (2) The nu mber of samples (N) must be such that the truncated spect rum, of width (N I )� Hz, has esse n t ially the same inve rse -
Fourier transform as FUw).
The res ult is summarised by Equ a t io n s 6.73 a n d 6.74. 6.13.1
Deriva tion of Fourier transform results
In the t rea t ment which fol l ows sampled signal s are represe n t e d by t rains of delta funct ions where as re al sampl ed systems employ n a rrow pulses. It ca n e asily be shown t hat samp l i n g w i t h a pulse of w i d t h d seconds is equ ivale n t t o impul sive sampl in g, provided that i n t e rest i s concen t ra t ed a t frequencies m u c h lower than I/d Hz and that a st re ngth of d (ra t he r t han u n i t y ) is all oc a t e d to the sampling d e l t a funct i o n s. The la t t e r m od i fication also serves to restore the correct dimensions ( t he d i mensions o f the d e l t a fu nct ion are t i me-I).
Lap/ace and Fourier transforms
105
Denoting the impulsively sampied form of f( t) by h(t), it follows that
fit) = f(t) 2:: oCt -i7) ;=-00
=
f(t) 1 7
�
i=-oo
ei(21Ti/r)t
(6.63)
The second half of Equation 6.63 follows from Equation 6.46. Hence the Fourier transform is given by
(6.64) Equation 6.64 is illustrated by Fig. 6.2 for the general case of f( t) complex. It can be seen that provided 1/7 is greater than the total bandwidth (Hz) of FGw), no spectral overlap occurs in Fs Gw) . Hence FGw) can be obtained by evaluating 7 FsGw). If, for an appropriate value of time displacement ( to ) , and large enough N, h(t) can be represented by
fs(t) Then FsUw)
� f(t) L oCt - to N-J ;=0
00
� f j(t) ,2:: oCt-to N -J
_00
=
1=0
N-l
(6.65)
-iT) e-iwt dt
2:: f(to +i7) e-iwto
;=0
- i7)
e-iw;r
(6.66)
Hence
Thus, if ai =
then 7 Fs
j(to +i7)
(J' 21Tn) � N7
e-j(2rrnto/Nr) 7A n
(6.67)
(6.68)
Note that it is often possible to choose the time origin such that to is zero .
Laplace and Fourier transforms
10 6 6.13.2
Derivation of inverse Fourier transform results
A similar proce d u r e can b e carried o u t fo r the case o f the inverse F o u r i e r t ransfo rm. By re-a r ranging Equations 6.34 and 6.43 it follows tha t if
F/jw)
=
FU w)
L
n=-oo
o(w-27Tn�)
(6.69)
then
(6 . 70) The units o f � are Hz. The rela tion ship given by Equation 6.70 is of the same form as that illust rat ed by Fig. 6.2, showing that .I(t) can b e o b t ained by evaluating 27T� .I�(t), provided that I / � is gre a t er than t he t o t al durat i o n o f f(t). The same reasoning used for t he t ime fu n c t ion case shows that if, for a n a ppropriat e val ue o f Wo a n d large en ough N , FrUw) can be represen ted by
Fr(jw) � FUw)
N -I
L
1/=0
o(w-wo - 27Tn�)
(6 . 7 1 )
t hen
Thus, if
(6.73 ) then
(6.74)
6.14
THE FAST FOURIER TRANSFORM (FFT) PROCESS
The fast Fourier tran sform (F FT) process is an e fficie n t method o f evaluat ing t he discre t e Fourier t ransform; it i s based u p o n two t h e ore ms which are de rive d in Sectio n 6.15.4. Reference t o the OFT de finition re l a tionship (Equa t i o n 6.47) shows t hat N operat io ns are req uired t o calculate a si ngle value o f the sequence {An}. Th us i t would appear that N2 ope rations are required to cal culate the full N val ues o f the sequence. H oweve r , it is not always
Lap/ace and Fourier transforms
107
necessary to carry out N'l operations. It is shown in Section 6.15.4 ( Equations 6.95 and 6 .96) that, if N is an even number (6.75 ) (6.76) Equations 6.75 and 6.76 are illustrated by Fig. 6 . 3, for the case of N = 8 . The multiplier W is equal to exp (-j2rr/N). It can be seen that the original N2 = 64 operations has been reduced to 2 x (!N)2 + N = 40 operations; furthermore there is no reason why the two 4-point DFT's should be evaluated by the direct method. If the FFT process is repeated each 4-point DFT will need 12 operations rather than 16. Thus the number of operations could be reduced from 64 to a total o f 3 2. The saving in processing time becomes more dramatic as N is increased. If N = 2 m the FFT process needs a total of (m + 1)N operations, rather than N2 .
ao a, az a3
bO
(a, + as)
b,
(aZ+a6 )
bz
(a3 +a7)
b3
8 8
4 Point OFT
a6
Fig. 6.3.
The
exp(-j21T/8)
reduction
An alternative 6.99 6.101.
Z
C3
(a3 - a7)
a7
=
C
OFT
of
an
procedure
8-point
DFT
to
two
Az
83 0
AO
=
8
C,
4 Point
(az - a6)
,
=
=
C
(a, - as)
as
0
Z
(a O-a4)
a4
W =
(ao + a4 )
A4 A6 =
A, =
A3 =
AS =
A7
4-point DFT's
can be obtained using Equations
-
(6.7 7 ) Hence, if hi = a2i and Ci = a2i+l, then A m = Bm + e-j(21TmIN)Cm,
A m+Np = Bm - e -j(2mn IN) Cm,
o �m �N/2-1
(6.78)
o �m �N/2-1
(6.79)
108
Lap/ace and Fourier transforms
The above relationships (for the case N = 8) are illustrated by Fig. 6.4. It can be seen that this method is a 'reverse' form of the first method.
a
b
2
a
b
4
a
1
B 1
+
0
B
+
0
B 3
+
0 3
B
-0
C
5
OFT
o
C,
3
4 Poin t
3
C
,
a
B +0 = 0 0
2
b
a 5 a
0
b
a o
2
B
4 Point
Fig.
6.4.
o 1
2
0
-0
1
B -0 2 2
OFT
( 3
a 7
2
1
B
An altcrnativc reduction of an 8-point
DFT W
=
3
exp(
-0
3
=
=
=
=
=
=
=
A
o
A 1 A 2 A
3
A 4 A A A
5 5 7
-j2rr/8)
Eq u ati o n s 6.75-6.79 a p p l y t o t h e cal cul a t ion of the OFT; very
s i m i l a r r e s u l t s a re a p p l i c a b l e to t h e c a l c u l a t io n o f the inverse OFT. The r e l e v a nt equati o n s a r e 6.97, 6.98.
6.102,6.103 and 6.104. To convert
Figs 6.3 a nd 6.4 to th e i n ve rse OFT c a l c u l a t i o n it is o n l y necessary to (1) Re p l ace ai by Ai a n d Ai by 2 ai. (2) Re p l ace th e DFT boxes by inve rse OFT boxe s.
(3) Cha nge the W m u ltipl i e r to W = expU27TjN), i.e. W = expU27T/8),
forN=8 .
Fo r
fu rth e r
i n fo rm a t i o n
F FT
a b out the
p ro c e s s
Gol d and
see
Ra d e r 19] .
6.15 PROOFS OF THE PROPERTIES OF THE DFT AND FFT
The formal p ro o fs of the
}
OFT tra n s fo r m p rope r t y a n d t he OFT
conv o l uti on th e o r e m d e p e nd u pon th e fol l ow i n g r e l a t i o n s hi p .V
�
I
11/=0
ei(2 rrkjN)J/1
O. ±N, ±2N, etc .
,V,
k
0,
k = a n y other i ntege r
=
In th e first case t h e a rg u m e nt o f th e summation i s
I fo r a l l
(6.80) m,
hence the
109
Laplace and Foun'er transforms
sum is clearly N. In the second case the summation is a geometric progression and it follows that N-l j21Tk 1_e----,L ej(21Tk/N)m = ----,N) i m =0 1 e (21Tk/ _
which is equal. to zero, for
k *-
0, ±N, etc.
6.15.1 The basic DFT property
Using the defini tion of the DFT given in Section 6.12, one can write _I
fl} N
N -I N-I
1
e-i(21T n/N)r ei(21Ti /N)n { f'jJ N(aJ} - N L L ar n=O r=O _
_
-
1
N-\ i ar L ei!21T( - r)/N-)n r=O n=O N-\
NL
(6.8 1)
Equation 6.8 0 shows that the inner summation is zero unless (i - r) is equal to 0, ±N, etc . Hence Equation 6.81 becomes (6.82) It also follows that ai = ai+N= ai+2N , etc. A similar procedure can be used to show that fl}N{ fl},-\/(An)}
=
(6.83)
An
6.15.2 The DFT convolution theorem
To prove the convolution theorem
I N-I LAm Bn N
m
-
m=O
iV-I
=
=
I L
N m=O
J.-
N-I
N-I
L
i=O N-I
L L
N i=O
k=O
aie-i(21T111/N)i
N-\
L bke-il21T(n-m)/Nlk
k=O
aibk e -i(21TI1/N)k
N-I
L
111=0
e -iI21T(i-k)/Nlm
Hence, using Equation 6.8 0
1 N-I N-I . . -J(21T1//N)/ = f!j {a/·b/·} a· ·e = � b B - � Am n-m / / N L N 11L i=O =0
1
(6.84)
110
Laplace and Fourier transforms
A simil ar procedure can be used to show that
l �N {AnBn} = For the case bi
=
N-l
L
m=O
ambi-m
COS(21Ti/N)
N-l = Bn .L 1=0
=
!
�
N-l
! iL
Hence, using Equation
Bn =
=O
( 6.85 )
]
ej(2rri/N) + e-j(2rri/N) e-j(2rrn/N)i
d[2rr(l -n)/Nli.+ e-j(21T(1 +n)/Nli
6.80
{ 12
NI2,
n = 1, 1±N, etc.
N ,
n = -1, -1±N, etc .
0,
otherwise
Similarly, for b i = sin(21TilN)
Bn
=
n = I, }±N, etc .
{-)N)N121,2,
n =
-1, -1±N, etc.
otherwise
0,
6.15.3 The relationship between the D FT and a delta function train
The proof of the relationship between the OFT and an infinite train of delta functions will now be given. Consider a delta function train f m(t) derived from a sequence {ai} of N sample values . Let fm(t) =
N \
i�-
( - ik'J)
aJ>k t
N
( 6 .86)
where
n = -oo
(6.87)
It can readily be shown that (6.88)
111
Laplace and Fourier transforms Hence
FmGw)
=
Since the
N-I
=
j
�
2rr / aj e-j(wik N) 821T/k(W) k
� (
8 W-
n= -oo
)
2rrn
2rr
k
k
f
N
l
j=O
21Tn/ j N) aj e-j(
i summation is periodic in n, it follows that
written in the form
FmGw) can be (6.89)
where An
By putting
w
=
N-J " L., a.I j=O
=
e-j(21Tn/N)j
2rrt, and using Equation
written in the form
F,nGw) =
1
k
N-J n
�
(6.90)
6.33,
Equation
6.89
can be
� �)
(6.91)
8J/dt)
(6
An8N/k
-
Using the relationship
G;-1{82ndw)}
=
�
2 C
it is possible to work backwards from Equation
which is equivalent to
fm(t) = If
Equation
6.93
N
-J
L
j=O
� - _·k) -
8k t
I
1 N -J
L
N N n=O
is compared with
6.89
92)
and say
An eK2ni/N)n
Equation
.
6.86
(6.93) (the defining
equation) it can be seen that the sample values are given by a· I
Thus Equations and its inverse.
6.90
=
and
- N-J
1 " ej(2ni/N)n L., An N n=O 6.94
(6.94)
form a logical definition for the
OFT
Laplace and Fourier transforms
112
6.15.4 Proof of the FFT theorems
The theorems leading to the FFT process will now be proved assuming that N is even. Equation 6.90 can be written in the form A
n
Nj2-I
L
=
i=O
N ai e-j(2rrn/ )i
+
N-I
L
i=Nj2
ai e-j(2rrnjN)i
which, with a change of variable in the second term, becomes A = n
=
Nj2 -I 2 " a·I e-j( rrnjN)i L i=O Nj2 -I
�
i=O
2) a·I+Nj 2 e-j(2rrnjN)(i+N/
- 1 -j(2rrnjN)i (a·I +a·I+Nj2e j1TI )e
" L
i=O
Hence A2r
A2r+1
Nj2-1
+ )'
NJ2 -I
" L
=
i=O
Nj2 -I
2 ijN) e-j]2rrrj( N j2) ]i (a i-ai+Nj2)e-j( rr
L
=
(a· +a· )e-j[2rrr/(NJ2)]i I I+N/2
i=O
(6.95)
(6.9 6)
A similar process, starting with Equation 6.94 gives a2r =
I
a2r+1 = -
�
Nj2 -I
N
" L
11
=0
NJ2-1
L
N
n=O
(A
n
+ An+NJ2)ej[2rrr/(N/2)]n
(2rrnjN)ej[2rrrj(NJ2)]n (AI1-An+Nj2) ei
(6 .97)
(6.98)
An alternative way of writing Equation 6.90 is AI1 =
NJ2-1
L
i=O
2 N)2i a2i e-j( rrl1 /
+
L:
NJ2-1 i=O
a2i+1 e-j(21mjN)(2i+1)
(6.99 ) Putting form A
bi = a 2i
and
Ci = a 2i+I,
Equation 6.99 can be written in the
-j( 2m n /N) C A 111 = Bm +e m,
e-j(2rrm/N)C , m + Nj2 - Bm m
o o
m
/2 - 1
(6.100)
m
/2 - 1
(6.101)
O t=0 tCt))]
(7.30)
f(t) = R e{o{ t) } = -! [ aCt)+a *(t) ]
(7.3 1 )
aCt)
Al so, from Equation 7. 1 5
f(t) = Re{fa(t) } = -! Ifa (t) +fa(t) ]
(7. 3 2)
From Section 6. 7
ff{g* Ct)} = G *(-jw) Th u s , Equations 7.3 1 and 7.3 2 give
FU w) = H dU w)+ d*(-jw) }
(7.3 3 )
FUw) = -! {Fa U w)+ [Fa * (-jw)] }
(7.34)
where
dUw)
=
,�{a(t) } ,
Fa G w)
= .?l" {fa(t) }
That Equations 7 . 3 3 and 7 . 34 d o not imply dOw) = FaOw), and hence aCt ) = fa Ct) can be seen by reference to Figs 7.1 and 7 . 2. Re-arrangi ng Equation 7.33 gives
slOw) = 2 FOw) - d *( -jw) For the special case of
Wo
high enough for
d Ow)
= 0,
w�o
it fol lows that
�/ (-jw )
=0 =
S'1 * ( -jw),
w�o
(7 .35 )
124
Hilbert transforms and complex analytic sig nals
and , from Equa t ion 7 . 3 3
FU w) = 0 ,
{
w=O
H e n c e , Equation 7 . 3 5 can b e w r i t t e n i n the fo rm
dG w ) =
2FG w ) ,
w>O
FG w ) .
w=0
0,
w (W2 + W4)
(7 .37b)
Note that the validity condition for Equation 7.3 7(b) is that the lowest frequency in f(t) must be greater than the highest frequency in g et). Since the spectral separation between let) and h(l) is equal to (2W4 - WI), the physical significance of the two signals is lost if WI> 2W4. However, the mathematical identity of let) and h(t) is still preserved by Equations 7.36 and 7.37.
EXAMPLE
A simple demonstration of the truth of Equations 7.36 and 7.37 can be obtained by considering single sinusoids. Let f(t) COS(W3t) and get)
=
co
=
s(w4r ). Then
j�(t)
&(r)
cos(wJr)+j
=
=
w ) r)
s in (
COS(W4r)+j sin(w4r)
Now
giving
1I(r)
=
l(r)
=
1 COS(WJ+W4)t 1 COS(WJ-W4)r
Thus
This can be seen
to be e q ual to �j�(r)gAr). Also
j�(r)g(�-(r) =
(OS(W3 -W4)r+j
sinew) - (4 )r
giving
Re{J�(r)g:(r)} j�(r)g�:(r) is only equal K{cos(wr)}:j: sin(wr) if w < o. Note that
to
=
21(r)
2Ia(r) if W3 > W4, this is because
127
Hilbert transforms and complex analytic signals 7.5.1 Proof of Equations 7.36 and 7.37
The general proof of the above results follows from the convolution theorem
�{ftt)g(t)}
=
f
I 2 1T
00
FOx)G[j(w-x)] dx
Expressing FOw), COw) in terms of complex analytic spectra, this becomes
ff{ftt)g(t)}
=
J
00
1
8 1T
{FaO x)+F;(-jx)}{Ca[j(w-x)]+C:[j(x-w)]}dx
Performing the multiplication yields
J
00
,9'{ftt)g(t)}
=
8I1T
+
1
8 1T
{FaO x)Ca[j(w-x)]+F;(-jx)C;[j(x-w)]} dx
J
00
{FaOx)Ca*[j(x-w)]+F;(-jx)Ca[j(w-x)]}
dx
_<X>
(7.38) For convenience of reference the first integral of Equation 7.38 will be defined as HOw) and the second integral as LOw). Thus Equation 7.38 becomes
.�{f(t)g(t)}
=
HOw)+ LOw)
(7.39)
If, in Equation 7.38, -w is substituted for w and the dummy variable is changed to -x it is clear that
H(-jw)
=
H * Ow)
L(-jw)
=
L * Ow)
Thus from Section 6.6, both HOw) and LOw) are Fourier transforms of real signals. The physical significance of HUw) and LOw) will now be shown. Using the convolution theorem (7 AO)
_i
<X>
y{fa(t)g;(t)}
=
211T
Fa U x)Ca[* j(x-w)] dx
(7 Al )
128
I 1 I
(.
Hilbert transforms and complex analytic signals
W2
I
I 1
'II
I
:W4
'jI 1
W3
'I·
(W, +W21
w-
Spectra involved in the multiplication > 0
7.4.
all
,_
X ------..
Fig.
,,
;
(W3+W41
w,
fa(t)ga(t). Note that
GaU(w - x) J is Ga( -jx ) moved to the right. for w
It can be seen from Fig. 7.4 that Equation 7.40 defines the spectrum of a complex analytic signal (since W3 + W4 > 0). Also
Re{!a(t)ga(t)}
1 ffa(t)ga(t) +fa'tt)g:(t)]
=
Hence
J F;\-ix)C;li(x - w)] dx} 00
+
Comparing the above with the first integral in Equation 7.38 shows .�
{Reffa(t)ga(t)]}
=
2HUw)
The above results, together with th e complex analytic nature of
!a(t)git), establishes the truth of Equations 7.36(a) and (b). The proof of Eq u ati o n 7.37(a) follows the same lines.
.7{fa (t)9 a"It)}
1 w2
(.
I I I.
(W3 - W4)
x_
Fig. 7. 5.
G�[i(x - w)J
Sp ectra
is
G�(jx)
I I'
.
I
1
I I
W,
I ( ' ( ·1, - W - W2 )
(
(W3,
'1
I
(w, tw21
( I
·1
4
w_ involved
in
the
multiplication
moved to the right for
w
> 0
fa( t)g; (t).
Note
that
Hilbert transforms and complex analytic signals
129
Figure 7.S shows that while g;{fa(t)g:(t)} defines an asymmetric spectrum which is lower in frequency than g;-{fa(t)ga(t)}, it is only one-sided if (W3 - W4 - W2 ) > O. This establishes Equation 7.3 7(b). 7.6 FILTERING EITHER REAL OR COMPLEX ANALYTIC SIGNALS
When working in terms of complex analytic signals it is essential to know how the results obtained relate to the corresponding operations using real signals. For the purposes of illustrating the effect of filtering it is convenient to define two hypothetical signal processors, shown in Fig. 7.6. The ASF (analytic signal former) converts a real signal into its complex analytic counterpart, while the RSF (real signal former) performs the inverse operation. f it)
----11
ASF
--o..---�_R_SF___' f-----. fit)
1-1
f It) a
=
fit)
+
jflt)
Fig. 7. 6. Hypothetical signal processors
It is also convenient to define the complex analytic transfer function
{
Ciiw) corresponding to a physical transfer function CUw). By definition
CaUw)
=:
2CUw),
w>O
CUw),
w=0
0,
w xll2b I
e i (w >w,}, de;
(7.63) From the definition of the Fourier transform 00
bet)
=
_I 21T
135
Hilbert transforms and complex analytic signals
00
Substituting Equation 7.64 for the inner integral in Equation 7.5 1 gives ha(t)=
1
ej1T/4 ejWo 4 rr
J(�) J EUx)b [ �: �] _
t+
00
+
which is Equation 7.52.
7.7.3
The case of high dispersion factor
Equation 7.52 can be evaluated numerically for specific functions. An example is given in Section 5.2.2. It is shown in Section 5.2 .4 that physical considerations are often such that the exponential term in the integral of Equation 7.52 can be considered equal to unity over the range of the other two functions. This leads to a substantial simplification; Equation 7.52 becomes
�rr ej·/4 e W ''J(�) {£(jx)b [;b �; t1 00
h,(t)"
j
+
+
dx
_
(7.65)
bet)
If the Fourier transform and its inverse are used to express in terms of and in terms of e(t), the integral in Equation 7.65 becomes
BUw),
00
II
j
1 e(y ) e- xy dy rr 2
EUw) 00
I BUw)
dw
ei[ (x/2b)+(W d/2b)+ tlw
dx
(7. 66)
Re-arranging expression 7.66 to put all the x variable under one integral gives 00
00
I
_00
e(y)
21rr I
ej(w/2b)-y)x dx
I BUw) 00
The integral in x will be recognised as Hence 7.67 can be written
I e(y)8 [2� -Y] I BUw) 00
_
00
a
5
l a- t\ b-t
-
expUat)
a>O
-j exp(jat)
sin(at) t
a>O
1 - cos(at) t
cos(at),
a>O
sin (at)
-get)
APPENDIX
e
g(t)
A SUMMARY OF THE MAIN NOTATION
The main notation is listed below. Any deviations are indicated in the appropriate sections of the text. Laplace transfonns
