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Unformatted text preview: 104 d Its Applications The Fourier Transform an otal strength 27m, d ‘ ' irn ulse of t
ulse. A Circular ring p and also comes u m mp g with annular slits ‘ 'n i .
34. Autocorrelation of n g n Optics in deatm scribed by 8(r  a), arises i
other fields. Show that 5(r  a) stir 8(r —~ a) :— (gﬂﬂl [l ,l (”21:32le 3(12). ' ' or examp ,
nci al features. F . . . .
rd 3 a why are these sin lanties unequal,
T 1‘ I he autocorrelation of Why does the ction of r and explain the p \ mph this fun finite at r a: 0 an «correlation become in an t #2“ z: 2‘ ' ‘ ‘ ' um value;
. hie exactly 2 a . M 20' what . the 1mm rn
why is the va 1 satrxgrtr I ‘ ”Ml
t are the va ue . f r two—dimensio
lily/2 ' 130,4. 31:)02/521‘25 it OCC 111'? (In this problem * "k St . d8 0
and at a V3 utocorre .tion.) I I u
a unction notat‘n, evaluate the foliowtng inte 35 Delta notatio As an exercise in delta f grals. [00 5(sin x)l'1(x ‘00 x
m ' "— dx. D
1)de [0° 5((305 x)il dx, L008(51n2x)il(4) there any objection to ‘ ‘ be uni , is
36 Integral of impulse. It ”1890 dx is _reed to ty _ l? D
09 x __ h '
I0 50C) d 2 one containing delta functions ’ ' si
37 Two variables. Use the method 4, integreting expres
I to arrive at the meaning of del e ( . y).
ted that
, uter. it s been sugges
ingerrnost pai of impulses ~6(x + 1} + 5
ing pair by pair s :gests that 1110:} sgn x has Fourier
38 Checking analysis by c (x ~ 1) has [2’1" “21
‘ transform ”"1" cot 11" s. sin 271' s; then transfo v i 21' sin 217 ks ~i cot it s.
k=i ore serious errors, compute both I . or ‘u
13 a mlssmg Eacptlcer 312:; sin ‘3 t sin 2° + sin 3". . . add up to
I To check iv a sign erro '
cu your finding. l> ‘ ' lue of s; for exam .
sides fo a single va and c115
% cot 1". Graph the sum of N terms versus N The Basic Theorems A small number of theorems play a basic role in thinking with Fourier trans
forms. Most of them are familiar in one form or another, but here we collect them
as simple mathematical properties of the Fourier transformation. Most of their dew
rivations are quite simple, and their applicability to impulsive functions can read
ily be verified by consideration of sequences of rectangular or other suitable
pulses. As a matter of interest, proofs based on the algebra of generalized fund
tions as given in Chapter 5 are gathered for iliustration at the end of this chapter. The emphasis in this chapter, however, is on illustrating the meaning of the
theorems and gaining familiarity with them. For this purpose a stockintrade of
particular transform pairs is first provided so that the meaning of each theorem
may be shown as it is encountered. A FEW TRANSFORMS FOR ILLUSTRATION Six transform pairs for reference are listed below. They are all well known, and
the integrals are evaluated in Chapter 7; we content ourselves at this point with
asserting that the following integrals may be verified. 2 2 oo ﬂ 2—? 7
5 77x8 I Wxsdx:e 11’s moo m m. 2 4»?!— w
and B 7758.1.”de 3;: 8 7m "'00 00 y 00 .
I sinc x 3‘12"” dx 3 H(s) and J ll(s)e”2”" ds : sine 3: *oo Moo
j sinc2 x 8"“{2m dx 2 Ms) and j A(s}e+"2m d5 = sine2 x ’00 ’00 Thus the transform of the Gaussian function is the same Gaussian function, the transform of the sinc function is the unit rectangle function, and the transform
of the sine2 function is the triangle function of unit height and area. 106 These formuias are illu Note that the second row 0 sinc x, could be supplemen terchanged, which would say that sinc s is the of the reciproca
pear redundant. HOWever, ap 1 property of The Fourier Transform and Its Ap strated as the f the figure, which says that ll ted by a second figure,
transform of formation, this extra figure Would the Fourier trans
the statement 00 first three transf with left 11(5) 3 L sinc x e'ﬂm‘s dx has quite a different character from nary functions is equal to Whether it is equa ever, it Is) exceeds 1g, the
comes to nothing and con This rathe
tegral conn hand, with
other. The second stateme Here an elementary definit
ction of the parameter s. Thu things. Two separate and
ated with ordinary fun
express different be seen to be associ Three further transforms re
e limiting sense discusse form pairs in th
d making a simp sian function, an As a > t}, the righthand
side is the Fourier trans
1 is the Fourier transform in the limit 0 w sinmc _.
j 6 :was dx 2 {
‘—00 M 11' .1.;,. Cnmmuln the. absolut The first statement tells us that the integral of l to say 0.3
situation changes abruptly,
dless of the precise value of s.
1 is) <
0 )5“) > r curious behavior is typical
ects ordinary, continuous,
awkward, abrupt functions sinc s = J unity for absolute va
or 0.35, the value of tinues to do so, regar ax nt may be rewritten "1
2 7T5 e integral of the e [00 eiwtaleeiihrsx (ix :1 (00 e value of a is used in order t quired for illustrating d earlier. Taking le substitution of variables, We have1 side represents a defin
forrn of What in the limi so ll(x)e“i2""3 doc. the product
lues of the constant s less than % plications orrn pairs in Fig. 6.1.
(s) is the transform of and right graphs in»
Elba). A consequence of certain rather ordi~ the integral is unchanged. How— because the integral now Thus i
2 i
‘2’» ' 'ons where the Fourier in t' 1
sun as _m j: (””de xponential f s the direct an
distinct physical meanings will later llalrieirrts/ o counterac _._F.L€nA ing sequence for 5
t is unity. it follows that ‘ le functions, on the one definition, on the unction is equai to an
d inverse transforms the basic theorems are trans— the result for the Gaus— a)? (s), the left—hand r 8(5). 1: the sign reversal associated with CHAPTER 6: The Basic Theorems 107 exp (”q”(2)
exP (was?)
2: Mi
Sine x 5
{1(5)
sinc2 x
A
A
5(5)
_MW____3____
_____.L__—
cos rrx Fig. 6.1 S . .
ome Fourler transform pairs for reference 111(5)
it
isinqrs
'\
1/ \
i \\
J
7} i
\ er Transform and Its Applications 108 ThelEiouri
The remaining two examples come from the verifiable relation
[00 eijeAiaxfe—iz‘ﬂsx dx 2 ”V1?“ eﬂwziisgiyﬂil’
we a whence
t of 5(5  %); s and the righthand sit—hand side into real a
e Fourier transform in the limit of d odd parts, cos mc is th
35(8 + i) + 35(5 r i) 2 “(8)
irnit of m is the Fourier transform in the limi e
or, splitting the 1 nd imaginary part side into even an and 1' sin am is the minus—i Fourier transform in the 1 sets + s + as  a z are. Summarizing the examples,
{”2 :3 (f
sinc x 3 [1(5)
sine2 x D Ms)
1 :3 6(5)
cos qrx 2) 11(5) E 158(5 + 3) + %3(s we 3)
sin rrx 3 i11{s):"§r1§i5(s + 3)  335(5 w 3)
1100 "_"> isin as.
chosen for illustration has physical interpretations,
Eater. Many properties appear among the transform eluding discontinuity, impulsiveness, iirnited extent,
les exhibiting complex or non 1152 Ail the transform pairs which will be brought out n for reference, in pairs chose
and oddness. The only examp nonnegativeness, symmetrical properties are
em 3 5(5 W 3) and 5(x  %) 3 e247”. 3333
SIMILARITY THEOREM then ﬂax) has the Fourier transform 1 ﬁx) has the Fourier transform 13(5),
Earle/a). Derivation:
1 co I .2
fmxkﬂf wéast/a) CKQX} ”0 *iZTrxs : f_
i )6ka dx Mi inst: "DO CHAPTER 6: The Basic Theorems
109 This theorem 1
sweil known in 't i ' 
compression of the ' 1 S apphcation to waveform
However, as one méﬂe SCale cormsponds to expansion of Shier}? Spectrawhere
not only contracts horichfitglflth: :Taiﬂsfolqn pair expands horizorigflyeriiy scfe.
\ Y Li a SO r  . , e 0t er
conslzapt thelarea beneath it, as shown if: gigs ggrmany m such a way as to keep
p€C1a case of ' > _ ' . . ..
6.3 shows, expansionirgtferea arises with periodic functions and im 1 '
a cosrnusoid leads simpbf to Shifts of thePu sesi. As Fig.
lmpu ses con Fig 6 2
' ' The effect of ch '
. anges in the .
Ian I or at: . ‘ scale of absassas a 
W sassa~scaimg, theorem. The shaded $332251 by the Simi'
ins constant. 110 stituting the tran would entail a in a more symme Ifftx) ha whereb S L: Then, as each function e Fig, 6.4) to compensate
constant, as will be seen Expansion of a cosinuso reduction in str 5 the Fourier transform 13(5) fheﬂ 151i; f *1 ADDITION THEOREM If fix) a not g(x) hot: 1 x 2.“ Linn 'CnHW'DT' sforrn. This is not simply a corn
ength of the impulses. tricai version o xpands or contra (in such a way th
later from the pOWer The Fourier f this theorem, cts it also shrink theorem). 8 the Fourier transforms RS) on transform HS) + C(s}. pression of the scaie of s, for (or) has the Fourier transform to at the integral of its square is Transform and Its Appiications 'd and corresponding shifts in it Eros), s or grows vertically (see maintained 61 Q5), respectively, then CHAPTER 6: The Basic Theorems
11} WM
14:01
70 percent ' percent
height height
L7 Width doubled ‘—> I
Fig 6 4 1+» Width halved *i
. . A ' '
symmetrical versron of the similarity theorem
Derivation: me {1:05) “i“ g{x)]e”j2m5 dx x If; f(x)e~f27rxs dx + J00 g(x)en1‘27rx5 d
3 F(S) "3" (3(5). This theorem Whi ‘ ‘ . . , h rs lllu t ablhty of the F  c 5 rated by an exam ie  . that af(x) has t1ourrer transform for dealing With 5min Fig. 6.5, reﬂects the suit
e transform oF(s), where o is a constan: PTO’Dlemg. A corollary is W
SHIFT THEOREM If f(x) has the Fit71H” lg? trans of E t
. 1 “I I 3 ~— t F
*2171asF( ) f ( )1 he}? f(x a) has he Gaffer franslfor Derivation: Ii x ~— *inrxs 9‘0
J f( (2)6 dx J f(x 7 ﬂag rrrrr iZT’i"—“)59Wi2¢rasd(x _ a) ‘00
‘00 : eiinrasFCS)‘ If a glifen fut! “(El IS ”ted H t E“) It 8 due in 1)}! a“. BHHHUI a “U
C h
S 1 he 7 112 The Fourier Transform and Its Applications f—t—gDFi—G. The addition theorem ges. According to
t proportional to
gle. This oc— iota ' confined to phase Chan
t is delayed in phase by an amoun s; that is, the higher the frequency, the greater the change in phase an
curs because the absolute shift a occupies a greater fraction of the period sf
harmonic component in proportion to its frequency. Hence the phase delay is a/s’l cycles or 217515 radians. The constant of proportionality describing the linear change
of phase with 5 is change of phase with frequency being greater as the shift a is greater.
e of those which are self The shift theorem is on
llei monochromatic light embodiment. Consider para
ture. To shift the diftracted beam through a small an incidence by that amount. But this is simply a way of causing the phase of the i1 lurnination to change linearly across the aperture; another way is to insert a thin
prism that iniects delays pro ' ess at each point of the portionai to the pr
aperture. These weiipunderstood proce ‘ ‘ the direction of a light
beam are shown in Chapter 13 to exemplify the shift theorem.
1“ Hm mmmnle of Fig. 6.6, a hose transform H5} is
_.L.:,J. 4a Aprimble by subjecting mponen 21711, the rate of
«evident in a chosen physical
falling normally on an aper»
gle, one changes the angle of function f(x) is shown w CHAPTER 6: The Basic Theorems
113 Imaginary
Imaginary Real mod F(s) pha H5) Imaginary
Imaginary Fis 6 6 Shift'
I 111g f(x) by one u rt > ,
90 deg per unit of 5' q a er unit of as subjects HS) to a uniform twist of plane containing HS) h
. as been def . .
of repreSentin a c ‘ .Orrned into a hehcoid. Th  . .
by showing tghe £13231? funiion Of 5 in a threedirnensi{iiifall)IiDaStticaZ dﬁflcumes
dim _ . :1 us an phage of p o are overCOme
aim; ddagmm Often gives a bfitter msightseparateiy; however, the three
_ econ example (see F' '
Sine functions andf ‘ ' 1g. .6‘7) Shows familiar res 1 .
the axis of x In thismcaii: titein‘ledlate cases which arise aslihse :{dls'the $251118 and
. _' e e icoidal f  me 5 1 85 511011
resentation in terms of sur ace Is not Sh g
' _ real and ima , . own. An aiternative
VEDthn introduced e ~ Smary parts 15 iv . . 1rEl?”
. arher of sh '  ' , 3 en, incorporatin th
shift evident} Ong the lma ma g 8 con—
.y leaves the real g ry part by a broken lin A
an odd ima inar ' part of the transform almost ‘ De. Small
shift of w/zgthereyiEaIfS'rW1gh further shift the imaginary par??? but Introduces
. ' ea art left. Th reases unti} at
SIgn untll at a shift p 8“ the real  a
of part rea  .
7r both components have undergone a iii??? Wit}; 0123 POSIte
. versa 0 phase. MODULATION THEOREM 114 Fig. 6.7 ﬁx) 2 cos x cos (x we)
(small shift) 188° shift Derivation: —‘ 7r 1
n no ‘ ‘ '2 x5 ,
$03 00 The Fourier Transiorm and its Ap The effect of shifting a cosinusoid. 00  r max a plications Rs) imaginary
PS) ‘LL
5 f{x)e_fwxe—i27rxs dx 00 —OO f(x)e—i2w(s+ 04’2le dx CHAPTER 6: The Basic Theorems 115 The new transform will be recognized as the convolution of HS) with
%6(5 + w/Z‘TT) + %5(s — w/27r) = (17/ w)n(rrs/ w). This is a special case of the convo
lution theorem, but it is important enough to merit special. mention. It is well
known in radio and television, where a harmonic carrier wave is modulated by
an envelope. The spectrum of the envelope is separated into two parts, each of
half the original strength. These two replicas of the original are then shifted along
the 5 axis by amounts ion/2W, as shown in Fig. 6.8. ﬂ
CONVOLUTION THEOREM As stated earlier, the convolution of two functions f and g is another function 11
defined by the integral he) = J: f(u)g(x _ u) du. A great deal is implied by this expression. For instance, Mr) is a linear functional
of f (x); that is, Mira) is a linear sum of values of ﬁx), duly weighted as described
by g(x). However, it is not the most general iinear functional, it is the particular
kind for which any other value h{x2) is given by a linear combination of values
of f(x) weighted in the same way. Another way of conveying this special property
of convolution is to say that a shift of ﬁx) along the x axis results simply in an
equal shift of Mr); that is, if 11(x) i f(x) * g(x), then f(x — a) * £06) 2 W m a) Suppose that a train is slowly crossing a bridge. The load at the point x is ﬁx),
and the deﬂection at x is hfx). Since the structural members are not being pushed
beyond the regime where stress is proportional to strain, it follows that the de
ﬂection at x] is a duly weighted linear combination of values of the load distri»
bution f (x). But as the train moves on, the deflection pattern does not move on
with it unchanged; it is not expressible as a convolution integral. All that can be
said in this case is that h{x) is a linear functional of ﬁx); that is, he) 2 f: f(u)g(x,u) du. It is the property of linearity combined with xshiﬂ invariance which makes
Fourier analysis so useful; as shown in Chapter 9, this is the condition that sim
ple harmonic inputs produce simple harmonic outputs with frequency unaltered. If the wellknown and widespread advantages of Fourier analysis are con»
comitant with the incidence of convoiution, one may expect in the transform
domain a simple counterpart of convolution in the function domain. This counter
part is expressed in the following theorem. 116 Fig. 6.8 An envelope function ﬁx) multiplied by cosinusoide cies, with the correspon The Fourier Transform and Its Applications ding spectra. F(s) of various frequen CHAPTER 6: The Basic Theorems 117 If f (x) has the Fourier transform Fis) and gix) has the Fourier transform Gis), then
f (x) * g(x) has the Fourier transform F(s)G(s); that is, convolution of two functions
means multiplication of their transforms. Derivation:
Jim “:0 f(r’)g(x — x’) dx’] 8“?” dx
a i: for) “foam — mom ax] dx’ [:0 f(x*)e—i2v’so(s) dx’
= F(s)G(s). Using bars to denote Fourier transforms, we can give compact statements of
the theorem and its converse. Thus 1! i7? 3 is
a r T * Er
Equivalent statements are, using a long bar for the inverse transform,
ﬁg—frg
ire ﬂ fg‘
We have stated earlier that
f r g 2 g * f (commutative)
f r (g r h) = (f r g) * h (associative)
f * (g + h) 3 f r g + f * h (distributive).
Further formulas are
W n "is"??? f*@[email protected]*m.
This powerful theorem and its converse play an important role in transform
ing a function which can be recognized as the convolution of two others or as the product of two others.
The foiiowing are statements in words of some of the above equations. 1. The transform of a convolution is the product of the transforms. 2. The transform of a product is the convolution of the transforms. 3. The convolution of two functions is the transform of the product of their trans—
forms. 4. The product of two functions is the transform of the convolution of their trans Farms: CHAPTER 6: The Basic Theorems
. 119 Three v owing.
1. The area under a convolution is equal to the product of the areas under the f *f (”:3 3 PG
”factors”; that 15, f(— 3 )3 PCP)
H30") 3 PPM“)
f *jérﬂ‘“) 3 PG*
“32: é)’*(~) 3 F(w)(;>2
W 1;: g* :3 Pp" (3* __
f*(") *3*{_) D 131cc: ( )
f*(_? 1': 3:1: 3 F*G*(w)
f" t g* :> P*(”)G*(m) The selfconvoiution formulas are 118 aluable properties often used for 2 The abscissas of the centers of gravrty add, that is, <x>ﬂgz<x>f+ (111; f( ”1511;: if ) ECO 36210:) dx f)“ M f*(— SURF 12
where (x) 7" ﬂﬂ' and f J”: * fa: 3 [WV )12 :1 E00 Mac) dx or autocorrelation we have 1‘ 11 f :3 PF ' *f( )*f( )3 F138
f'()* *( )D P*P*() r O; in general,
f* if»: D F*F*(— ) 3. The second moments add 1t(x)f or {3% %
RAYLEIGH’S THEOREM {953113 3 {ﬂ}? J“ {ﬁg + 2<x>f<x>gr < 2) 1: 2&me
DC 31: I 100 km dx Th I
e mte
gmi of the squared modulus of a function is equal to th '
e Integral red modulus of its spectrum; that 1'5 [11 1f(x)2dxz I” \Hsﬂzds. 9f where 1t {chews that the variances must add (p 158). We have enunciated the convo‘lution theorem in the form Derivation: ﬁx) a: 813‘} has the Fourier transform F{S}G(s)1 which, written in E1111, becomes either CC .ojmﬂﬁlﬂx ' u) GILL 5 3—00 FC5)G(S)ef2mcs d5 F(s)G(s) : {Dc K: f(u)g(x  11):?"2 onvolution theorem is con
gn reversals of the “5 du dx. OI? 20 versions in which the c .111 Ma run EGWQI' than
. m111111mps. and 51 ...
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