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Unformatted text preview: 3 Applications The Fourier Transform and It 50c) ' esenta ‘ ’
Graphl(zaéyrli‘lgo1 8w as a 5131 e of log Slxlflxraldx‘flmal' d s
an gm; can be empties ed b ' y writing
‘a the convolution mte The resemblance loo 506 (x r X’W’ : )DO ‘ ' tation,
or, in asterisk no (x) I ﬁx) 3‘: 5“ ﬂ ﬁx) d m such as
'a ra
~ little tho ht devoted to a‘ 1‘ tgm value of
If ﬁx) has a jump at x M at int.,rai will have a llml g
‘ v (m e general to write "lg(0+) 4w flO—li 3H) + fﬂ‘
50:) * flxl 2 null f
the rightha i use differ from ﬁx) only by a
ssion on . is does not alter
The expre I . . inarﬂy not imprtaﬂt Th s o w
ment 1  different in value mflxl that llet) t flell can ' 41mg, H pry/Ti entiOHEd abOV
f(x+}. At pomts use of this asymme
analysis, where dis
men, the ch ' . se .
fish; (:2qu An important one which mus
pro I .
rnanipul ion is 1
5013c) : #7 50¢),
13; ' ea f the
d by a factor a, thus reducmg the ar . 9 scale of x is Compresse a then the streﬂgth of the impulse is re
 ' had unit are r
hich prevmusly
pulses w , the pIOPe‘ftY
l i 1 5 Si n allows f0]?
bv the lactor le The mom” g_, \ _ W unction,
the fact 8 will be seen i discontinuity of f(x),‘the
ent result. in transrent amaneous" ' mo of which can be
caret ly in algebraic clued CHAPTER 5: The Impulse Symbol 8E om this it would seem that the impulse symbol has the property of evenne :;
ho ever, we gave an equation earlier involving a sequence of displaced re an
gle fu '~ tiOns which were not themselves even (Prob. 19, Chapter 5).
It ca . asin be shown by considering sequences of pulses that we 2 ay write,
if f(x) is con ‘nuous at x m 0, f (x) 5(36) 3 f (0) 5m.
From the sifting pro rty, putting f {x} 2 x, we have {60 x6(x) dx 2 O. "'00 One generally writes
(X) E ., although if the prelimit graphs are cont plated, it will be seen that this equa
tion conceals a nonvanishing compo nt 1: ‘ iniscent of the Gibbs phenomenon
in Fourier series. Thus it is true th 11m « "1H = O for . ix;
790 7 lim [JCT—1E w %,
THO T max and, moreo A er, the limit of the minimum value is “i Consequently, am ' ; those
function which are identically zero, 9550:) is rather curious, and one has th feel—
ing t a if it couid be applied to the deﬂecting eiectrodes of an oscilloscope, o e
w a cl see spikes. nevertheless, THE SAMPLING OR REFLICATING SYMBOL IIHx) Consider an infinite sequence of unit impulses spaced at unit interval as shown
in Fig. 5.4. Any reservations that apply to the impulse symbol 5(x) apply equally Figi 54 The shah symbol Illix). k‘\ .\ 83 CHAPTER Impulse Symbol 8?.
e we have to deal with an
Ill(x)f(x) indeed, even more may be needed becaus er of infinite discontinuities and a nonconvergent infinite integral.
all the conditions for existence of a Fourier transform are violated.
an infinite sequence of impulses proves, however, to be ex» d easy to manipulate algebraically.
ntroduce the sham2 symbol lll(x) and write in this case ;
infinite numb
For example,
The conception of tremely usefulw—an
To describe this conception we '1 X
‘ x CD Flg. 5 5 he Sam Elng Pr" erty 0f ointed out: me} * f(x) z i W W n): 21:7“, 5 obvious properties may be p 1H “L 5( — 35)
(ax) H 5111 a HIVx) S Ill(x)
Hip: + n) :— l.ll(x) n integral Variou as shown in Fig 5 6
. . . A  , the function 
mﬁmtum in b  . flx) appears in re 1‘ ‘ 
unit wid thOth‘dIIEChOﬂS. Of course, if f (X) s readp ma at amt mtervais Of x M
Th Ell ere ls Overlapping. P 8 Over a base more than one 8 s mbol f ~ .
Y is thus also applicable wherever there are periodic { S ructures. {no « t) z are: + t) 111m dx x 1 111(x) : 0 x s5 it.
dic with unit period. ty follows as a generalization of the sitting inte
with the impulse symbol. Thus multiplica ly samples it at unit int Evidently, lll(x) is perio A periodic sampling proper
gral already discussed in connection
tion of a function ﬁx) by lli(x) effective 00 il'l(x)f(x) : 2 ﬁn) 5(x * n). integers where lli(x) z 0 is the intervals between
f ﬁx) at integral values of x (x) in
e values 0 The intormation about f
duct; however, th not contained in the pro
are preserved (see Fig. 5.5). The sampling property makes lll(x) a valued tool in the study of a wide va—
riety of subjects (for example, the radiation patterns of antenna arrays, the dif
fraction patterns of gratings, raster scanning in television and radar, pulse mod» t discrete tabular intervals). ulation, data sampling, Fourier series,
just as important as the sampling proper tiplication is a replicat—
' function f(x). tug property exhibited when lll(x) enters int
Thus cter ill, which is said to have been modeted 1The symbol Ill is pronounced shah after the Cyrillic Chara
on the Hebrew letter w (slain), which in turn may derive from the Egyptian E g , a hiero« This twofold ch 
_ aracter IS not ac ‘d 
its own Fourier . C1 ental, but is connected '
as it OtherW' transform (In the limit), which of C With the fact that III is
Th Else Would have been muse makes 1t twme as useful
e se reci rocal .
b When a Shal: functfdgﬁgrgédldldeiithe Fourier tranSformation is derive“ t
eing written Imzx) th ‘ eze 1n the xdirection b E a or
. , e impulses are . y ‘3 aC‘EOI‘ 2, the result
conslstency the im u} packed tWice as cl 1
588 are red, at ‘ ose Y‘ But for 611 b ‘
derstand this ' p “CE in stren th b h be mm
In terms of the se g y t 9 Same factor One (3
pulse; each rectan . quence of rectangie fun t. _ " an un
. gle 15 squeezed t h . C Ions deflmng Each '
1115? as a a : ~1 0 alf the Width 13’“
l 3‘) I513 5(96) (lat aPl3€ars in case a < G) sadld half the area' Themfore’ 111m) 2 .512— 5(x g This is a known ‘ " .ll‘ap for stud t stretched so th t   . 399,8 WC} may be restat d resents impuls; ﬂespth‘ilsgt'sgacmg mailgigs from unify tgsxfotlilgliIEf/ng) is
'  . g ut i ‘ ’ x re _ Preseron foras Spaced X isygfflﬁaiggger of unit strength X. Tide e}; for)
HHX) >:< ﬁx) x
x Fi . 5. '
g 6 The replicating property of HIM}. ...
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