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Unformatted text preview: F(x)eixt dx A pply s ymmetry p roperty t o p airs 1, 3, a nd 9 ( Table 4.1) t o show t hat  00 = = aw
( a) jt~a
211"e u(  w) ( b) t2~aa2
211"e a1wl
(e) o(t + to) + o(t  to)
2 cos tow \ l F (x)ejxtdx 4.33 C hanging t t o w yields Eq. (4.31). = The Scaling Property If
f (t) • E xample 4 .8
In this example we apply the symmetry property [Eq. (4.31)J to the pair in Fig. 4.17a.
From Eq. (4.23) we have
rect ( ;) = r sinc (~T) "v' (4.32) '.....' f (t) { => F (w) then, for any real constant a,
f eat) I!I { => F (~) (4.34) Proof: For a positive real constant a, F (w) Also, F (t) i s t he same as F(w) with w replaced by t, and f (  w) is the same a s f (t) with
t replaced by  w. Therefore, the symmetry property (4.31) yields 1 00 F [f(at)] == f (at)e jwt dt ==  00 (~)
~
F(t)
r sinc 255 =  27rrect (  ;) = 27rrect (~) (4.33) a 1 00 f (x)e(jw/a)x d x ==  00 Similarly, we can d emonstrate t hat if a < 0, 2 " f (w) [n Eq. (4.33) we used the fact t hat reet ( x) = rect (x) because rect is an even function.
Figure 4.17b shows this pair graphically. Observe the interchange of the roles of t and w ~ f eat) Hence follows Eq. (4.34). { => (w) 1
 ;;F ; ~F (~)
a a 256 4 C ontinuousTime Signal Analysis: T he F ourier Transform IU) ~ "2 o 4.3
•  I 2 Some P roperties o f t he F ourier Transform 2 57 E xample 4 .9
F ind t he Fourier transforms o f e atu(_t) a nd e  a1tl .
A pplication of Eq. (4.35) t o p air 1 (Table 4.1) yields 1 Also IU) F (oo) Therefore 2~ ea1tl = a + jw + _1_ = a + w
_ 1_
~
a jw
2 o (4.36) 2 T he signal e  a1tl a nd i ts s pectrum a re illustrated in Fig. 4.19. • 00 F (ro)=+, F ig. 4 .18 a +'" Scaling p roperty of t he Fourier transform. ( b) Significance o f the Scaling Property The f unction f rat) r epresents t he f unction f (t) c ompressed in t ime by a factor
a (see Sec. 1 .32). Similarly, a f unction F(~) r epresents t he f unction F (w) e xpanded
in frequency by t he s ame factor a. T he s caling p roperty s tates t hat t ime compression
o f a s ignal r esults in i ts s pectral e xpansion, a nd t ime expansion o f t he s ignal r esults
i n i ts s pectral compression. Intuitively, compression in t ime b y a factor a m eans
t hat t he s ignal is varying rapidly by t he s ame factor. To synthesize such a signal,
t he f requencies of its sinusoidal components m ust b e increased by t he f actor a,
i mplying t hat i ts frequency s pectrum is e xpanded by t he f actor a. Similarly, a
s ignal e xpanded in t ime varies more slowly; hence t he frequencies of its components
a re l owered, implying t hat i ts frequency s pectrum is compressed. For instance, t he
s ignal cos 2 wot is t he s ame as t he s ignal cos wot t imecompressed by a factor of 2.
Clearly, t he s pectrum o f t he former (impulse a t ± 2wo) is a n e xpanded version of t he
s pectrum o f t he l atter ( impulse a t ± wo). T he effect of t his scaling is d emonstrated
i n Fig. 4.18.
Reciprocity o f Signal Duration and Its Bandwidth
T he s caling p roperty...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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