Signal Processing and Linear Systems-B.P.Lathi copy

# 3 some p roperties of t he fourier transform 253 t

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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&quot;e u( - w) ( b) t2~aa2 211&quot;e- a1wl (e) o(t + to) + o(t - to) 2 cos tow \ l F (x)e-jxtdx 4.3-3 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) &quot;-v-' (4.32) '-.....-' f (t) { =&gt; F (w) then, for any real constant a, f eat) I!I { =&gt; 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 &lt; 0, 2 &quot; 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). { =&gt; (w) -1 - ;;F ; ~F (~) a a 256 4 C ontinuous-Time Signal Analysis: T he F ourier Transform IU) -~ &quot;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~ e-a1tl -= 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 +'&quot; 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 .3-2). 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 ime-compressed 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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