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

# 2 transforms o f s ome useful functions f or

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Unformatted text preview: g L ' H6pital's r ule, we find sinc (0) = 1 . 4. s inc (x) is t he p roduct of a n o scillating signal sin x (of p eriod 211&quot;) a nd a monotonically d ecreasing function l /x . T herefore, s inc(x) e xhibits sinusoidal oscillations of p eriod 211&quot;, w ith a mplitude d ecreasing continuously as l /x. ..1... 2 t-- F igure 4 .9a shows sinc (x). O bserve t hat s inc (x) = 0 for values of x t hat a re p ositive a nd n egative integral multiples o f 11&quot;. F igure 4 .9b shows sinc T he en· tsinc (x) is a iw denoted by Sa (x) in the literature. Some authors define sinc (x) as sinc ( x) = sin 11&quot;X 11&quot;X F ig.4.10 A gate pulse J (t), its Fourier spectrum F(w), amplitude spectrum IF(w)l, and phase spectrum LF(w). 248 4 C ontinuous-Time S ignal A nalysis: T he F ourier T ransform J (t)=5(1) 4.2 F (ro)= I o o F ig. 4 .11 o U nit impulse and its Fourier spectrum. 249 -- j (t)=1 (b) (a) F (rol= 21t 5(ro) o t -+ F ig. 4 .12 A c onstant (dc) signal a nd i ts Fourier spectrum. Bandwidth o f r ect (*) 0.+ f (t) = 1, we need a single everlasting exponential e jwt w ith w = T his results in a spectrum a t a single frequency w = O. A nother way of looking a t t he s ituation is t hat f (t) = 1 is a dc signal which has a single frequency w = 0 (dc). • T he s pectrum F(w) i n Fig. 4.10 p eaks a t w = 0 a nd d ecays a t h igher f requencies. T herefore, r ect ( *) is a lowpass s ignal w ith m ost o f t he s ignal e nergy i n lower f requency c omponents. S trictly s peaking, b ecause t he s pectrum e xtends f rom 0 t o 0 0, t he b andwidth is 0 0. H owever, m uch o f t he s pectrum is c oncentrated w ithin t he f irst l obe ( from w = 0 t o w = ~). T herefore, a r ough e stimate o f t he b andwidth o f a r ectangular p ulse o f w idth T s econds is ~ r ad/s, o r ~ H z.t N ote t he r eciprocal r elationship o f t he p ulse w idth w ith i ts b andwidth. W e s hall o bserve l ater t hat t his r esult is t rue i n g eneral. • T ransforms o f S ome U seful F unctions w= • I f a n i mpulse a t w = 0 is a s pectrum o f a d c s ignal, w hat d oes a n i mpulse a t r epresent? W e s hall a nswer t his q uestion i n t he n ext e xample. Wo E xample 4 .5 F ind t he inverse Fourier transform of 6 (w - wo). Using the sampling property of t he impulse function, we o btain E xample 4 .3 F ind t he F ourier transform of t he u nit impulse 6(t). Using t he s ampling p roperty of the impulse [Eq. (1.24)J, we o btain F[6(t)J = I: 6 (t)e- jwt (4.24a) Therefore (4.24b) dt = 1 or or 6 (t)=1 Figure 4.11 shows 6(t) a nd its spectrum. • • (4.26a) E xample 4 .4 F ind the inverse Fourier transform of 6 (w). O n t he b asis of Eq. (4.8b) and the sampling property of the impulse function, r 1 [6(w)J = ~ 2~ Therefore 1 1 00 6(w)e - 00 - =6(w) or 2~ jwt 1 This result shows t hat t he s pectrum of an everlasting exponential e jwot is a single impulse a t w = woo We reach t he s ame conclusion by qualitative reasoning. To represent the everlasting exponential e Jwot , we need a si...
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## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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