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

6 56 106 10 fourier analysis o f d iscrete time s

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Unformatted text preview: tions in the s tudy o f LTID system analysis. Actually i t i s t he first limitation t hat is also t he cause of t he second limitation. Because D TFT i s i ncapable o f handling growing signals, i t is incapable of handling unstable or marginally stable systems. t O ur goal is, therefore, t o e xtend t he c oncept o f D TFT s o t hat i t can handle exponentially growing signals. We may wonder what causes this limitation on D TFT s o t hat i t is incapable of h.andling exponentially growing signals. Recall t hat in D TFT, we a re synthesizing a.n a rbitrary signal I[kJ using sinusoids or exponentials of t he form e jClk . These signals are sinusoids with constant amplitudes. T hey a re incapable of synthesizing exponentially growing signals no m atter how many such components we add. O ur h.ope, therefore, lies in trying t o synthesize I[kJ using exponentially growing sinus()ids o r exponentials. This goal c an be accomplished by generalizing t he frequency variable j n t o a + j n; t hat is, by using exponentials of t he form e(u+jCl) i nstead o f e xponentials e jCl . T he p rocedure is a lmost identical t o t hat used in extending t he Fourier transform t o t he Laplace transform in Sec. 6.1. T he i ntuitive argument i s identical t o t hat discussed in Sec. 6.1, a nd t he r eader may wish t o review it t o r efresh his memory. Here we s hall go s traight t o t he a nalytical development. As in t he case of Fourier t o Laplace, it is desirable t o use t he n otation F (jn) i nstead o f F (n) for t he D TFT i n order to unify t he D TFT a nd t he generalized t ransform (z-transform). Thus, L I[kJ e - jClk (10.81) k =-oo a nd (10.82) Consider now t he D TFT of I[kJ e - uk ( a real ) L Hence, t he inverse D TFT of F (a + j n) is I[kJ e - uk . T herefore I[kJ e - uk = - 1 21T I[kJ e - uk e - jClk (10.83) j"" F(a + jn) ejClk dn _ I[kJ = ~ j" F(a + jn) e(u+jCl)k dn L et us define a new variable z as l nz=a+jn a nd 1 - dz = j d n z (10.88) Because z = eu+jCl is complex, we c an express it as z = r e jCl , w here r = eU • T hus, z lies on a circle of radius r , a nd as n varies from -1T t o 1T, z c ircumambulates along this circle, completing exactly one rotation in counterclockwise direction, as illustrated in Fig. 10.19. Changing t o variable z in Eq. (10.87) yields ~ 21TJ f F(lnz) z k-l d z (1O.89a) a nd from Eq. (10.85) we o btain 00 F (lnz) = 00 L (10.87) 21T _ " k =-oo = (10.86) Multiplying b oth sides o f t he above equation by e uk yields J[kJ = 00 D TFT [I[kJ e-ukJ = C ontour o f i ntegration f or t he z -transform. so t hat 00 F un) = F ig. 1 0.19 I[kJ e-(u+jCl)k (10.84) L I [kJz- k (1O.89b) k =-oo k =-oo I t follows from Eq. (10.81) t hat t he above sum is F (a + j n). T hus 00 D TFT [I[kJ e-ukJ = L l[kJ e-(u+jCl)k = F (a + j n) (10.85) k =-oo tR.ecall t hat t he o utput o f a n u nstable system grows exponentially. Also, t he o utput o f a marginally s table s ystem t o c haracteristic m ode i nput grows with time. where t he i ntegral f indicates a contour integral around a circle of radius r in counterclockwise direction. T he above two equations are t he desired extensions. T hey are, however, in a clumsy form. For t he sake of convenience, we mak...
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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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