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

T he no point d ft of j k gives exact values of t he

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Unformatted text preview: . 5.17, a nd show t hat i t is equivalent t o t he linear convolution found in p art ( b). ( d) F ind t he c ircular convolution of I [k] a nd g[k] o btained in p art ( a) using O FT. ( e) F ind t he linear convolution of I [k] a nd g[k] o btained i n p art ( b) using O FT. 669 11.1 T he Z - Transform where t he symbol f i ndicates a n i ntegration in counterclockwise direction around a closed p ath in t he complex plane (see Fig. 11.1). As in t he case of t he Laplace transform, we need not worry a bout t his integral a t t his point because inverse z transforms of many signals o f engineering interest can be found in a z-transform Table. T he direct a nd inverse z-transforms can b e expressed symbolically as F[z] = or simply as f [k] D iscrete-Time S ystem Analysis U sing t he Z-Transform f [k] = a nd Z { J[k]} z -t {F[z]} ~F[z] Note t hat z -t [Z{J[k]}] = a nd f [k] Z [ Z-t{F[z]}] = F[z] Following t he e arlier argument, we c an find a n LTID system response t o a n i nput f [k] using t he s teps as follows: T he c ounterpart o f t he Laplace transform for discrete-time systems is t he z transform. T he Laplace transform converts integro-differential equations into algebraic equations. In t he s ame way, t he z -transforms changes difference equations into algebraic equations, thereby simplifying t he analysis o f d iscrete-time systems. T he z -transform method o f analysis o f d iscrete-time systems parallels t he Laplace transform m ethod of analysis o f c ontinuous-time systems, with some minor differences. I n fact, we shall see t hat t he z -transform is t he Laplace t ransform in disguise. T he behavior o f discrete-time systems (with some differences) is s imilar t o t hat o f continuous-time systems. T he frequency-domain analysis o f d iscrete-time systems is based on t he fact (proved in Sec. 9.4-2) t hat t he response of a linear t ime-invariant discrete-time (LTID) system t o a n e verlasting exponential zk is also t he s ame exponential (within a multiplicative constant), given by H[z]zk. We t hen express a n i nput f [k] as a s um o f (everlasting) exponentials o f t he form zk. T he s ystem r esponse t o f[k] is t hen found as a sum o f t he system's responses t o all these exponential components. T he tool which allows us t o r epresent a n a rbitrary i nput f [k] as a sum of (everlasting) exponentials o f t he form zk is t he z -transform. 11.1 t he s ystem r esponse t o zk is H[zJzk f [k] = ~ 27rJ a nd y[k] f ~f ~ = f F [z]zk-t dz F[z]H[z]zk-t dz = 27rJ Y[z]zk-t dz where Y[z] L f [k]z-k 27rJ f ¢ =} t hen F[z] Y[z] = y[k] ¢ =} Y[z] F[z]H[z] (11.1) L inearity o f t he Z - Transform k =-oo ~ F[z]H[z] We shall derive this result more formally later. < Xl J[k] = = In conclusion, we have shown t hat for a n LTID system with transfer function H[z], if t he i nput a nd t he o utput a re J[k] a nd y[k], respectively, a nd if f [k] F[z] '= shows y[kJ a s a s um o f r esponses t o e xponential c omponents 27rJ The Z- Transform I n t he l ast Chapter, we extended t he discrete-time Fourier transform t...
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