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

# 13 we c an d etermine a a nd a2 b y clearing

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Unformatted text preview: = z5(z - 1) 6 Therefore 00 =L z f [rJz-(r-m) F[z] = (z _ 1)2 - z5(z - 1)2 - z5(z - 1) r =m 6z + 5 z5(z - 1)2 Z6 - - • r=m E xercise E l1.5 U sing o nly t he f act t hat u[kJ ¢ =? z :'1 a nd t he r ight-shift p roperty [Eq. (11.15)], find t he z -transforms o f t he s ignals i n F igs. 1 l.2 a nd 11.3. T he a nswers a re g iven i n E xample 11.2d a nd £::" m -l 00 = zm [ ~f[rJz-r - ~ f [rJz-r E xercise E Il.la. m -l = zmF[zJ - zm L f [rJz-r r=O 'V C onvolution T he t ime c onvolution p roperty a nd t he f requency convolution p roperty s tate t hat i f h [kJ Fl[ZJ a nd h [kJ F2[ZJ, = f lkl = t hen ( time c onvolution) 5 (11.18) a nd ( frequency convolution) (11.19) o 2 F ig. 1 1.4 3 4 5 Signal for Example 11.4. P roof: T hese p roperties a pply t o c ausal as well a s n oncausal s equences. F or t his r eason, we s hall p rove t hem for t he m ore g eneral c ase o f n oncausal s equences, w here t he c onvolution s um r anges f rom - 00 t o 0 0. T o p rove t he t ime c onvolution, we have • E xample 1 1.4 F ind the z-transform of the signal f[kJ depicted in Fig. 11.4. T he signal f[kJ can be expressed a s a p roduct of k a nd a gate pulse u[kJ - u[k - 6J. T herefore f[kJ = k {u[kJ - u[k - 6]} = ku[kJ - ku[k - 6J 00 = L k =-oo 00 z -k L m =-oo h [mJh[k - mJ 684 11 Discrete-Time Systems Analysis Using t he Z - Transform Interchanging the order of summation, I::. 00 Z [h[k] * h[kJ] = 11.3 Z -Transform Solution of Linear Difference Equations 685 E xercise E ll.6 U sing E q. ( 11.21), derive P airs 7 a nd 8 i n T able 11.1 from P airs 2 a nd 3, respectively. 00 h Im] L h [k - m]z-k m =-oo k=-oo \l L 00 L = Multiplication by k (Scaling in t he z-Domain 00 hIm] L m =-oo If h[r]z-(r+m) J[k]u[k] r =-oo 00 m =-oo ¢ => F[z] t hen 00 r =-oo kJ[k]u[k] d ¢ => - z-F[z] dz (11.22) Proof: T o prove t he frequency convolution, we s tart w ith -z~F[z] = -z~ f f[k]z-k = - z dz 00 Z {h[k]h[k]} = L h [k]h[k]z-k k=-oo dz k=O f - kf[k]z-k-l k=O 00 = L kf[k]z-k = Z {kf[k]u[k]} k=O I::. E xercise E ll.7 U sing E q. (11.22), derive P airs 3 a nd 4 i n T able 11.1 from P air 2. Similarly, derive P airs 8 I nterchanging t he o rder of summation a nd i ntegration a nd 9 f rom P air 7. \l Initial and Final Value = 2!j f Ftlu]F2 [~] For a causal f[k], u- 1 J[O] = lim F[z] (11.23a) z~oo du T his result follows immediately from Eq. (11.9) We can also show t hat i f (z - l)F{z) has no poles outside the u nit circle, t hen LTID System Response I t is interesting t o a pply t he t ime convolution p roperty t o t he LTID inputoutput equation y[k] = f[k]*h[k]. I n Eq. (11.14), we have shown t hat h[k] ¢ => H[z]. Hence, according t o Eq. (11.18), i t follows t hat Y[z] = F[z]H[z] (11.20) lim f {N) = lim{z - l)F{z) N --+oo 1 1.3 z --+l (11.23b) Z- Transform Solution o f linear Difference Equations E arlier in t he c hapter, we derived this i mportant r esult using informal arguments. Multiplication by 'Yk If f[k]u[k] ¢ => F[z] 'Yk f[k]u[k] ¢ => F t hen [~] (11.21) T he time-shifting (left- or right-shift) pr...
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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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