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

# T hus a t t he i nput of a discrete time processor a

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Unformatted text preview: ce. Earlier, we showed t hat (11.51) I n c ontrast, t he z -transform of t he signal - -/u[-(k + 1)), illustrated in Fig. 11.12a, is hi (11.52) A comparison o f Eqs. (11.51) w ith (11.52) shows t hat t he z -transform of - lurk] is identical to t hat o f - -lU[-(k + 1)]. T he regions of convergence, however, a re different. In t he former case, F[z] converges for Izl &gt; hi; in t he l atter, F[z] converges for Izl &lt; hi (see Fig. 11.12b). Clearly, t he inverse transform o f F[z] is n ot u nique unless t he region o f convergence is specified. I f we r estrict all o ur signals t o b e causal, however, this ambiguity does n ot arise. T he inverse transform o f z/(z - ,) is , ku[k] even without specifying t he region o f convergence. Thus, i n t he u nilateral transform, we c an ignore t he region of convergence in determining t he inverse z-transform of F[z] . • E xample 1 1.11 Determine the z-transform of f[kJ = (O.9)ku[kJ = + (1.2)k u [-(k + I)J /I[kJ + h[kJ 11 7 06 D iscrete- Time S ystems A nalysis U sing t he Z - Transform f [k) 1 7 07 1 1.7 T he B ilateral Z -Transform to k- 5 10 k- 15 __~~~~~~~~~~~o f [k) f [k) - 20 5 - 5 x 10 _106 (b) (a) F ig. 1 1.13 Signal f [k) for Example 11.11. f [k) F rom t he results in Eqs. (11.51) a nd (11.52), we have Z Izl &gt; 0.9 Fl[Z) = Z _ 0.9 Izl &lt; 1.2 T he common region where b oth Ft[z) a nd F2[Z) converge is 0.9 &lt; Hence (Fig. 11.13a). - 10 -5 o 5 10 k -- (c) z z = Izl &lt; 1.2 z - 0.9 - z - 1.2 F ig. 1 1.14 T hree possible inverse transforms o f F[z) in Example 11.12. - 0.3z (z - 0.9)(z - 1.2) 0.9 &lt; Izl &lt; 1.2 (11.53) Since t he region o f convergence is Izl &gt; 2, b oth t erms c orrespond t o c ausal sequences a nd T he sequence f [k) a nd t he region of convergence of F[z) a re depicted in Fig. 11.13 . • • E xample 1 1.12 T his sequence a ppears in Fig. 11.14a. F ind t he inverse z-transform of ( b) I n t his case, Izl &lt; 0.8, which is less t han t he m agnitudes of b oth poles. Hence, b oth t erms c orrespond t o a nticausal sequences, a nd - z(z + 0.4) F[z) = (z - 0.8)(z - 2) i f t he region of convergence is ( a) Izl &gt; 2 ( a) F[z) z ( b) Izl &lt; 0.8 - (z + 0.4) (z - 0.8)(z - 2) 2 z - 0.8 ( c) 0.8 &lt; Izl &lt; 2. J[k) = [_(0.8)k + 2(2)k] u [ -(k + 1») T his sequence a ppears in Fig. 11.14b. ( c) I n t his case, 0.8 &lt; Izl &lt; 2; t he p art o f F[z) c orresponding t o t he pole a t 0.8 is a causal sequence, a nd t he p art c orresponding t o t he pole a t 2 is a n a nticausal sequence: z- 2 f [k) = (0.8)ku[k) a nd F[z) = _ z_ _ 2 _z_ z - 0.8 z- 2 T his sequence a ppears in Fig. 11.14c. + 2 (2)ku[-(k + 1») • - 60 708 f'o. 11 D iscrete- Time S ystems A nalysis U sing t he Z - Transform E xercise E l1.13 Find the inverse z - transform F[z] = of z z 2 5 ~ I + SZ + S &gt; Izl &gt; ! 1 709 11. 7 T he B ilateral Z - Transform Since the system is causal, t he region of convergence of H[zJ is Izl &gt; 0.5. T he region of convergence of F[zJ is 0.8 &lt; Izl &lt; 2. T he common region of convergence for...
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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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