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# ch07 - Solutions to Additional Problems 7.17 Determine the...

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Solutions to Additional Problems 7.17. Determine the z -transform and ROC for the following time signals: Sketch the ROC, poles, and zeros in the z -plane. (a) x [ n ] = δ [ n k ] , k > 0 X ( z ) = n = −∞ x [ n ] z n = z k , z = 0 Im Re k multiple Figure P7.17. (a) ROC (b) x [ n ] = δ [ n + k ] , k > 0 X ( z ) = z k , all z Im Re k multiple Figure P7.17. (b) ROC 1

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(c) x [ n ] = u [ n ] X ( z ) = n =0 z n = 1 1 z 1 , | z | > 1 Im Re 1 Figure P7.17. (c) ROC (d) x [ n ] = ( 1 4 ) n ( u [ n ] u [ n 5]) X ( z ) = 4 n =0 1 4 z 1 n = 1 ( 1 4 z 1 ) 5 1 1 4 z 1 = z 5 ( 1 4 ) 5 z 4 ( z 1 4 ) , all z 4 poles at z = 0, 1 pole at z = 0 5 zeros at z = 1 4 e jk 2 π 5 k = 0 , 1 , 2 , 3 , 4 Note zero for k = 0 cancels pole at z = 1 4 2
Im Re 0.25 Figure P7.17. (d) ROC (e) x [ n ] = ( 1 4 ) n u [ n ] X ( z ) = 0 n = −∞ 1 4 z 1 n = n =0 (4 z ) n = 1 1 4 z , | z | < 1 4 Im Re 0.25 Figure P7.17. (e) ROC (f) x [ n ] = 3 n u [ n 1] X ( z ) = 1 n = −∞ ( 3 z 1 ) n = n =1 1 3 z n 3

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= 1 3 z 1 1 3 z = 1 1 3 z 1 , | z | < 3 Pole at z = 3 Zero at z = 0 Im Re 3 Figure P7.17. (f) ROC (g) x [ n ] = ( 2 3 ) | n | X ( z ) = 1 n = −∞ 3 2 z 1 n + n =0 2 3 z 1 n = 1 1 3 2 z 1 + 1 1 2 3 z 1 = 5 6 z ( z 3 2 )( z 2 3 ) , 2 3 < | z | < 3 2 2 3 3 2 Im Re Figure P7.17. (g) ROC (h) x [ n ] = ( 1 2 ) n u [ n ] + ( 1 4 ) n u [ n 1] 4
X ( z ) = n =0 1 2 z 1 n + 1 n = −∞ 1 4 z 1 n = 1 1 1 2 z 1 + 1 1 2 3 z 1 , | z | > 1 2 and | z | < 1 4 No region of convergence exists. 7.18. Given the following z -transforms, determine whether the DTFT of the corresponding time signals exists without determining the time signal, and identify the DTFT in those cases where it exists: (a) X ( z ) = 5 1+ 1 3 z 1 , | z | > 1 3 ROC includes | z | = 1, DTFT exists. X ( e j ) = 5 1 + 1 3 e j (b) X ( z ) = 5 1+ 1 3 z 1 , | z | < 1 3 ROC does not include, | z | = 1, DTFT does not exist. (c) X ( z ) = z 1 (1 1 2 z 1 )(1+3 z 1 ) , | z | < 1 2 ROC does not include, | z | = 1, DTFT does not exist. (d) X ( z ) = z 1 (1 1 2 z 1 )(1+3 z 1 ) , 1 2 < | z | < 3 ROC includes | z | = 1, DTFT exists. X ( e j ) = e j (1 1 2 e j )(1 + 3 e j ) 7.19. The pole and zero locations of X ( z ) are depicted in the z -plane on the following figures. In each case, identify all valid ROCs for X ( z ) and specify the characteristics of the time signal corresponding to each ROC. (a) Fig. P7.19 (a) X ( z ) = Cz ( z 3 2 ) ( z + 3 4 )( z 1 3 ) There are 4 possible ROCs (1) | z | > 3 4 x [ n ] is right-sided. 5

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(2) 1 3 < | z | < 3 4 x [ n ] is two-sided. (3) | z | < 1 3 x [ n ] is left-sided. (b) Fig. P7.19 (b) X ( z ) = C ( z 4 1) z ( z 2 e j π 4 )( z 2 e j π 4 ) There are 2 possible ROCs (1) | z | > 2 x [ n ] is right-sided. (2) | z | < 2 x [ n ] is two-sided. (c) Fig. P7.19 (c) X ( z ) = ( z 1 2 )( z + 1)( z 2 + 9 16 ) C, | z | < x [ n ] is stable and left-sided. 7.20. Use the tables of z -transforms and z -transform properties given in Appendix E to determine the z -transforms of the following signals: (a) x [ n ] = ( 1 2 ) n u [ n ] 2 n u [ n 1] a [ n ] = 1 2 n u [ n ] z ←−−−→ A ( z ) = 1 1 1 2 z 1 , | z | > 1 2 b [ n ] = 2 n u [ n 1] z ←−−−→ B ( z ) = 1 1 2 z 1 , | z | < 2 x [ n ] = a [ n ] b [ n ] z ←−−−→ X ( z ) = A ( z ) B ( z ) X ( z ) = 1 1 1 2 z 1 1 1 2 z 1 , 1 2 < | z | < 2 (b) x [ n ] = n (( 1 2 ) n u [ n ] ( 1 4 ) n u [ n 2] ) a [ n ] = 1 2 n u [ n ] z ←−−−→ A ( z ) = 1 1 1 2 z 1 , | z | > 1 2 6
b [ n ] = 1 4 n u [ n ] z ←−−−→ B ( z ) = 1 1 1

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ch07 - Solutions to Additional Problems 7.17 Determine the...

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