4 a 10 points find the transfer function of h z and

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Unformatted text preview: ] . 4 a) (10 points) Find the transfer function of H (z ) and indicate the region of convergence. b) (5 points) Is the system causal? Is it stable? b) (5 points) Write the di↵erence equation that characterizes the system. (Sam) a). We have X (z ) = 1 1− z = z− 1 −1 2z − 1 2 − 1 1 − 2 z −1 (14) z z−2 (15) −3z 1 2 , < |z | < 2 1 2 (z − 2 )(z − 2) 6 6 Y (z ) = 1 −1 − 1 − 2z 1 − 3 z −1 4 = = (z − −3 2z 1 2 )(z − 3) 4 |z | > , 3 . 4 (16) (17) (18) Then H (z ) = Y (z ) z−2 = X (z ) z−3 4 (19) = (20) 10 1 − 2z −1 1 − 3 z −1 4 3 The ROC of H (z ) is either |z | > 3 or |z | < 4 . Furthermore, the region of 4 convergence of Y (z ) must include at least the intersection of H (z ) and X (z ), therefore the ROC of H (z ) must be |z | > 3 . 4 (21) b). Thus the system is stable (since the ROC includes the unit circle) and causal (since the ROC is the exterior of a circle). c). 3 y [n] − y [n − 1] = x[n] − 2x[n − 1] 4 2 (22)...
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