Transfer Functions-Digital Signal Processing-Assignment Solution, Exercises of Digital Signal Processing

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183 G.1. We proceed by obtaining the transfer functions for each of the networks. For network 1, ¥(z) = 2r cos @2“¥ (2) — 22-°¥ (z) + X(z) or ¥(z) | 1 X(z} 1 = 2rcos@z-! 4722-7 For network 2, define W(z) as in the figure below: By(z)= Wiz) zo rood =rsing rsind rcosé fate me vip] wo then W(z) = X(z) —rsin6z7'¥ (z) + rcos@z 7 W(z) and ¥(z) =rsin@z7'W{z) + 6 cos 82~'Y(z) Eliminate W(z) to get ¥(2) rsin@z— Xz) = 2p cosBz7) + r22-2 Bence the two networks have the same poles. Ay(z) = 6.2. The only input to the y{n] node is a unity branch connection from the z[n] node. The rest of the network does not affect the input-output relationship. The difference equation is y(n] = z[n]. 6.3. 24327! BQ Ta System (d) is recognizable as a transposed direct form I] implementation of H(z). 6.4, (a) From the flow graph, we have: ¥ (2) = 2X(2) + GX) ~ Pe) + Sy(eet