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630_rec_2_sol

# 630_rec_2_sol - ENEE630 ADSP 1 RECITATION 2 w solution...

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ENEE630 ADSP RECITATION 2 w/ solution September 17/18, 2009 1. Consider the structures shown in Fig. R2.1, with input transforms and filter responses as indicated. Sketch the quantities Y 0 ( e ) and Y 1 ( e ). Figure R2.1: Solution: Comment: For a down-sampled signal, it’s still possible to recover the original signal (not necessarily low-pass) using filters and multirate building blocks as long as there is no aliasing ( H 1 in this problem). 2. For each case shown in the Fig. R2.2, prove or disprove whether the left system is equivalent to the right system? Assume M, L, K are all integers larger than 1. Solution: Assume the input, output, intermediate signals are x ( n ) , y ( n ), and u ( n ), respectively. (a). (FD) U ( z ) = X ( z M ), Y 2 ( z ) = 1 M M - 1 k =0 U ( z 1 /M W k M ) = 1 M M - 1 k =0 X ( zW kM M ) = X ( z ). (TD) u ( n ) = x ( n M ) if n is a multiple of M , and u ( n ) = 0 otherwise. Then, y 2 ( n ) = u ( Mn ) = x ( n ). (b). (FD) U ( z ) = 1 M M - 1 k =0 X ( z 1 /M W k M ), Y 2 ( z ) = U ( z M ) = 1 M M - 1 k =0 X ( zW k M ). (TD) u ( n ) = x ( Mn ), y 2 ( n ) = u ( n M ) = x ( n ) only if n is a multiple of M , i.e., y 2 ( n ) = ( x ( n ) n is multiple of M 0 otherwise.

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630_rec_2_sol - ENEE630 ADSP 1 RECITATION 2 w solution...

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