630F09_hw2 - ENEE 630 F09 Homework#2 Due Friday October 2...

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ENEE 630 F’09 Homework #2 Due: Friday October 2, 2009 at 6:00 PM Problem 1 Consider the structure shown in Fig P-1(a), where W is the 3 × 3 DFT matrix. This is a three channel synthesis bank with three filters F 0 ( z ), F 1 ( z ), F 2 ( z ). (For example F 0 ( z ) = Y ( z ) /Y 0 ( z ) with y 1 ( n ) and y 2 ( n ) set to zero.) a) Let R 0 ( z ) = 1 + z - 1 , R 1 ( z ) = 1 - z - 2 , R 2 ( z ) = 2 + 3 z - 1 . Find expressions for the three synthesis filters F 0 ( z ), F 1 ( z ), F 2 ( z ). b) The magnitude response of F 1 ( z ) is sketched in Fig. P-1(b). Plot the responses | F 0 ( e ) | and | F 2 ( e ) | . Does the relation between F 0 ( z ), F 1 ( z ), and F 2 ( z ) depend on choices of R k ( z )? y 0 (n) y 1 (n) y 2 (n) W R 0 (z 3 ) R 1 (z 3 ) R 2 (z 3 ) z -1 z -1 y(n) Figure: P-1(a) 1 π /3 2 π /3 π 2 π ϖ | F 1 (e j ϖ ) | 0 Figure: P-1(b) 1
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Problem 2 For a uniform DFT analysis bank, we know that the filters are related by H k ( z ) = H 0 ( zW k ), 0 k M - 1, with W = e - j 2 π/M . Let M = 5 and define two new transfer functions G 1 ( z ) = H 1 ( z ) + H 4 ( z ) and G 2 ( z ) = H 2 ( z ) + H 3 ( z ). Let h 0 ( n ) denote the impulse response of H 0 ( z ), assumed to be real for all n.
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