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090206HomeWork3Solution

# 090206HomeWork3Solution - 1 2 2 2 1 1 2 2 2 1 Y z z H z z H...

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1 4. (Mitra, problem 13.40) Consider the system shown below. a) Show that the overall system is alias free, and calculate the transfer function ( ) ( ) ( ) Y z T z X z = in terms of H 0 ( z ) and H 1 ( z ). b) What is T ( z ) if 2 2 0 1 1 1 ( ) [ ( ) ( )], ( ) [ ( ) ( )]. 2 2 H z H z H z H z z H z H z = + - = - - c) If H ( z ) is a length 2 K filter, what are the lengths of the filters H 0 ( z ) and H 1 ( z )? d) How many multiplies and adds are required to compute one output sample if the length of H ( z ) is 2 K samples? Solution: Part (a): Output of top branch before the summer is 1 2 2 1 2 2 0 0 1 1 ( ) ( ) ( ) ( ) ( ) ( ) 2 2 z z H z X z z z H z X z - - - - - - - - Output of the middle branch before the summer is 1 2 2 2 1 2 2 2 0 1 0 1 1 1 ( )[ ( ) ( )] ( ) ( )[ ( ) ( )] ( ) 2 2 z z H z H z X z z z H z H z X z - - - - + + - + - Output of the bottom branch of the summer is 1 2 2 1 2 2 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) 2 2 z z H z X z z z H z X z - - - - - - - - - Combining all 3 branches,
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Unformatted text preview: 1 2 2 2 1 1 2 2 2 1 ( ) [ ( ) ( )] ( ) ( ) ( ) ( ) ( ) ( ) Y z z H z z H z X z Y z T z z H z z H z X z----= + = = + Part (b): Under the given conditions, 1 ( ) ( ) T z z H z-= Part (c): Each of the filters H ( z ) and H 1 ( z ) will be of length K . Part (d): For each output sample, the top branch will have K /2 multiplies and ( K-1)/2 + 1 adds, the middle branch will have K /2 multiplies and ( K-1)/2 + 2 adds, and the bottom branch will have K /2 multiplies and ( K-1)/2 + 1 adds. Putting it all together, the overall structure requires 3 K /2 multiplies and 3( K +5)/2 adds....
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090206HomeWork3Solution - 1 2 2 2 1 1 2 2 2 1 Y z z H z z H...

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