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EE 341 Spring 2009 Final Exam Solutions corrected pages 3 and 4

# EE 341 Spring 2009 Final Exam Solutions corrected pages 3 and 4

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EE 341 Spring 2009 Prof. Atlas Page 2 of 10 Problem 1 Consider the block diagram below, where x[n] and y[n] are the system’s input and output sequences, respectively. For your convenience, we label intermediate nodes’ sequences as [ 1], [ ], and [ 1] w n w n w n + . The numbers in circles represent scalar multiplication. For example, [ ] 3 [ ] y n w n = . a) Derive the difference equation for the above system as a function of input sequence x[n] and output sequence y[n]. (As you proceed to your solution, you should end up removing all temporary sequences, such as w[n].) (15 points) Solution: ( ) 1 3 1 1 2 2 1 1 2 4 [ ] 3 [ ] [ ] [ ] [ 1] [ ] [ ] ( 1) [ ] [ ] ( 1) y n w n w n y n w n x n w n w n x n w n w n = = + = + = Plug y[n] in for temporary variable w[n] 1 1 1 1 1 3 2 3 4 3 1 1 2 4 [ 1] [ ] ( ) ( 1) [ 1] 3 [ ] ( ) ( 1) y n x n y n y n y n x n y n y n + = + = 1 1 2 4 [ 1] ( ) ( 1) 3 [ ] y n y n y n x n + + + = b) Find the input/output transfer function, ( ) H z , for the above system.
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EE 341 Spring 2009 Final Exam Solutions corrected pages 3 and 4

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