SOL03 - Signals and Systems Fall 2012 Solution to Homework...

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Signals and Systems Fall 2012 Solution to Homework Assignment #3 1. (a) y ( k 1) y ( k ) x ( k ), y (0) 0, x ( k ) 0 , 0 0 , 1 k k . We can rewrite the equation as follows: ( 1) ( ) ( ) y k y k x k   By using the initial condition, we have: (0) 0 (1) (0) (0) 0 1 1 (2) (1) (1) 1 1 0 (3) (2) (2) 0 1 1 y y y x y y x y y x              After taking a few iterations, it can be easily seen that the sequence continues to oscillate between 0 and 1. (b) y ( k 1) y ( k ) 0, y (0) 1. The system can be described as: 1 ( ) y k y k   Taking a few iterations:   ( ) 0 1 (2) (1) 1 (3) (2) 1 1 (4) (3) 1 y y y y y y y y             Similar to (a), after a few iterations, the sequence will oscillate alternatively between -1 and 1. (c) y ( k 2) y ( k 1) 2 y ( k ) x ( k 1) x ( k ), y (0) y (1) 0, x ( k ) 0 , 0 0 , 1 k k . The equation can be written as below: ( ) 2 1 2 1 ( ) ( ) y k y k y k x k x k Taking a few iterations: (0) 1 (0) 0 (1) 0 (1) 0 x y x y
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