A5 - Signals and Systems Issued 23 September 2015 Fall 2015...

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Signals and Systems Issued: 23 September 2015 Fall 2015 Due: Tuesday 29 September 2015 Homework Assignment #5 1. Are the following systems BIBO stable? Justify your answer. (a) (E 2 + 3E + 2){ y ( k )} = x ( k ) (b) (10E 2 + 3E + 2){ y ( k )} = (E 2){ x ( k )} (c) (E 2 + 1.2E + 0.2){ y ( k )} = x ( k ) (d) (E 3 + 3E 2 + 3E + 1){ y ( k )} = (E 2 0.1E){ x ( k )} 2. For each of the following systems, find an expression for the impulse response h ( k ) by iterating until you see a pattern. (a) y ( k ) = 25 . 0 1 E { x ( k )} [Ans: h ( k ) = (0.25) k 1 u s ( k 1).] (b) y ( k ) = ) 25 . 0 ( 1 E E { x ( k )} (c) y ( k ) = 1 2 1 2 + + E E { x ( k )} 3. Find the simulation diagrams and transfer functions of the systems having the following impulse responses: (a) h ( k ) = ¯ ® ­ = 2 , 0 2 , 1 k k (b) h ( k ) = ¯ ® ­ < 0 , 0 0 , 1 k k (c) h ( k ) = ¯ ® ­ < 1 , 0 1 , 1 k k (d) h ( k ) = ¯ ® ­ < 2 , 0 2 , 1 k k (e) h ( k ) = ° ° ¯ ° ° ® ­ = = = otherwise 0 2 , 3 1 , 2 0 , 1 k k k 4. Use a convolutional sum to find the zero-state response of an LTI system to a unit step sequence if the system impulse response is (a) h ( k ) = ¯ ® ­ < 0 , 0 0 , 1 k k ,
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(b) h ( k ) = ° ° ¯ ° ° ® ­ = = = otherwise 0 3 , 3 2 , 2 1 , 1 k k k , (c) h ( k ) = ¯ ® ­ < 0 , 0 0 , ) 9 . 0 ( k k k . [Ans: y ( k ) = 10 9(0.9) k , k = 0,1,2, ⋅⋅⋅ .]
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Signals and Systems Fall 2015 Solution to Homework Assignment #5 1. a) (E 2 + 3E + 2){ y ( k )} = x ( k ) The characteristic polynomial is 2 ( ) 3 2 ( 1)( 2) D E E E E E = + + = + + It can be seen that we have two roots 1, 2 , which are not inside the unit circle. The system is unstable. Therefore, the system is not BIBO stable.
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