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Quiz2Soln

# Quiz2Soln - Name ECE 301 Quiz 2 Purdue University Summer...

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Name: ECE 301: Quiz 2 Purdue University, Summer 2009 1. Let the response y ( t ) to a continuous-time system be related to the input x ( t ) by the following relationship: y ( t ) = 2 x ( t 2 ) . With yes/no answers (no proofs necessary) state whether the system is: (a) memoryless No. y (2) = 2 x (4) so that y ( t ) at time t = 2 depends on future values of the input x ( t ). (b) causal No. By the same argument as above. (c) invertible No. The response y ( t ) depends on the input x ( t ) only for positive values of t . Therefore consider two inputs x 1 ( t ) and x 2 ( t ) such that x 1 ( t ) = x 2 ( t ) for t 0 but x 1 ( t ) negationslash = x 2 ( t ) for t < 0. These two inputs will produce identical outputs, and therefore the system is not invertible. (d) stable Yes. If the input is bounded, there exists some bound B < such that | x ( t ) | < B for all t . Therefore the magnitude of the output | y ( t ) | = | 2 x ( t 2 ) | = 2 | x ( t 2 ) | ≤ 2 B < . (e) linear Yes. Let x 1 ( t ) and x 2 ( t ) are inputs with corresponding outputs y 1 ( t ) = 2 x 1 ( t 2 ) and y 2 ( t ) = 2 x 2 ( t 2 ), respectively. Now let a third input x 3 ( t ) = ax 1 ( t ) + bx 2 ( t ) for some constants a and b . We have that the system’s response to x 3 ( t ) is y 3 ( t ) = 2 x 3 ( t 2 ) = 2( ax 1 ( t 2 ) + bx 2 ( t 2 ) = ay 1 ( t ) + by 2 ( t ). (f) time invariant No.

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Quiz2Soln - Name ECE 301 Quiz 2 Purdue University Summer...

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