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EL 611 HW 6 - sin n n x o = results in the output sin o o o...

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EL611 3 Homework 5 1) Given an LTI system with real impulse response (i.e. the impulse response ) ( n h has no imaginary part), show that a) Any real input gives rise to a real output. Use the convolution sum to show this. b) If the response to a complex input, ) ( n x , is ) ( n y , then the response to the input ) ( * n x is ) ( * n y . Use the convolution integral again to show this. c) If the response to a complex input, ) ( n x , is ) ( n y , then the response to the input { } ) ( Re n x is { } ) ( Re n y , and the response to the input { } ) ( Im n x is { } ) ( Im n y . 2) Given an LTI system with transfer function ) ( z H , we know that the input n j o e n x ! = ) ( results in the output )) ( ( ) ( ) ( ) ( ) ( ) ( o o o o o o n j o n j j o n j j e A e e A e e H n y ! " ! ! ! " ! ! ! ! + = = = . a) Using the results of problem (1), part (c) above, show that the input ) cos( ) ( n n x o ! = results in the output )) ( cos( ) ( ) ( o o o n A n y ! " ! ! + = . b) Using the results of problem (1), part (c) above, show that the input
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Unformatted text preview: ) sin( ) ( n n x o = results in the output )) ( sin( ) ( ) ( o o o n A n y + = . 3) Given the function ) cos 8 17 )( cos 6 10 ( cos 4 5 ) (cos " + + = V . a) Is this the magnitude function of any rational ) ( z H ? b) If the answer to part (a) is yes, find any ) ( z H that has the magnitude function given. 4) Let x[n] be a causal, N-point sequence that is zero outside the range 0 <= n <= N-1. When this x[n] is the input to the causal LTI system described by: ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! a. Is y[n] also a causal N point sequence? b. Show that this system describes an all-pass filter....
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