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probset-three - EE533 Problem Set-3 1 Consider the length-9...

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EE533 Problem Set-3 1. Consider the length-9 sequence x ( · ) = { 2 , 3 , ( - 1) , 0 , - 4 , 3 , 1 , 2 , 4 } . Evaluate the following functions of X ( e ) without computing the transform itself: (a) X ( e j 0 ) (b) X ( e ) (c) π - π X ( e ) (d) π - π | X ( e ) | 2 (e) π - π dX ( e ) 2 Determine the following: (a) Inverse-DTFT of X ( e - ) (b) Inverse-DTFT of X ( - e ) 2. An LTI system with impulse response { h ( n ) } and frequency response H ( e ) has the following properties: (a) the system is causal; (b) H ( e - ) = H * ( e ); (c) the DTFT of the sequence { h ( n + 1) } is real; (d) H ( e ) = 0; (e) π - π H ( e ) = 4 π . From the above information, determine h ( · ) completely. 3. Determine the Inverse-DTFT of (a) X ( e ) = cos N ω , for an arbitrary integer N > 0. (b) Y ( e ) = cos N ( ω/ 2), for an arbitrary even integer N > 0. Express the cosine function in terms of complex exponentials, and apply the binomial theorem. 4. Determine the RoC of the z -transform of each of the following sequences without explic- itly finding X ( z ), and specify whether the DTFT exists in each case: (a) x ( n ) = [(1 / 2) n - ( - 4 / 3) n ] u ( n - 10)
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