solution7

solution7 - ECE 101 Linear Systems Fundamentals Problem Set...

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Unformatted text preview: ECE 101 Linear Systems Fundamentals Problem Set # 7 Solutions December 3, 2009 (send questions/comments to psiegel@ucsd.edu) Problem 1. 10.22(b) n n n x = 2 1 ] [ . First, let n n x = 2 1 ] [ 1 . Therefore, we have ] [ ] [ 1 n nx n x = . We can therefore use the differentiation in z-domain property to solve for X(z). But first, we need X 1 (z). Notice that we can rewrite x[n] as such: ( ) ( ) ( ) = + = < = ] 1 [ 2 ] [ 2 1 ] 1 [ 2 ] [ 2 1 ) 2 / 1 ( ) 2 / 1 ( ] [ 1 n u n u n u n u n for n for n x n n n n n n Using lines from the common transform pairs table, we get 1 1 1 ) 2 ( 1 1 ) 2 / 1 ( 1 1 ) ( = z z z X 2 2 1 : < < z ROC Using the differentiation in z-domain property, we can now compute X(z): = = 2 1 2 2 1 2 1 ) ) 2 ( 1 ( ) 2 ( ) ) 2 / 1 ( 1 ( ) 2 / 1 ( ) ( ) ( z z z z z dz z dX z z X 2 1 1 2 1 1 ) ) 2 ( 1 ( ) 2 ( ) ) 2 / 1 ( 1 ( ) 2 / 1 ( ) ( + = z z z z z X 2 2 1 : < < z ROC Since the ROC includes the unit circle |z| = 1, the DTFT does exist To draw the pole-zero plot, first write X(z) as a single ratio on the order of z instead of z-1 . If you do the math correctly, you should get:...
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solution7 - ECE 101 Linear Systems Fundamentals Problem Set...

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