HW2_2005

Digital Signal Processing: A Practical Approach (2nd Edition)

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Homework Set #2: z-Transform 1 . Please find z-transform and its region of convergence of each of the following sequences: (a) x [ n ] = ( 1) n u [ n 2] + u [ n 1]; (b) x [ n ] = ( −2 ) n u [ n 3] + ( −3 ) n u [ n 1]; (c) x [ n ] = −2 δ [ n 2]+ 3 [ n + 2], (d) x [ n ] = ( 0.2) n ( n 1) u [ n 1]; (e) x [ n ] = ( 2) n u [ n 2]; (f) x [ n ] = n u [ n ] for n 7; x [ n ] = 7 for n > 7 2. Let x [ n ] be the sequence with poles z = 5 1 4 1 j + , z = 5 1 4 1 j , z = 3 1 and zeros at z = 1, z = 1, find poles and zeros of ( a ) y [ n ] = ] 3 [ ) 2 1 ( ] 1 [ ) 3 1 ( + n x n x n n ; (b) w [ n ] = ] 2 [ ) 2 cos( ] 2 [ ) 2 sin( π + π n x n n x n ; (c) ) s [ n ] = ] 2 [ ) 1 ( ] 2 [ ) 2 ( + + n x n x n n ; (d) w [ n ] = 2 ] 3 [ ) exp( n x jn ; 3. Determine the inverse z-transform of (a) ] 1 includes ROC ), cos( ) ( > = z z z X ; (b) X (z) = 1 ) 1 2 ( -1 )( z z for l z l > 1; (c) X
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