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Unformatted text preview: ECE 421 - Sum 2010 Detailed Solutions - Set 3: Z-Transforms 1 3-1. A signal s [ m has the analytical form s [ m = A m u [ m . Compute the z-transform S z if A has the values given below. (a) A = : 45, (b) A = : 72, (c) A = : 3 e j : 1 , (d) A = : 5 j : 2 DETAILED SOLUTION: The z-transform, S z , of the s [ m given in the problem statement is shown below: s [ m = A m u [ m S z = 1 1 Az 1 = z z A a Thus, for this problem we have S z = z z A b Therefore, we simply need to substitute the appropriate value of A into equation b to obtain the desired z-transform. (a) A = : 45 Substitute A = : 45 into the z-transform given in equation b , obtaining S z = z z : 45 c (b) A = : 72 Substitute A = : 45 into the z-transform given in equation b , obtaining S z = z z : 72 d Simplify the expression on the right side of d to obtain the desired form for the transform: S z = z z : 72 e ECE 421 - Sum 2010 Detailed Solutions - Set 3: Z-Transforms 2 3-1 (cont.) (c) A = : 3 e j : 1 Substitute A = : 3 e j : 1 into the z-transform given in equation b , obtaining the desired form for the transform: S z = z z : 3 e j : 1 f (d) A = : 5 j : 2 This transform is simplified if we write A in the polar form A = M e j . It is straightforward to compute the magnitude M and phase angle : M = : 5 2 : 2 2 = : 538 g = tan 1 : 2 : 5 = : 380 = : 121 (in radians) h The results in g and h allow us to write A as A = : 538 e j : 121 i Finally, substituting i into equation b gives the desired form for the...
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This note was uploaded on 09/07/2010 for the course ECE 421 taught by Professor Hallen during the Summer '08 term at N.C. State.
- Summer '08
- Signal Processing