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Wztrans3

Wztrans3 - ECE 421 Sum 2010 Detailed Solutions Set 3...

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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 0 45, (b) A 0 72, (c) A 0 3 e j 0 1 π , (d) A 0 5 j 0 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 0 45 Substitute A 0 45 into the z -transform given in equation b , obtaining S z z z 0 45 c (b) A 0 72 Substitute A 0 45 into the z -transform given in equation b , obtaining S z z z 0 72 d Simplify the expression on the right side of d to obtain the desired form for the transform: S z z z 0 72 e

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ECE 421 - Sum 2010 Detailed Solutions - Set 3: Z-Transforms 2 3-1 (cont.) (c) A 0 3 e j 0 1 π Substitute A 0 3 e j 0 1 π into the z -transform given in equation b , obtaining the desired form for the transform: S z z z 0 3 e j 0 1 π f (d) A 0 5 j 0 2 This transform is simplified if we write A in the polar form A M e j φ . It is straightforward to compute the magnitude M and phase angle φ : M 0 5 2 0 2 2 0 538 g φ tan 1 0 2 0 5 0 380 0 121 π (in radians) h The results in g and h allow us to write A as A 0 538 e j 0 121 π i Finally, substituting i into equation b gives the desired form for the z -transform: S z z z 0 538 e j 0 121 π j
ECE 421 - Sum 2010 Detailed Solutions - Set 3: Z-Transforms 3 3-2.

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