Question

5. The z transform is a very useful tool for studying difference equations. Often difference and differential equations are used to describe causal systems and only the causal solution is of interest. This is the "initial condition" problem of a differential equations course. But both difference and differential equations describe more than just the causal system. For instance, "backwards" solutions and "two point boundary value" solutions. One way in which to think about the problem is the ROC of the transform: different choices of ROC give different solutions. Consider the difference equation (173) (a) Compute the system function for Eq. 173, i.e., H(z)--Y(z)/X(z) (b) Compute the poles and zeros of the system function. (Hint: Poles at 1/2 and 2, zeros at 3/4 and 3/2) (c) Plot the poles and zeros of the system function in the complex z plane (d) A ROC is an annular region, because it is a condition on , bounded by a pole, because the pole locations are the values of where the system function is infinite. On three different plots, show the three possible ROCs for Eq. 173 i. Let z[n] = an u[n]. Compute the z transfor!n X(z) including the ROC. ii. Let a[n]-an . Compute the z transform X(2) ncluding the ROC. (e) Notice that X(z) is the same, but the ROCs are different! (f) Use synthetic division and partial fractions expansion to express the system function in the form (174) 1- az (g) What is the ROC for the term c? (h) What are the two possible ROCs for the term (175) - az-1 (i) What are the two possible ROCs for the term (176) 1-bz-1 G) The ROC for H(z) has to be the intersection of the ROCs for each term. Counting gives 1x2x2- 4 ROCs for H(z). Why did you only find 3 ROCs in Item 5d? (k) For each of the 3 ROCs in Item 5d: i. Write down the ROCs for each of the terms in Eq. 174. i. Determine hinl, the impulse response, which is the inverse z transform of H(). iii. Is hn BIBO stable? iv. Is h[n] causal (i.e., h[n-0 for n 0), or non-causal (i.e., there are values of n that are both positive and negative for which hln] is nonzero).

Answer 1

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