MAS160: Signals, Systems & Information for Media TechnologyProblem Set 8DUE: December 1st, 2003Instructors: V. Michael Bove, Jr. and Rosalind Picard T.A. Jim McBrideProblem 1: Return of the rabbits!HINT :This problem is supposed to demonstrate that z-transforms are useful outside the realmof signal processing. Luckily this problem can be solved using just the things we’ve taughtyou so far. The idea is to transform the equation for r[n] into an equation for R[z] by takingthe transform. You should be able to solve for R[z] directly after the transform.There are 2 potentially tricky things about this problem. The first is figuring out whathappens to the terms that look like r[n − 1]. You’ll want to use the z-transform rules tomake them look more like R[z]. The second problem is that you need to specify the initialcondition, namely r[1] = 1. Think δ function.Problem 2: Inverse z-TransformsHINT :The inverse z-transforms are all solved in the same fashion: do a partial fraction ex-pansion and perform the inversion on the terms individually. Since the inverse is typicallynot unique, you’ll need to use the requirements (like causal or stable) to pick out which ofthe possible inverse functions we’re looking for.Problem 3: Utilizing the z-transform (DSP First 8.12)HINT :In part (a) you’ll determining the system function H (z) of the filter described in thisproblem. When applying the filter, resist the temptation to take an inverse transform ofH(z) directly. It will be much easier to take the forward transform of the input signal,multiply, and then invert the transform of the whole expression.PS 8-1Signals, Systems & Information : Problem Set 8 Hints PS 8-2Problem 4: MAS 510 Additional ProblemHINT :If you want the overall system function to be unity (i.e. the identity function) you’relooking forˆh(n) such thatˆh(n) ∗ h(n) = 1. In the z domain however this just meansˆH(z)H(z) = 1. If you know H(z) you can solve this directly.Problem 5: Discrete Fourier Transforms (DSP First 9.2)Problem 6: Inverse DFT (DSP First 9.3)HINT :Both of these problems make liberal use of the geometric series summation formula andthe simple values of ej2πx(like ejπ= −1). In every case the summation can be evaluated toeither give the answer in either a direct or case by case way.Problem 7: Convolution revisitedHINT :Just follow the problem. Don’t get ahead of yourself and start padding the lists withzeros. That’s the next problem!Problem 8: MAS 510 Additional ProblemHINT :This is a problem designed to teach you about windowing and the trade off betweengetting good resolution in different domains. Think about how the shape of the Hanningwindow will effect the FFT. It will help to look at the shape of the Hanning window directly(i.e. look at hanning(32) as well as hanning(32).*x). Multiplication in the time domainis convolution in the frequency domain, so you can think of windowing functions as filtersthat are applied in the frequency domain.PS
View Full Document
Unlocking...