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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!Consider the case of the Fibonacci rabbits: They become fertile at one month of age, they have a gestation periodof one month, each litter is one male and one female, and they live and breed “forever,” from our perspective.From that description, we could write the following difference equation for the output of a system:r[n] = r[n − 1] + r[n − 2],where r is pairs of rabbits and n is months, and we will introduce 1 pair of rabbits into the system at n = 1(moreexplicitly, for n ≤ 0, r[n] = 0, and r[1] = 1).(a) Let’s think about this a bit more explicitly in z-transform terms. Where are the poles and zeroes of thissystem?(b) Suppose we add a new twist: only P percent of the fertile (i.e. at-least-two-month-old) rabbits give birthin any given month. Rewrite the difference equation. Now where are the poles and zeroes? If it makesyour life easier, from this point on you can forget that r must always be an integer. If you are looking fora challenge, try to retain that constraint!Problem 2: Inverse z-TransformsDetermine all possible signals that can have the following z-transforms with the given conditions.(a)11−32z−1+12z−2(b)2−32z−11−32z−1+12z−2, causal(c)31−103z−1+z−2, stable(d)1−12z−11+12z−1, right-handedProblem 3: Utilizing the z-transform (DSP First 8.12)Problem 4: MAS 510 Additional Problem(All-Pass System) Consider the causal linear shift-invariant system with system functionH(z) =1 − a−1z−11 − az−1where a is real(a) For what range of values of a is the system stable?(b) If 0 < a < 1, plot the pole-zero diagram and shade the region of convergence. Do the same for a > 1.(c) H(z) is to be cascaded with a systemˆH(z) so that the overall system function is unity. With 0 < a < 1andˆH(z) specified to be a stable system, determine its impulse responseˆh(n).PS 8-1Problem 5: Discrete Fourier Transforms (DSP First 9.2)Problem 6: Inverse DFT (DSP First 9.3)Problem 7: Convolution revisited(a) In matlab, create two vectors as follows:x1 = [1 1 1 1 0 0 0 0];x2 = [0 0 1 1 1 1 0 0];Now convolve the two using the conv function and plot the result using stem.(b) Now create two new vectors as follows:x3 = [1 1 1 1 1 0 0 0];x2 = [0 0 1 1 1 1 1 0];Again convolve the two using the conv function and plot the result using stem.(c) Remember convolution in the time domain is equivalent to multiplication in the Fourier (frequency) domain.So now calculate the convolution from part (a) by taking the FFT of both vectors (using the fft function),multiplying the FFTs (using .*), and inverse transforming back to the time domain (using the ifftfunction). You may need to take the real part of the inverse transform since very small numerical errorswill accumulate in MATLAB during the FFT and IFFT process. How does your result compare with part(a)?(d) Repeat this procedure for the vectors in part (b). How does your result compare with part (b)? Explainwhat appears to be happening.Hint 1: We’ve seen a similar phenomenon earlier in the semester.Hint 2: What is the FFT/DFT doing in the frequency domain?Problem 8: MAS 510 Additional ProblemStart in matlab with the following commands:n = 0:31;x = cos(2*pi*0.11*n);X = abs(fft(x));(a) Explain what the last line accomplishes. Plot x and X using stem. What information does the plot of Xconvey? Does it look like what you’d expect?(b) Now zero-pad x to a length of 2048 using the following command:x1 = [x zeros(1,2016)];Again take the FFT and plot the magnitude of the result. Explain what is going on.(c) Now window x with a Hanning window and zero-pad to a length of 2048 using the following command:x2 = [x.*hanning(32)’ zeros(1,2016)];Take the FFT of x2 and plot the magnitude of the result. Compare this to your plot from part(b). Howis it similar and how is it different? Explain what is going on.PS


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MIT MAS 160 - Problem Set 8

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