MAS160: Signals, Systems & Information for Media TechnologyProblem Set 5DUE: November 5, 2003Instructors: V. Michael Bove, Jr. and Rosalind Picard T.A. Jim McBrideProblem 1: Unit-step and running average (DSP First 5.5)Problem 2: ConvolutionFor each of the following sets of signals, compute their convolution (1) graphically by hand,(2) with MATLAB (you may use the conv function), and (3) by expressing the signalsin terms of δ[n] and computing the convolution sum. In matlab, plot your results withstem, but be sure to fix the n-axis appropriately (use stem(n,y) where n is a vector of theappropriate range).For each of the following of signals, compute their convolution with x[n] = cos(2π(116)n)using matlab (you may use the conv function). Use stem to plot your result over the range[0:99], assuming the sinusoid exists for all time. Compare each convolution with x[n].(a) h[n] =12δ[n] +12δ[n − 1](b) h[n] = δ[n] − δ[n − 1]PS 5-1Problem 3: Time-domain response of FIR filters (DSP First 5.6)Problem 4: LTI SystemsConsider the interconnection of LTI systems as shown below.h1[n]h4[n]h3[n]h2[n]+x[n]y[n]-(a) Express the overall impulse response, h[n], in terms of h1[n], h2[n], h3[n] and h4[n].(b) Determine h[n] whenh1[n] = {12,14,12}h2[n] = h3[n] = (n + 1)u[n]h4[n] = δ[n − 2]Problem 5: Block Diagrams (DSP First 5.9)Problem 6: MAS.510 Additional ProblemIt is possible to determine the impulse response for a LTI system using a system of equations,given enough information about the system. For example, if we know that the system isFIR and has no delay and that y[0] = 1 if x[n] = δ[n], theny[n] = ax[n]y[0] = ax[0]1 = a ∗ 1a = 1Using systems of equations, compute the impulse response given the following systemdescriptions and input-output pairs(a) FIR and single delay, x[n] = δ[n], y[0] = 2, y[1] = −2(b) FIR and double delay, x[n] = δ[n], y[0] = 3, y[1] = −4, y[2] = 3/2(c) FIR and double delay, x[n] = 4δ[n] + δ[n − 1], y[0] = 2, y[1] = 2, y[2] = −1(d) Calculate y[3] for each of the preceding systems.PS
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