Homework 12

Homework 12 - EE 4541 Digital Signal Processing Fall 2007...

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Unformatted text preview: EE 4541 Digital Signal Processing Fall 2007 Problem Set 12 Due: December 6, 2007 1. Problem 10.12 P&M 2. Problem 10.19 (a) (c) P&M 3. An analog signal y (t ) = x1 (t ) + x 2 (t ) is sampled at Fs and is to be filtered to remove the undesired component x 2 (t ) . You are given the following information about the signal components: 0, > 2 30 rad / s x1 (t ) : X 1 ( j) = 1, < 2 30 rad / s 0, < 2 40 & > 2 60 rad / s x 2 (t ) : X 2 ( j) = 1, 2 40 < < 2 60 rad / s The sampled signal y[n] is filtered by a digital lowpass filter with the following objectives: To attenuate the undesired component, x 2 [n] , such that the magnitude of its Fourier transform in the domain is less than 0.001 of . To pass the desired component, x1 [n] , with a tolerance of 0.01 . Two options are considered for the sampling frequency: (1) Fs=120 Hz, and (2) Fs=100 Hz. Furthermore, we will consider both ParksMcClellan and Kaiser window design methods. a) Which sampling frequency will result in a lower order filter? Consider all design specifications, not just edge frequencies. Give specific values and justify your answer. You can answer this question based on the Kaiser window design method. b) For the sampling frequency selected in Part a), which filter design method results in a lower order filter? Give specific values and justify your answers. c) Which filter design method is more appropriate for the design specifications given in this problem? Explain your answer. 4. Let x(t ) be the signal x(t ) = a1 cos(2 F t + 1 ) + a2 cos(2 1.2 F t + 2 ) . The frequency F is unknown, but is known to lie in the range 1000 Hz < F < 2000 Hz (with strict inequalities). The amplitudes and phases of the two sinusoids are real, but otherwise unknown. a) Specify the lowest sampling frequency that will allow us to reconstruct x(t ) unambiguously from its samples for the given range of values for f. An answer based on straight application of the Nyquist Sampling Theorem will be acceptable. EE 4541 Digital Signal Processing Fall 2007 Problem Set 12 Due: December 6, 2007 b) We wish to estimate a1 and a 2 using sizeN DFT. Given that 0.0025 < a2 < 1 , suggest a1 an approach using the Kaiser window for estimating these parameters from the DFT spectrum. Give the sampling interval, T, and window size, L, and the Kaiser window parameter, , assuming worst case analysis and show the details of the calculations. Find N assuming radix2 DIT algorithm used for computing the DFT. c) An alternative procedure is to generate the signal y (t ) = x(t ) cos(2 F0 t ), and pass it through an ideal LPF with an appropriately cutoff frequency, Fc. Suggest appropriate values of F0 and Fc. d) Let z (t ) be the output of the filter. Give the values of T, N, L, and for this case. Assume worstcase scenarios in your calculations. The following may be useful in solving the problem: Kaiser Window parameters can be obtained using the following equations: 0, Asl < 13.26, 0.4 = 0.76609( Asl - 13.26) + 0.09834( Asl - 13.26), 13.26 < Asl < 60, 0.12438( Asl + 6.3), 60 < Asl < 120. L 24 ( Asl + 12) +1 155 ml where Asl is the maximum sidelobe level (in dB) relative to the mainlobe level and ml is the mainlobe width (symmetric distance between central zerocrossings). ...
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