hw5_600 - N d ∈ Z 3 A signal x t containing 3 real-valued...

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ECE-600 Introduction to Digital Signal Processing Winter 2011 Homework #5 Feb. 12, 2011 HOMEWORK ASSIGNMENT #5 Due Fri. Feb. 18, 2011 (in class) 1. Consider two sequences: x = [1 , 2 , 3] and w = [1 , 2 , 3 , 4]. (a) Compute the linear convolution of these sequences on paper (show your work), and verify using Matlab’s conv command. (b) Compute the 4-circular convolution of these sequences on paper (show your work). (c) In Matlab, perform ifft(fft([1,2,3],4).*fft([1,2,3,4],4)) , recalling that fft(x,N) performs an N -DFT, ifft(x,N) performs an N -IDFT, and .* performs pointwise multipli- cation. Does the output match the result of your 4-circular convolution? If so, why? (d) Compute the 6-circular convolution of these sequences on paper (show your work). (e) In Matlab, perform ifft(fft([1,2,3],6).*fft([1,2,3,4],6)) . Does the output match the result of your 6-circular convolution? What about your linear convolution? If so, why? 2. Prove the DFT shift/modulation properties x [ a n d A N ] N -DFT ←→ e - j 2 π N kd X [ k ] , d Z e j 2 π N nd x [ n ] N -DFT ←→ X [ a k d A
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Unformatted text preview: N ] , d ∈ Z . 3. A signal x ( t ) containing 3 real-valued sinusoids has been sampled at rate 120 Hz to produce the samples in the ±le hw5.txt . You are to estimate the frequencies of the 3 components (not their amplitudes) using the spectral analysis method outlined in lecture. (a) For each of the following windows: rectangular triangular Hann Hamming Blackman Blackman-harris, calculate the windowed spectrum using Matlab’s fft command, and then plot the spectral magnitude in amplitude units of dB and frequency units of Hz (from 0 to 60 Hz). Identify which (if any) of the 3 sinusoids are apparent from each plot. ( Hint : fft(x,N) zero-pads x to length-N before computing the DFT.) (b) Repeat part (a) using Kaiser windows with di²erent choices of parameter β . Make plots only for the useful values of β (i.e., those which allow positive identi±cation of some frequency component). P. Schniter, 2011 1...
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This note was uploaded on 02/21/2011 for the course ECE 600 taught by Professor Clymer,b during the Spring '08 term at Ohio State.

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