Solutions_Assignment4_w2010

Solutions_Assignment4_w2010 - Carleton University...

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1 of 7 Carleton University Department of Systems and Computer Engineering SYSC-4405: Digital Signal processing Winter Semester 2010 Solutions of Assignment # 4 Date Due: March 23, 2010, before 1:00 p.m. ______________________________________________________________________________ Problem 1 use matlab to answer this question Consider a system that is described by the difference equation where b =[0.0076 0.0455 0.1138 0.1517 0.1138 0.0455 0.0076] and a = [ 1.0000 -1.4851 1.6036 -0.9241 0.3592 -0.0756 0.0073] The transfer function of such a system is given by The frequency response of this system is given by . a) From the system coefficients given above, determine the poles and zeros of the system, and show that it is stable. [ You may use the “roots” command of matlab to determine the poles and zeroes of the system]. Now use the “zplane” command to generate the pole-zero diagram for this system. >> b=[0.0076 0.0455 0.1138 0.1517 0.1138 0.0455 0.0076]; >> a=[1.0000 -1.4851 1.6036 -0.9241 0.3592 -0.0756 0.0073]; >> poles=roots(a); zero=roots(b); poles_mag=abs(poles); >> poles_mag' ans = 0.7836 0.7836 0.4581 0.4581 0.2380 0.2380 We see that the magnitudes of all poles are less than one. The system is stable. ai () yn i i 0 = 6 bi xn i i 0 = 6 = Tz z i i 0 = 6 z i i 0 = 6 ------------------------- = He j ω ze j ω = =
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2 of 7 b) Determine and plot the frequency response of this system, for frequencies in the range . Provide separate plots for the magnitude and phase responses of this system. From your observation of this result what kind of symmetry seems to have? [ to calculate the frequency response you may use matlab’s “freqz” command] >> w=-pi:pi/128:pi; H=freqz(b,a,w); subplot(2,1,1); plot(w,abs(H)) >> xlabel('freq, rad/sample') >> subplot(2,1,2) >> plot(w,angle(H)) >> xlabel('frequency, rad/sample') >> ylabel('phase, radians') >> axis([-pi pi -4 4]) -1.5 -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Real Part Imaginary Part pole zero diagram π ωπ ≤≤ He j ω ()
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Solutions_Assignment4_w2010 - Carleton University...

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