# ps3 - 2.171 Problem Set 3 Assigned Mon Due Wed Oct 4 2006...

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2.171 Problem Set 3 Assigned: Mon. Sept 25, 2006 Due: Wed. Oct 4 , 2006, in class Reading: F,P,&W Ch. 4, starred sections optional; F,P,&W 5.1, 5.2, 5.3 (Sections 5.4 and 5.5 are only of historical interest at this point in time; these techniques are largely obsolete in the present era.) Problem 1 FPW 4.19 Use Matlab to plot the discrete-time Bode plot for this plant assuming a DC gain of unity. Also, use Matlab to plot the system unit step response. Problem 2 FPW 4.28 Problem 3 FPW 4.29 Problem 4 Consider a continuous-time signal y a ( t ) = sin ωt which is sampled at a rate of T seconds/sample to yield y ( k ) = y a ( kT ). a) Show that y ( k ) = sin Ω k where Ω = ωT . b) One can notice that although the sampled sinusoid is a pure harmonic at frequency Ω, the time waveform appears to consist of beats as Ω approaches π . Some insight into this appearance may be gained as follows. Specifically, show that a discrete-time signal y ( k ) = sin Ω k may be written as y ( k ) = sin Δ k cos πk , and thus as a beat tone between frequencies Δ and π , where Δ π Ω.

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