Use MATLAB residuez command to determine partial fractions, no long division, ma

Question

Use MATLAB residuez command to determine partial fractions, no long division, may use MATLAB zplane command to plot poles and zeros, just set the axes so they are visible

4. lhe continuous-time signal Xc(t) = sin (20m) + cos (40m) is sampled with a sampling period T to obtain the discrete-time signal 211 sin (-n) + cos (22) . x[n] (a) Determine a choice for T consistent with this information. (b) Is your choice for T in part (a) unique? If so, explain why. If not, specify another choice of T consistent with the information given.

Explanation / Answer

a) Given, the continuous time signal Xc(t) = sin(20*pi*t ) + cos(40 * pi*t)

for a sampling period of T ; replacing t =n *T in the above signal

X(n) = sin(20*pi *n*T) + cos(40*pi*n*T) which is given as X(n) = sin(pi*n/5) + cos(2*pi*n/5)

therefore, 20*T = 1/5 ; T = 0.01 sec = 10 milli seconds (sampling period)

b) Since, the sinusoids repeats values for every phase shift of 2*pi ;

for the sampling frequency of 2pi + pi/5 = 11pi/5 rad/sec with sampling period of T= 2pi / (11pi/5) = 10/11 = 0.91sec

so the choice of T in part (a) can not be unique.

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