Question

Digitization of signals Consider the analog signal s(t) = 5sin(500t +p 5)+ cos(200t +p 4) to be transferred over a digital communications system. (a) Compute the maximum allowable sampling period that the analog-to-digital converter (ADC) must use to ensure the perfect reconstruction of the signal at the receiver. (b) What theorem governs the choice made in part (a)? (c) How many samples of the analog signal do we need to store in order to reproduce 10p seconds of it (i.e., the analog signal)? (d) Use MATLAB code to verify whether or not the samples collected can be used to uniquely reproduce the analog signal.

Answer 1

(a) The maximum frequency sontent in the given signal is wmax = 500 rad /sec. => fmax = 500/(2*pi) = 79.6 Hz (approximately).

Minimum sampling rquired is fs = 2*fmax = 159.2 samples/sec.

maximum allowable sampling period is given by T = 1/fs = 0.0063 sec = 6.3ms

(b) The minimum sampling rate requirement in the above solution is taken from NYQUIST SAMPLING THEOREM.

Which states that for faithful reproduction of signal from its samles will occur if the sampling rate is atleast twice the maximum frequency content in the signal.

(c) In 10 seconds time 10/T samples = 10/0.0063 = 1587 samples are stored.

(d) MATLAb code is given below in bold letters.

T = 0.0063;
fs = 1/T;
t = 0:T:10;
s = 5*sin(500*t+pi/5)+cos(200*t+pi/4);
figure;plot(t,s);title('s(t) sampled at minimum sampling requirement');

it is observed that the signal content is preserved.