Question

3 Accuracy of Sampling Let us sample the function gt)cos(2fo) at sampling interval dt- 1, for frequency values of f 0,0.25,0.5,0.75, 1.0 hertz. In each case, plot on the screen the points of the resulting time series (as isolated red crosses) to see how well it approximates g(t) (plotted as a blue-dotted line, try a very small dt fine sampling). Submit only plots for frequencies of 0.25 and 0.75 Hertz, use xlabel, ylabel, title commands to annotate each plot. For each frequency that you investigated, do you think the sampling time series is a fair representation of the original time series g )? What is the apparent frequency for the sampling time series? (Figure out after how many points (N) the series repeats itself, then the apparent frequency 1/(N*dt). You can do this either mathematically or by inspection. A flat time series has apparent frequency = 0.) Can you guess with a sampling interval of dt = 1 , what is the maximum frequency f of g() such that it can be fairly represented by the discrete time series? (Please attach your Python code.)

Answer 1

The required MATLAB CODE is as follows:

fs=0.01; %Very high sampling rate
f=1; %Frequency of sinusoid
nCyl=5; %generate five cycles of sinusoid
maxf=nCyl*f;
t=0:fs:maxf; %time index
x=cos(2*pi*f*t); % continious wave

fs1=0.25; %1/0.25 sampling rate
t1=0:fs1:maxf; %time index
x1=cos(2*pi*fs1*t1);

fs2=0.75; % 1/0.75 sampling rate
t2=0:fs2:maxf; %time index
x2=cos(2*pi*fs2*t2);

figure
subplot(2,1,1);
plot(t,x);
hold on;
stem(t1,x1); % plot the first sample
xlabel('Time(s)');
ylabel('Amplitude')
title('f=0.25');

subplot(2,1,2);
plot(t,x);
hold on;
stem(t2,x2); % plot the first sample
xlabel('Time(s)');
ylabel('Amplitude')
title('f=0.75')

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