Final answer:
The question pertains to down sampling a discrete-time signal using MATLAB. The explanation covers the possible effects of down sampling, including aliasing and the loss of signal detail. A MATLAB script example is provided for practical demonstration.
Step-by-step explanation:
The student's question involves down sampling a discrete-time signal and observing the effects of this process. The provided signal is x[n] = cos(2πn/7). The process of down sampling by a factor of m involves taking every mth sample from the original signal, which results in the compressed signal y₁[n] = x[mn]. A MATLAB script can be written to perform this, and the resulting signals can be compared graphically. The act of down sampling effectively reduces the sampling rate and can lead to aliasing if the original signal contains frequencies higher than half of the new sampling rate (Nyquist rate).
Here is a simple example MATLAB script that can be adjusted for the specific down sampling factor m:
n = 0:30;
m = 3; % Example down sampling factor
original_signal = cos(2 * pi * n / 7);
compressed_signal = cos(2 * pi * (m*n) / 7);
figure;
subplot(2,1,1);
stem(n, original_signal);
title('Original Signal x[n]');
xlabel('n');
ylabel('Amplitude');
subplot(2,1,2);
stem(n(1:m:end), compressed_signal(1:m:end));
title('Compressed Signal y1[n]');
xlabel('n');
ylabel('Amplitude');
The result of down sampling is that certain signal details might be lost, and depending on the factor m, different patterns of compression will emerge.