How to use LUT in matlab to generate DTFT signals with different combinations of low and high frequencies

To use LUT (Look-Up Table) in MATLAB to generate DTFT signals with different combinations of low and high frequencies, you can follow the steps below:

1. Define the frequency range and step size for the DTFT. For example, you can use the following code to define the frequency range from -pi to pi with a step size of 0.01:

w = -pi:0.01:pi;

2. Create a LUT for the DTFT using the function exp() to calculate the complex exponential. For example, you can use the following code to create a LUT for the DTFT:

N = length(w);
LUT = zeros(N,N);
for n = 1:N
    for k = 1:N
        LUT(n,k) = exp(-1i*w(k)*n);
    end
end

This code creates a LUT LUT(n,k) for the DTFT with n and k representing the indices for the frequency and time domains, respectively. The exp() function calculates the complex exponential with the given frequency and time values.

3. Define the frequency components for the DTFT signal. For example, you can create a low-frequency sinusoid and a high-frequency sinusoid using the following code:

low_freq = 0.2*pi;
high_freq = 0.8*pi;
x1 = sin(low_freq*w);
x2 = cos(high_freq*w);

4. Calculate the DTFT of the signal by multiplying the LUT with the signal. For example, you can use the following code to calculate the DTFT of the signal:

X1 = LUT*x1.';
X2 = LUT*x2.';

This code multiplies the LUT with the signal using the transpose (.') to ensure that the signal is in the correct orientation for the multiplication. The resulting DTFT signals X1 and X2 are complex-valued vectors representing the frequency-domain representation of the low-frequency and high-frequency sinusoids, respectively.

5. Plot the DTFT signals using the plot() function. For example, you can use the following code to plot the DTFT signals:

figure;
subplot(2,1,1);
plot(w,abs(X1));
title('DTFT of Low-Frequency Sinusoid');
xlabel('Frequency (rad/sample)');
ylabel('Magnitude');

subplot(2,1,2);
plot(w,abs(X2));
title('DTFT of High-Frequency Sinusoid');
xlabel('Frequency (rad/sample)');
ylabel('Magnitude');

This code plots the magnitude of the DTFT signals using the abs() function and the plot() function, with the first subplot showing the DTFT of the low-frequency sinusoid and the second subplot showing the DTFT of the high-frequency sinusoid. The resulting plot shows the frequency-domain representation of the signals with the low-frequency sinusoid having a peak at the low-frequency and the high-frequency sinusoid having a peak at the high-frequency.