Explore DFT Resolution, need help with answer interpretation

Hello there, I’m relatively new to Matlab and Digital Signal Processing. I’m trying to analyze the frequency resolution of the DFT as a function of the DFT size assuming a rectangular window. Using the a sum of sinusoids sampled at 5 kHz vary the frequency spacing
of the sinusoids to show the frequency resolution for at least 4 different DFT lengths, 256, 512, 1024, and 4096.

I got two versions of code from chatgpt. But I don’t know how to interpret them.

1st version
% MATLAB code to analyze the frequency resolution of the DFT using dftmtx and a rectangular window
fs = 5000; % Sampling frequency (5 kHz)
t_duration = 1; % Signal duration in seconds
t = 0:1/fs:t_duration-1/fs; % Time vector
% DFT sizes to analyze
N_dft = [256, 512, 1024, 4096];
% Frequencies of the sum of sinusoids (vary frequency spacing)
f1 = 1000; % Frequency of the first sinusoid (1 kHz)
f_spacing = [1, 5, 10, 20]; % Frequency spacing between the two sinusoids
f_end = f1 + f_spacing; % Frequency of the second sinusoid
% Prepare figure
figure;
hold on;
for N = N_dft
figure; % Create a new figure for each DFT size
hold on;
for spacing = f_spacing
f2 = f1 + spacing; % Second sinusoid frequency

% Generate the sum of two sinusoids with frequencies f1 and f2
x = sin(2*pi*f1*t) + sin(2*pi*f2*t);

% Apply rectangular window (by taking the first N samples)
x_windowed = x(1:N); % Select the first N samples for DFT calculation

% Generate DFT matrix for size N using dftmtx
DFT_matrix = dftmtx(N);

% Manually compute the DFT using the DFT matrix
X = DFT_matrix * x_windowed(:); % Compute DFT of the windowed signal

% Compute the frequency axis for the current DFT
freq_axis = (0:N-1)*(fs/N);

% Plot the magnitude of the DFT
plot(freq_axis, abs(X), ‘DisplayName’, [‘Spacing = ‘, num2str(spacing), ‘ Hz’]);
end

% Add labels and legend
xlabel(‘Frequency (Hz)’);
ylabel(‘Magnitude’);
title([‘DFT Magnitude Spectrum (N = ‘, num2str(N), ‘)’]);
legend(‘show’);
grid on;
hold off;
end
which gives me these resulted figures.

All the peaks are so similar that I can’t distinguish them at all.

2nd version of code is here:
% Parameters
fs = 5000; % Sampling frequency (Hz)
T = 1; % Signal duration (seconds)
t = 0:1/fs:T-1/fs; % Time vector
% Sinusoids: Frequencies and amplitudes
f1 = 400; % Frequency of first sinusoid (Hz)
f2 = 450; % Frequency of second sinusoid (Hz)
f3 = 500; % Frequency of third sinusoid (Hz)
% Create the signal: sum of three sinusoids
x = sin(2*pi*f1*t) + sin(2*pi*f2*t) + sin(2*pi*f3*t);
% Define DFT lengths
DFT_lengths = [256, 512, 1024, 4096];
% Plot frequency resolution for each DFT length using a rectangular window
figure;
for i = 1:length(DFT_lengths)
N = DFT_lengths(i); % DFT size

rectangular_window = rectwin(N)’;
% Apply rectangular window (implicitly applied since no windowing function)
% Select only the first N samples for each DFT calculation
x_windowed = x(1:N).*rectangular_window;

% Manually compute the DFT
% Compute the N-point DFT
X = (dftmtx(N)*x_windowed(:)).’; % Compute the N-point DFT
X_mag = abs(X); % Magnitude of DFT

% Frequency vector
f = (0:N-1) * (fs / N);

% Plot the DFT magnitude (first half)
subplot(length(DFT_lengths), 1, i);
plot(f(1:N/2), X_mag(1:N/2));
title([‘DFT Magnitude Spectrum with Rectangular Window, N = ‘, num2str(N)]);
xlabel(‘Frequency (Hz)’);
ylabel(‘|X(f)|’);
grid on;
end
% Adjust plot layout
sgtitle(‘Frequency Resolution for Different DFT Sizes Using Rectangular Window’);
%Part 5
% MATLAB code to analyze zero-padding in the DFT using a rectangular window
fs = 5000; % Sampling frequency (5 kHz)
t_duration = 1; % Signal duration in seconds
t = 0:1/fs:t_duration-1/fs; % Time vector
% Window sizes to analyze
window_sizes = [256, 512];
% Zero-padded DFT sizes to analyze
N_dft = [1024, 2048, 4096];
% Frequencies of the sum of sinusoids (vary frequency spacing)
f1 = 1000; % Frequency of the first sinusoid (1 kHz)
f_spacing = [5, 10]; % Frequency spacing between the two sinusoids
f_end = f1 + f_spacing; % Frequency of the second sinusoid
% Prepare figure
for window_size = window_sizes
figure; % Create a new figure for each window size
hold on;
for N = N_dft
for spacing = f_spacing
f2 = f1 + spacing; % Second sinusoid frequency

% Generate the sum of two sinusoids with frequencies f1 and f2
x = sin(2*pi*f1*t) + sin(2*pi*f2*t);

% Apply rectangular window (by taking the first window_size samples)
x_windowed = x(1:window_size); % Select the first window_size samples

% Zero-pad the signal if DFT size is larger than window size
x_padded = [x_windowed, zeros(1, N – window_size)];

% Generate DFT matrix for size N using dftmtx
DFT_matrix = dftmtx(N);

% Manually compute the DFT using the DFT matrix
X = DFT_matrix * x_padded(:); % Compute DFT of the windowed and zero-padded signal

% Compute the frequency axis for the current DFT
freq_axis = (0:N-1)*(fs/N);

% Plot the magnitude of the DFT
plot(freq_axis, abs(X), ‘DisplayName’, [‘Spacing = ‘, num2str(spacing), ‘ Hz, N = ‘, num2str(N)]);
end
end

% Add labels and legend
xlabel(‘Frequency (Hz)’);
ylabel(‘Magnitude’);
title([‘Zero-Padded DFT Magnitude Spectrum (Window Size = ‘, num2str(window_size), ‘)’]);
legend(‘show’);
grid on;
hold off;
end

with the second version of code I can see the differences among different sample sizes, as more sampling yields more sharp signals, but why is that and what differences it will make?Hello there, I’m relatively new to Matlab and Digital Signal Processing. I’m trying to analyze the frequency resolution of the DFT as a function of the DFT size assuming a rectangular window. Using the a sum of sinusoids sampled at 5 kHz vary the frequency spacing
of the sinusoids to show the frequency resolution for at least 4 different DFT lengths, 256, 512, 1024, and 4096.