Explore DFT frequency accuracy, got stuck

I’m trying to analyze the frequency accuracy of the DFT as a function of DFT size assuming a rectangular
window. Using single sinusoidal waveform sampled at 5 kHz vary its frequency in small
increments to show the how the frequency accuracy changes for at least 4 different DFT length,
256, 512, 1024, and 4096.

Here is my code:

%Part 1
% MATLAB code to analyze the frequency accuracy of the DFT manually
fs = 5000; % Sampling frequency 5 kHz
f_start = 1000; % Start frequency of sinusoid (1 kHz)
f_end = 1050; % End frequency of sinusoid (1.05 kHz)
f_step = 1; % Frequency step size (small increments)
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];
% Prepare figure
figure;
hold on;
for N = N_dft
freq_accuracy = []; % Store accuracy results for each DFT size
for f = f_start:f_step:f_end
% Generate sinusoidal waveform
x = sin(2*pi*f*t);

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
X = x_windowed*dftmtx(N);

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

% Find the index of the peak in the magnitude of the DFT
[~, peak_idx] = max(abs(X));

% Estimated frequency from the DFT
f_estimated = freq_axis(peak_idx);

% Store the frequency error (absolute difference between true and estimated frequency)
freq_error = abs(f_estimated – f);
freq_accuracy = [freq_accuracy, freq_error];
end

% Plot the frequency accuracy for the current DFT size
plot(f_start:f_step:f_end, freq_accuracy, ‘DisplayName’, [‘N = ‘, num2str(N)]);
end
% Add labels and legend
xlabel(‘True Frequency (Hz)’);
ylabel(‘Frequency Error (Hz)’);
title(‘Frequency Accuracy of the DFT for Different DFT Sizes (Manual DFT)’);
legend(‘show’);
grid on;
hold off;

However, the result I got is really weird.

At some frequency the estimate error between true frequency can be 3000, while most of the time it’s under 1.

However, it I substitue this line: X = x_windowed*dftmtx(N);
with this line: X = fft(x_windowed, N);

Then I got this image

It seems that this is the correct figure.
I read from this website that fft and dftmtx function can be interchangeble
https://www.mathworks.com/help/signal/ref/dftmtx.html
But my results doesn’t approve it. I want to know why.

Besides, am I applying windowing function in a correct way?
The code is as following:
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;I’m trying to analyze the frequency accuracy of the DFT as a function of DFT size assuming a rectangular
window. Using single sinusoidal waveform sampled at 5 kHz vary its frequency in small
increments to show the how the frequency accuracy changes for at least 4 different DFT length,
256, 512, 1024, and 4096.

Here is my code:

% Manually compute the DFT
X = x_windowed*dftmtx(N);

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

% Find the index of the peak in the magnitude of the DFT
[~, peak_idx] = max(abs(X));

% Estimated frequency from the DFT
f_estimated = freq_axis(peak_idx);

% Store the frequency error (absolute difference between true and estimated frequency)
freq_error = abs(f_estimated – f);
freq_accuracy = [freq_accuracy, freq_error];
end

However, the result I got is really weird.

At some frequency the estimate error between true frequency can be 3000, while most of the time it’s under 1.

However, it I substitue this line: X = x_windowed*dftmtx(N);
with this line: X = fft(x_windowed, N);

Then I got this image

Besides, am I applying windowing function in a correct way?
The code is as following:
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; I’m trying to analyze the frequency accuracy of the DFT as a function of DFT size assuming a rectangular
window. Using single sinusoidal waveform sampled at 5 kHz vary its frequency in small
increments to show the how the frequency accuracy changes for at least 4 different DFT length,
256, 512, 1024, and 4096.

Here is my code:

% Manually compute the DFT
X = x_windowed*dftmtx(N);

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

% Find the index of the peak in the magnitude of the DFT
[~, peak_idx] = max(abs(X));

% Estimated frequency from the DFT
f_estimated = freq_axis(peak_idx);

% Store the frequency error (absolute difference between true and estimated frequency)
freq_error = abs(f_estimated – f);
freq_accuracy = [freq_accuracy, freq_error];
end

However, the result I got is really weird.

At some frequency the estimate error between true frequency can be 3000, while most of the time it’s under 1.

However, it I substitue this line: X = x_windowed*dftmtx(N);
with this line: X = fft(x_windowed, N);

Then I got this image

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