How To design a Butterworth low-pass filter in MATLAB, follow these steps for both 1D signals and 2D images.

% Example: Filter a noisy sine wave
Fs = 1000;          % Sampling frequency (Hz)
t = 0:1/Fs:1;       % Time vector
f_signal = 10;      % Signal frequency (Hz)
f_noise = 200;      % High-frequency noise (Hz)

% Generate signal + noise
x = sin(2*pi*f_signal*t) + 0.5*sin(2*pi*f_noise*t);

% Step 1: Define filter parameters
n = 4;              % Filter order
cutoff = 50;        % Cutoff frequency (Hz)
Wn = cutoff/(Fs/2); % Normalized cutoff (0 < Wn < 1)

% Step 2: Design Butterworth filter
[b, a] = butter(n, Wn, 'low');

% Step 3: Apply filter (zero-phase)
y = filtfilt(b, a, x);

% Step 4: Plot results
figure;
subplot(2,1,1);
plot(t, x, 'b', t, y, 'r', 'LineWidth', 1.5);
legend('Noisy Signal', 'Filtered Signal');
title('Time Domain');
xlabel('Time (s)');

% Frequency response
[h, freq] = freqz(b, a, 1024, Fs);
subplot(2,1,2);
plot(freq, 20*log10(abs(h)), 'LineWidth', 1.5);
title('Frequency Response');
xlabel('Frequency (Hz)');
ylabel('Gain (dB)');
xlim([0 Fs/2]);
grid on;

% Example: Remove high-frequency noise from an image
img = im2double(imread('cameraman.tif'));
[M, N] = size(img);

% Add noise
noisy_img = imnoise(img, 'salt & pepper', 0.05);

% Step 1: Design Butterworth filter
n = 4;              % Filter order
D0 = 30;            % Cutoff frequency (pixels)

% Create grid of frequencies
[u, v] = meshgrid(-N/2:N/2-1, -M/2:M/2-1);
D = sqrt(u.^2 + v.^2);          % Distance from center
H = 1 ./ (1 + (D./D0).^(2*n));  % Butterworth LPF formula

% Step 2: Apply filter in frequency domain
F = fftshift(fft2(noisy_img));
F_filtered = F .* H;
filtered_img = real(ifft2(ifftshift(F_filtered)));

% Step 3: Display results
figure;
subplot(1,3,1), imshow(img), title('Original');
subplot(1,3,2), imshow(noisy_img), title('Noisy Image');
subplot(1,3,3), imshow(filtered_img), title('Filtered Image');