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

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The Butterworth filter is characterized by a maximally flat frequency response in the passband and is widely used for noise reduction while preserving edges.

Design for 1D Signals (Time Series)

Steps:

  1. Define filter specifications: Order (n), cutoff frequency (Wn), and sampling frequency (Fs).
  2. Design the filter: Use MATLAB’s butter function to compute filter coefficients.
  3. Apply the filter: Use filtfilt for zero-phase distortion.
  4. Visualize results: Plot the original/filtered signals and frequency response.

MATLAB Code:

matlab
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% 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;

Design for 2D Images

Steps:

  1. Create a frequency-domain mask using the Butterworth formula.
  2. Apply the filter in the frequency domain via FFT.
  3. Invert the FFT to get the filtered image.

MATLAB Code:

matlab
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% 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');

Key Parameters:

  • Order (n): Higher order = sharper roll-off (steeper transition between passband and stopband).
  • Cutoff (Wn or D0): Frequency beyond which attenuation begins.
  • Sampling Frequency (Fs): Critical for normalizing Wn in 1D.

Notes:

  • Zero-Phase Filtering: Use filtfilt instead of filter to eliminate phase distortion in 1D.
  • Frequency Domain for Images: The 2D filter is applied via FFT for computational efficiency.
  • Adjust Parameters: Tune n and D0/Wn based on your data’s noise characteristics.
  • Toolbox Requirement: Requires the Signal Processing Toolbox for butter and filtfilt.

This approach works for smoothing signals, denoising images, or preprocessing data for analysis. For real-time applications, consider using designfilt for more control over filter properties.

Published by howdy