MATLAB for Signals

Using MATLAB for signal processing — essential functions, filter design, spectral analysis, and system simulation with the Signal Processing Toolbox.

Introduction

MATLAB (Matrix Laboratory) has been the dominant tool in signal processing education and research for decades. Its Signal Processing Toolbox provides hundreds of specialized functions for filter design, spectral analysis, statistical signal processing, and system simulation. While Python has gained ground as a free alternative, MATLAB remains the standard in many academic courses and industrial applications due to its comprehensive documentation, intuitive syntax for matrix operations, and specialized toolboxes.

This guide covers the essential MATLAB functions and workflows you need for a signals and systems course — from basic signal generation through advanced filter design and spectral analysis. Each section includes runnable code that you can paste directly into MATLAB or GNU Octave for experimentation.

Signal Generation

matlab example

fs = 8000;                      % Sampling rate
t = 0:1/fs:1-1/fs;            % Time vector (1 second)
N = length(t);                  % Number of samples

% Basic signals
x_sin = sin(2*pi*440*t);       % 440 Hz sine (concert A)
x_cos = cos(2*pi*1000*t);     % 1 kHz cosine
x_exp = exp(-5*t);             % Decaying exponential
x_step = heaviside(t - 0.3);  % Unit step at t=0.3

% Complex signals
x_chirp = chirp(t, 100, 1, 2000);  % Linear chirp 100-2000 Hz
x_noise = randn(1, N);              % White Gaussian noise
x_am = (1 + 0.5*sin(2*pi*5*t)) .* cos(2*pi*500*t);  % AM signal

% Impulse train (for sampling demonstrations)
x_impulse = zeros(1, N);
x_impulse(1:100:end) = 1;           % Impulse every 100 samples

% Multi-tone signal (sum of sinusoids)
x_multi = sin(2*pi*200*t) + 0.5*sin(2*pi*500*t) + 0.3*sin(2*pi*1200*t);

Key points: Always define fs and t first. The time vector 0:1/fs:1-1/fs gives exactly fs samples in one second. Use .* for element-wise multiplication (not matrix multiplication).

Core Signal Processing Functions

Convolution

matlab example

y = conv(x, h);          % Linear convolution
y = conv(x, h, 'same');  % Same length as x (centered)
y = conv(x, h, 'valid'); % Only fully-overlapping region

% Example: Convolve exponential with rectangle
h_rect = ones(1, 50)/50;        % 50-point moving average
y_smooth = conv(x_noisy, h_rect, 'same');

% Circular convolution (periodic, length N)
y_circ = ifft(fft(x, N) .* fft(h, N));

Filtering

matlab example

y = filter(b, a, x);     % Apply IIR/FIR filter (causal, real-time)
y = filtfilt(b, a, x);   % Zero-phase filtering (non-causal, stored data)

The difference between filter and filtfilt is critical: filter is causal (introduces phase distortion but works in real-time), while filtfilt runs the filter forward and backward, canceling phase distortion but requiring the entire signal in memory.

Correlation

Example

[Rxx, lags] = xcorr(x); % Autocorrelation [Rxy, lags] = xcorr(x, y); % Cross-correlation [Rxx, lags] = xcorr(x, 'coeff'); % Normalized (correlation coefficient) % Plot autocorrelation stem(lags, Rxx); xlabel('Lag'); ylabel('R_{xx}'); title('Autocorrelation Function');

Filter Design

Example

% Butterworth low-pass (maximally flat passband) [b, a] = butter(6, 1000/(fs/2)); % 6th order, 1 kHz cutoff % Chebyshev Type I (equiripple passband, sharper transition) [b, a] = cheby1(4, 0.5, 800/(fs/2)); % 4th order, 0.5 dB ripple % Chebyshev Type II (equiripple stopband) [b, a] = cheby2(4, 40, 800/(fs/2)); % 4th order, 40 dB stopband % Elliptic (sharpest transition for given order) [b, a] = ellip(4, 0.5, 40, 800/(fs/2)); % Combines ripple specs % FIR filter (Parks-McClellan optimal equiripple) b_fir = firpm(50, [0 0.2 0.3 1], [1 1 0 0]); % 50-tap, LP % Window-based FIR design b_win = fir1(64, 0.25, 'low', hamming(65)); % 64-tap with Hamming % Frequency response visualization freqz(b, a, 1024, fs); % Plots magnitude and phase

Filter selection guide:

Butterworth: maximally flat magnitude, gentle rolloff — good for general use
Chebyshev I: sharper rolloff than Butterworth at cost of passband ripple
Elliptic: sharpest possible rolloff for a given order, has both passband and stopband ripple
FIR (Parks-McClellan): linear phase guarantee, higher order needed for sharp transitions

Spectral Analysis

Example

% FFT spectrum X = fft(x, N); f = (0:N-1)*(fs/N); plot(f(1:N/2), abs(X(1:N/2))/N); xlabel('Frequency (Hz)'); ylabel('|X(f)|'); title('Single-Sided Amplitude Spectrum'); % Zero-padded FFT for interpolated frequency display Nfft = 4*N; % Zero-pad to 4x length X_zp = fft(x, Nfft); f_zp = (0:Nfft-1)*(fs/Nfft); plot(f_zp(1:Nfft/2), 20*log10(abs(X_zp(1:Nfft/2))/N)); ylabel('Magnitude (dB)'); % Power Spectral Density (Welch's method) [Pxx, f] = pwelch(x, hamming(256), 128, 256, fs); plot(f, 10*log10(Pxx)); xlabel('Frequency (Hz)'); ylabel('PSD (dB/Hz)'); % Spectrogram (time-frequency representation) spectrogram(x, 256, 200, 256, fs, 'yaxis'); colorbar; title('Spectrogram');

Practical note: Zero-padding the FFT does NOT increase frequency resolution — it only interpolates between existing frequency samples, producing a smoother-looking spectrum. To increase actual resolution, you need more data (longer observation time).

System Analysis

Example

% Transfer function definition num = [1 2]; den = [1 3 2]; % H(s) = (s+2)/(s^2+3s+2) sys = tf(num, den); % Impulse and step response figure; impulse(sys); title('Impulse Response'); figure; step(sys); title('Step Response'); % Pole-zero map figure; pzmap(sys); title('Pole-Zero Map'); % Bode plot (magnitude and phase vs frequency) figure; bode(sys); grid on; % Nyquist plot (for stability analysis) figure; nyquist(sys); % Discrete-time system sys_d = tf([1 0.5], [1 -0.8 0.15], 1/fs); figure; impulse(sys_d); % State-space representation A = [0 1; -2 -3]; B = [0; 1]; C = [1 0]; D = 0; sys_ss = ss(A, B, C, D);

Z-Transform Analysis

Example

% Pole-zero plot for discrete system b = [1 -0.5]; a = [1 -1.2 0.36]; figure; zplane(b, a); % Shows unit circle with poles and zeros title('Pole-Zero Plot (z-plane)'); % Stability check poles = roots(a); is_stable = all(abs(poles) < 1); fprintf('System is %s (pole magnitudes: %s)\n', ... char(["unstable" "stable"](is_stable+1)), ... mat2str(abs(poles), 3)); % Partial fraction expansion (for inverse Z-transform) [r, p, k] = residuez(b, a); fprintf('Residues: %s\n', mat2str(r, 4)); fprintf('Poles: %s\n', mat2str(p, 4)); % Frequency response of discrete system [H, w] = freqz(b, a, 512); plot(w/pi, 20*log10(abs(H))); xlabel('\omega/\pi (normalized frequency)'); ylabel('Magnitude (dB)');

Useful Workflow Patterns

Complete Analysis Pipeline

Example

% 1. Load/generate signal [x, fs] = audioread('signal.wav'); % 2. Visualize in time domain figure; plot((0:length(x)-1)/fs, x); xlabel('Time (s)'); ylabel('Amplitude'); % 3. Compute and plot spectrum figure; pwelch(x, [], [], [], fs); % 4. Design appropriate filter [b, a] = butter(4, [300 3000]/(fs/2), 'bandpass'); % 5. Apply filter y = filtfilt(b, a, x); % 6. Verify result in time and frequency domains figure; subplot(2,1,1); plot((0:length(y)-1)/fs, y); subplot(2,1,2); pwelch(y, [], [], [], fs);

Remove 60 Hz Interference

Example

x_noisy = x_clean + 0.5*sin(2*pi*60*t); [b, a] = butter(2, [58 62]/(fs/2), 'stop'); % Notch at 60 Hz x_filtered = filtfilt(b, a, x_noisy); % Verify removal figure; subplot(2,1,1); pwelch(x_noisy, [], [], [], fs); title('Before'); subplot(2,1,2); pwelch(x_filtered, [], [], [], fs); title('After');

Resampling and Interpolation

matlab example

% Downsample by factor M
M = 4;
x_down = decimate(x, M);   % Includes anti-aliasing filter

% Upsample by factor L
L = 3;
x_up = interp(x, L);       % Includes interpolation filter

% Resample to arbitrary rate
x_new = resample(x, 16000, fs);  % From fs to 16000 Hz

Key Takeaways

MATLAB's Signal Processing Toolbox provides comprehensive, well-documented functions
conv for convolution, filter/filtfilt for real-time/offline filtering
butter, cheby1, ellip, firpm for filter design — choose based on ripple tolerance
fft + proper frequency axis for spectral analysis; pwelch for PSD estimation
tf, impulse, step, bode, pzmap for continuous-time system analysis
zplane, freqz, residuez for Z-transform and discrete-system analysis
Zero-padding FFT interpolates but does not improve resolution
filtfilt gives zero phase distortion; filter is causal (real-time capable)
Most MATLAB code runs directly in GNU Octave with minimal modification

Introduction

Signal Generation

matlab example

fs = 8000;                      % Sampling rate
t = 0:1/fs:1-1/fs;            % Time vector (1 second)
N = length(t);                  % Number of samples

% Basic signals
x_sin = sin(2*pi*440*t);       % 440 Hz sine (concert A)
x_cos = cos(2*pi*1000*t);     % 1 kHz cosine
x_exp = exp(-5*t);             % Decaying exponential
x_step = heaviside(t - 0.3);  % Unit step at t=0.3

% Complex signals
x_chirp = chirp(t, 100, 1, 2000);  % Linear chirp 100-2000 Hz
x_noise = randn(1, N);              % White Gaussian noise
x_am = (1 + 0.5*sin(2*pi*5*t)) .* cos(2*pi*500*t);  % AM signal

% Impulse train (for sampling demonstrations)
x_impulse = zeros(1, N);
x_impulse(1:100:end) = 1;           % Impulse every 100 samples

% Multi-tone signal (sum of sinusoids)
x_multi = sin(2*pi*200*t) + 0.5*sin(2*pi*500*t) + 0.3*sin(2*pi*1200*t);

Key points: Always define fs and t first. The time vector 0:1/fs:1-1/fs gives exactly fs samples in one second. Use .* for element-wise multiplication (not matrix multiplication).

Core Signal Processing Functions

Convolution

matlab example

y = conv(x, h);          % Linear convolution
y = conv(x, h, 'same');  % Same length as x (centered)
y = conv(x, h, 'valid'); % Only fully-overlapping region

% Example: Convolve exponential with rectangle
h_rect = ones(1, 50)/50;        % 50-point moving average
y_smooth = conv(x_noisy, h_rect, 'same');

% Circular convolution (periodic, length N)
y_circ = ifft(fft(x, N) .* fft(h, N));

Filtering

matlab example

y = filter(b, a, x);     % Apply IIR/FIR filter (causal, real-time)
y = filtfilt(b, a, x);   % Zero-phase filtering (non-causal, stored data)

Correlation

Example

Filter Design

Example

Filter selection guide:

Butterworth: maximally flat magnitude, gentle rolloff — good for general use
Chebyshev I: sharper rolloff than Butterworth at cost of passband ripple
Elliptic: sharpest possible rolloff for a given order, has both passband and stopband ripple
FIR (Parks-McClellan): linear phase guarantee, higher order needed for sharp transitions

Spectral Analysis

Example

System Analysis

Example

Z-Transform Analysis

Example

Useful Workflow Patterns

Complete Analysis Pipeline

Example

Remove 60 Hz Interference

Example

Resampling and Interpolation

matlab example

% Downsample by factor M
M = 4;
x_down = decimate(x, M);   % Includes anti-aliasing filter

% Upsample by factor L
L = 3;
x_up = interp(x, L);       % Includes interpolation filter

% Resample to arbitrary rate
x_new = resample(x, 16000, fs);  % From fs to 16000 Hz

Key Takeaways

MATLAB's Signal Processing Toolbox provides comprehensive, well-documented functions
conv for convolution, filter/filtfilt for real-time/offline filtering
butter, cheby1, ellip, firpm for filter design — choose based on ripple tolerance
fft + proper frequency axis for spectral analysis; pwelch for PSD estimation
tf, impulse, step, bode, pzmap for continuous-time system analysis
zplane, freqz, residuez for Z-transform and discrete-system analysis
Zero-padding FFT interpolates but does not improve resolution
filtfilt gives zero phase distortion; filter is causal (real-time capable)
Most MATLAB code runs directly in GNU Octave with minimal modification

MATLAB for Signals

Introduction

Signal Generation

Core Signal Processing Functions

Convolution

Filtering

Correlation

Filter Design

Spectral Analysis

System Analysis

Z-Transform Analysis

Useful Workflow Patterns

Complete Analysis Pipeline

Remove 60 Hz Interference

Resampling and Interpolation

Key Takeaways

Continue learning this concept

MATLAB for Signals

Introduction

Signal Generation

Core Signal Processing Functions

Convolution

Filtering

Correlation

Filter Design

Spectral Analysis

System Analysis

Z-Transform Analysis

Useful Workflow Patterns

Complete Analysis Pipeline

Remove 60 Hz Interference

Resampling and Interpolation

Key Takeaways

Continue learning this concept