Hi @Shamim,
The code below performs a comparative time-frequency analysis using complex Morlet wavelets. It illustrates two methods—FFT-based convolution and time-domain convolution—to extract how power varies across time and frequency in a synthetic multi-component signal.
A signal is constructed using known sinusoidal components combined with noise to simulate real-world behavior. A Morlet wavelet is generated for each frequency from 0.1 Hz to 20 Hz. In the first method, both signal and wavelet are transformed via FFT, multiplied in the frequency domain, and inverted to yield power over time. In the second method, the signal is directly convolved with each wavelet using MATLAB’s “conv” function.
Both resulting time-frequency representations (tf for FFT-based, tf_conv for time-domain) are visualized using “imagesc”, allowing direct comparison.
% Sampling parameters
srate = 100; % Sampling rate in Hz
duration = 10; % Duration in seconds
t = 0:1/srate:duration; % Time vector
n = length(t);
% Generate random signal with multiple components
rng(1); % For reproducibility
% Components: sinusoids + random noise
signal = ...
0.7 * sin(2*pi*2*t) + ... % 2 Hz
0.5 * sin(2*pi*10*t + pi/4) + ... % 10 Hz with phase shift
0.3 * sin(2*pi*15*t) + ... % 15 Hz
0.2 * randn(size(t)); % White noise
% Optional: visualize the signal
figure;
plot(t, signal);
xlabel('Time (s)');
ylabel('Amplitude');
title('Randomly Initialized Synthetic Signal');
% Define frequency range
freqs = 0.1:0.1:20; % in Hz
% Define time vector long enough to capture low frequencies
srate = 100; % sampling rate in Hz
t = -5:1/srate:5; % symmetric time vector (adjust as needed)
nTime = length(t);
nFreq = length(freqs);
% FFT of signal
nData = length(signal);
signalFFT = fft(signal, nTime); % Zero-pad to match wavelet size
% Output matrix
tf = zeros(nFreq, nTime); % stores time-frequency representation
% Loop through frequencies
for fi = 1:nFreq
f = freqs(fi);
sigma_f = 0.7;
s = sigma_f / (2*pi*f); % standard deviation in time
% Create complex Morlet wavelet
wavelet = exp(2*1i*pi*f*t) .* exp(-t.^2/(2*s^2));
% FFT of wavelet
waveletFFT = fft(wavelet, nTime);
waveletFFT = waveletFFT / max(abs(waveletFFT)); % normalize
% Convolve via frequency domain multiplication
convolutionResult = ifft(waveletFFT .* signalFFT, nTime);
% Store power or phase
tf(fi, :) = abs(convolutionResult).^2; % or angle(...) for phase
end
% Time-frequency representation is stored in tf(frequency, time)
% Frequencies for analysis
freqs = 0.1:0.1:20; % Frequency range (Hz)
nFreq = length(freqs);
% Morlet wavelet parameters
sigma_f = 0.7;
tf_conv = zeros(nFreq, n); % Time-frequency result (conv)
% Loop over each frequency
for fi = 1:nFreq
f = freqs(fi);
s = sigma_f / (2*pi*f); % std dev in time domain
% Define time for wavelet (ensure it's long enough)
t_wavelet = -5:1/srate:5;
% Create complex Morlet wavelet
wavelet = exp(2*1i*pi*f*t_wavelet) .* exp(-t_wavelet.^2/(2*s^2));
% Time-domain convolution
convResult = conv(signal, wavelet, 'same');
% Store power
tf_conv(fi, :) = abs(convResult).^2;
end
% Time-frequency plot - Time-Domain Convolution
figure;
imagesc(t, freqs, tf_conv);
axis xy;
xlabel('Time (s)');
ylabel('Frequency (Hz)');
title('Time-Frequency Power (Time-Domain Convolution)');
colormap jet;
colorbar;
% If FFT-based
:
figure;
imagesc(t, freqs, tf);
axis xy;
xlabel('Time (s)');
ylabel('Frequency (Hz)');
title('Time-Frequency Power (FFT-Based Convolution)');
colormap jet;
colorbar;
Please refer to the MATLAB documentation below for information on Morlet Wavelet:
I hope this is beneficial!
