Empirical wavelet transform
the multiresolution analysis (MRA) components corresponding to the empirical wavelet
transform (EWT) of
mra = ewt(
ewt to decompose
signals using an adaptable wavelet subdivision scheme that automatically determines the
empirical wavelet and scaling filters and preserves energy.
By default, the number of empirical wavelet filters is automatically determined by
identifying peaks in a multitaper power spectral estimate of
ewt(___) with no output arguments plots the original
signal with the empirical wavelet MRA in the same figure. For complex-valued data, the
real part is plotted in the first color in the MATLAB® color order matrix and the imaginary part is plotted in the second
Load and visualize a nonstationary continuous signal composed of sinusoidal waves with a distinct change in frequency. The signal is sampled at 250 Hz.
fs = 250; load nonstatdistinct t = (0:length(nonstatdistinct)-1)/fs; plot(t,nonstatdistinct) xlabel('Time (s)') ylabel('Signal') axis tight
ewt to obtain a multiresolution analysis (MRA) of the signal.
mra = ewt(nonstatdistinct);
Use the MRA components with the
hht function and plot the Hilbert spectrum.
The frequency versus time plot is a sparse plot with a vertical color bar indicating the instantaneous energy at each point in the MRA. The plot represents the instantaneous frequency spectrum of each component decomposed from the original mixed signal.
Create a nonstationary continuous signal composed of sinusoidal waves with a distinct change in frequency. The signal is sampled at 1000 Hz.
Fs = 1000; t = 0:1/Fs:4; x1 = sin(2*pi*50*t) + sin(2*pi*200*t); x2 = sin(2*pi*25*t) + sin(2*pi*100*t) + sin(2*pi*250*t); x = [x1 x2] + 0.1*randn(1,length(t)*2); t1 = (0:length(x)-1)/Fs; plot(t1,x) xlabel('Time (s)') ylabel('Amplitude') title('Signal')
ewt and obtain the MRA of the signal. Display the normalized peak frequencies identified in the signal, and the approximate frequency passbands of the filter bank. Because the frequencies are in cycles per sample, normalize by the sampling frequency. Note that the peak frequencies correspond to the frequencies of the sinusoidal waves.
[mra,~,wfb,info] = ewt(x); Fs*info.PeakFrequencies
ans = 5×1 249.9375 200.0750 100.1000 50.1125 25.1187
ans = 5×2 223.6941 500.0000 141.5896 223.6941 70.8573 141.5896 35.4911 70.8573 0 35.4911
Plot the magnitude spectrum of the signal, and the filter bank. The locations of the peaks determine the filter passbands.
f = 0:Fs/length(x):Fs-1/length(x); plot(f,wfb) ylabel('Magnitude') grid on yyaxis right plot(f,abs(fft(x)),'k--','linewidth',1.5) ylabel('Magnitude') xlabel('Hz')
Because the empirical wavelets form a Parseval tight frame, the analysis filter bank is equal to the synthesis filter bank. Therefore, squaring the magnitudes at each frequency summed over the filters equals 1. If the sum was not equal to 1, perfect reconstruction would not be possible.
Load an ECG signal. The signal is sampled at 180 Hz.
ewt to obtain a multiresolution analysis (MRA) of the signal, and the corresponding analysis coefficients. Use the four largest peaks to determine the filter passbands.
mp = 4; [mra,cfs] = ewt(wecg,'MaxNumPeaks',mp);
Plot the signal and the MRA components.
fs = 180; subplot(mp+1,1,1) t = (0:length(wecg)-1)/fs; plot(t,wecg) title('MRA of Signal') ylabel('Signal') axis tight for k=1:mp subplot(mp+1,1,k+1) plot(t,mra(:,k)) ylabel(['MRA ',num2str(k)]) axis tight end xlabel('Time (s)')
Verify that summing the MRA components results in perfect reconstruction of the signal.
ans = 8.8818e-16
Verify energy preservation of the EWT analysis coefficients.
cfsenergy = sum(sum(abs(cfs).^2)); [cfsenergy norm(wecg,2)^2]
ans = 1×2 298.2759 298.2759
x— Input data
Input data, specified as a real- or complex-valued vector or as a single-variable
timetable containing a single column vector.
x must have at least
Complex Number Support: Yes
comma-separated pairs of
the argument name and
Value is the corresponding value.
Name must appear inside quotes. You can specify several name and value
pair arguments in any order as
ewt(x,'MaxNumPeaks',5,'SegmentMethod','localmin')obtains the MRA of
xusing the five largest peaks and the first local minimum between adjacent peaks.
'PeakThresholdPercent'— Threshold percentage of maximum peak
70(default) | real number in the interval (0, 100)
Threshold percentage of maximum peak used to determine which peaks to retain in
the multitaper power spectrum of
x, specified as a real number in
the interval (0, 100). Local maxima in the multitaper power spectral estimate of
x are normalized to lie in the range [0,1] with the maximum
peak equal to 1. All peaks with values strictly greater than
PeakThresholdPercent of the maximum peak are retained.
'SegmentMethod'— Segmentation method
Segmentation method used to determine the EWT filter passbands, specified as:
'geomean' — Geometric mean of adjacent peaks
'localmin' — First local minimum between adjacent
If no local minimum is identified between adjacent peaks, the function uses the geometric mean.
'MaxNumPeaks'— Maximum number of peaks
Maximum number of peaks used to determine the EWT filter passbands. If
ewt finds fewer peaks than the number specified in
MaxNumPeaks, it uses the maximum number of peaks available. If it
does not find any peaks,
ewt uses a level-one discrete wavelet
transform (DWT) filter bank.
You cannot specify both
'FrequencyResolution'— Frequency resolution bandwidth
Frequency resolution bandwidth of the multitaper power spectral estimate, specified as a real number less than or equal to 0.25..
The value of
FrequencyResolution determines how many sine
tapers are used in the multitaper power spectrum estimate. The bandwidth of a sine
multitaper power spectral estimate is
(K+1)/(N+1), where K is the
number of tapers and N is the length of the signal. The minimum
FrequencyResolution is 2.5/N, where
N is the maximum of the signal length and 64.
'LogSpectrum'— Logarithm of spectrum
Logarithm of spectrum logical used to determine the peak frequencies. If
LogSpectrum is set to
true, the log of the
multitaper power spectrum is used. Consider setting
true if using the
segmentation method and there is a dominant peak frequency that is significantly
larger in magnitude than other peaks.
mra— Multiresolution analysis
Multiresolution analysis (MRA), returned as a matrix or timetable.
x is a vector,
mra is a matrix
where each column stores an extracted MRA component.
x, the MRA components are ordered
by decreasing center frequencies. The final column in
mra corresponds to the lowpass scaling filter.
x, the MRA components start near
−½ cycles per sample and decrease in center frequency until the lowpass
scaling coefficients are obtained. The frequency then increases toward +½
cycles per sample.
x is a timetable,
mra is a
timetable with multiple single variables where each variable stores an MRA
info structure array for a description of
the frequency bounds for empirical wavelet and scaling filters.
x has less than 64 samples,
on a zero-padded version of
x of length 64. The MRA components are
truncated to the original length.
cfs— EWT analysis coefficients
EWT analysis coefficients, returned as a matrix. If the input data is real-valued,
cfs is a real-valued matrix. Otherwise,
cfs is a complex-valued matrix. Each column of
cfs stores the EWT analysis coefficients for the corresponding
MRA component. The frequency bands of the analysis coefficients are identical to the
ordering of the MRA components. If
x has less than 64 samples,
cfs contains the analysis coefficients obtained from the
zero-padded version of
wfb— Empirical wavelet filter bank
Empirical wavelet filter bank, returned as a matrix. The center frequencies of the
wfb match the order in
cfs. Because the empirical wavelets form a Parseval tight frame,
the analysis filter bank is equal to the synthesis filter bank. Therefore, summing the
MRA components results in perfect reconstruction of the signal.
info— Filter bank information
Filter bank information, returned as a structure with the following fields:
PeakFrequencies — The peak normalized frequencies in
cycles/sample identified in
x as a column vector. For
x, the frequencies are positive in the interval
(0, ½) in decreasing order. For complex-valued
frequencies are ordered from (−½, ½). If
ewt did not find any peaks and a default one-level
discrete wavelet transform (DWT) subdivision is used.
FilterBank — A table with two variables:
MRAComponent is the column index of the MRA component in
Passbands is a
L-by-2 matrix where L is the number of MRA
components. Each row of
Passbands is the approximate frequency
passband in cycles/sample for the corresponding EWT filter and MRA
Complex Number Support: Yes
 Gilles, Jérôme. “Empirical Wavelet Transform.” IEEE Transactions on Signal Processing 61, no. 16 (August 2013): 3999–4010. https://doi.org/10.1109/TSP.2013.2265222.
 Gilles, Jérôme, Giang Tran, and Stanley Osher. “2D Empirical Transforms. Wavelets, Ridgelets, and Curvelets Revisited.” SIAM Journal on Imaging Sciences 7, no. 1 (January 2014): 157–86. https://doi.org/10.1137/130923774.
 Gilles, Jérôme, and Kathryn Heal. “A Parameterless Scale-Space Approach to Find Meaningful Modes in Histograms — Application to Image and Spectrum Segmentation.” International Journal of Wavelets, Multiresolution and Information Processing 12, no. 06 (November 2014): 1450044. https://doi.org/10.1142/S0219691314500441.
Usage notes and limitations:
Timetable input data is not supported.