Time-averaged wavelet spectrum
returns the time-averaged wavelet power spectrum of the signal
tavgp = timeSpectrum(
using the continuous wavelet transform (CWT) filter bank
tavgp is obtained by time-averaging the magnitude-squared
scalogram over all times. The power of the time-averaged wavelet spectrum is normalized to
equal the variance of
timeSpectrum(___) with no output arguments plots the
time-averaged wavelet power spectrum in the current figure.
Time-Averaged Wavelet Spectrum of Oceanic Eddy Data
Load the NPG2006 dataset . The data is the trajectory of a subsurface float trapped in an eddy. Plot the eastward and northward displacement. The triangle marks the initial position.
load npg2006 plot(npg2006.cx) hold on grid on xlabel('Eastward Displacement (km)') ylabel('Northward Displacement (km)') plot(npg2006.cx(1),'^','markersize',11,'color','r',... 'markerfacecolor',[1 0 0 ])
Create a CWT filter bank that can be applied to the data. Use the default Morse wavelet. The sampling period for the data is 4 hours.
fb = cwtfilterbank('SignalLength',length(npg2006.cx),'SamplingPeriod',hours(4));
Obtain the time-averaged wavelet power spectra and the center periods.
[tavgp,centerP] = timeSpectrum(fb,npg2006.cx); size(tavgp)
ans = 1×3 73 1 2
The first page is the time-averaged wavelet spectrum for the positive scales (analytic part or counterclockwise component), and the second page is the time-averaged wavelet spectrum for the negative scales (anti-analytic part or clockwise component). Plot both spectra.
subplot(2,1,1) plot(centerP,tavgp(:,1,1)) title('Counterclockwise Component') ylabel('Power') xlabel('Period (hrs)') subplot(2,1,2) plot(centerP,tavgp(:,1,2)) title('Clockwise Component') ylabel('Power') xlabel('Period (hrs)')
If you omit the output arguments and execute
timeSpectrum(fb,npg2006.cx) on the command line, the scalograms and time-averaged power spectra are plotted in the current figure. Note that the clockwise rotation of the float is captured in the clockwise rotary scalogram and the time-averaged spectrum.
Normalize Time-Averaged Wavelet Spectrum
Load a time series of solar magnetic field magnitudes recorded hourly over the south pole of the sun by the Ulysses spacecraft from 21:00 UT on December 4, 1993 to 12:00 UT on May 24, 1994. See  pp. 218–220 for a complete description of this data. Create a CWT filter bank that can be applied to the data. Plot the scalogram and the time-averaged wavelet spectrum.
load solarMFmagnitudes fb = cwtfilterbank('SignalLength',length(sm),'SamplingPeriod',hours(1)); timeSpectrum(fb,sm)
Obtain the time-averaged wavelet spectrum of the signal using default values. By default,
timeSpectrum normalizes the power of the time-averaged wavelet spectrum to equal the variance of the signal. Verify that the sum of the spectrum equals the variance of the signal.
tavg = timeSpectrum(fb,sm); [var(sm) sum(tavg)]
ans = 1×2 0.0448 0.0447
Obtain the time-averaged wavelet spectrum of the signal, but instead normalize the power as a probability density function. Verify that the sum is equal to 1.
tavg = timeSpectrum(fb,sm,'Normalization','pdf'); sum(tavg)
ans = 1.0000
If you set
timeSpectrum normalizes the weighted integral of the wavelet spectrum according to the value of
Normalization. The spectrum mimics a probability density function whose integral, numerically evaluated, equals the value specified by
Plot the scalogram and the time-averaged wavelet spectrum with spectrum type
To confirm the integral of the spectrum equals 1, first obtain the time-averaged wavelet spectrum with
'density' spectrum type and
tavg = timeSpectrum(fb,sm,'SpectrumType','density','Normalization','pdf');
By default, the filter bank uses the analytic Morse (3,60) wavelet. Obtain the admissibility constant for the wavelet and numerically integrate the wavelet spectrum using the trapezoidal rule. Keep in mind that the CWT uses L1 normalization. Confirm that the integral equals 1.
ga = 3; tbw = 60; be = tbw/ga; anorm = 2*exp(be/ga*(1+(log(ga)-log(be)))); cPsi = anorm^2/(2*ga).*(1/2)^(2*(be/ga)-1)*gamma(2*be/ga); rawScales = scales(fb); numInt = 2/cPsi*1/length(sm)*trapz(rawScales(:),tavg./rawScales(:))
numInt = 1.0000
fb — Continuous wavelet transform filter bank
Continuous wavelet transform (CWT) filter bank, specified as a
x — Input data
Input data, specified as a real- or complex-valued vector. The input data
x must have at least four samples.
Complex Number Support: Yes
Specify optional pairs of arguments as
the argument name and
Value is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name in quotes.
500],'Normalization','none') returns the time-averaged wavelet spectrum averaged
over the time limits specified in samples without normalizing the spectrum.
Normalization — Normalization
'var' (default) |
Normalization of the time-averaged wavelet spectrum, specified as a
comma-separated pair consisting of
'Normalization' and one of the following:
SpectrumType — Type of wavelet spectrum
'power' (default) |
Type of wavelet spectrum to return, specified as a comma-separated pair consisting
'SpectrumType' and either
'density'. If specified as
averaged sum of the time-averaged wavelet spectrum over all times is normalized
according to the value specified in '
'density', the weighted integral of the wavelet
spectrum over all times is normalized according to the value specified in
With regards to the numerical implementation of the continuous wavelet transform, the integral over scale is performed using L1 normalization. With L1 normalization, if you have equal amplitude oscillatory components in your data at different scales, they will have equal magnitude in the CWT. Using L1 normalization provides a more accurate representation of the signal. For more information, see L1 Norm for CWT.
TimeLimits — Time limits
[1 length( or
[1 size( (default) | two-element vector
Time limits over which to average the wavelet spectrum, specified in samples.
TimeLimits is specified as a comma-separated pair consisting of
'TimeLimits' and a two-element vector with nondecreasing
elements. When you specify the input data as a signal, the elements are between 1 and
the length of
x. When you specify the input data as CWT
coefficients, the elements are between 1 and
tavgp — Time-averaged wavelet power spectrum
real-valued vector | real-valued 3-D array
Time-averaged wavelet power spectrum, returned as a real-valued vector or
real-valued 3-D array. If
x is real-valued,
tavgp is an F-by-1 vector, where
F is the number of wavelet center frequencies or center periods in
the CWT filter bank
tavgp is an F-by-1-by-2 array,
where the first page is the time-averaged wavelet spectrum for the positive scales
(analytic part or counterclockwise component), and the second page is the time-averaged
wavelet spectrum for the negative scales (anti-analytic part or clockwise
f — Center frequencies or center periods
column vector | duration array
Center frequencies or center periods for the time-averaged wavelet spectrum,
returned as a column vector or duration array, respectively. If the sampling frequency
is specified in
fb, then the elements of
the center frequencies ordered from high to low. If the sampling period is specified in
fb, then the elements of
f are the center
 Lilly, J. M., and J.-C. Gascard. “Wavelet Ridge Diagnosis of Time-Varying Elliptical Signals with Application to an Oceanic Eddy.” Nonlinear Processes in Geophysics 13, no. 5 (September 14, 2006): 467–83. https://doi.org/10.5194/npg-13-467-2006.
 Torrence, Christopher, and Gilbert P. Compo. “A Practical Guide to Wavelet Analysis.” Bulletin of the American Meteorological Society 79, no. 1 (January 1, 1998): 61–78. https://doi.org/10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2.
 Percival, Donald B., and Andrew T. Walden. Wavelet Methods for Time Series Analysis. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge ; New York: Cambridge University Press, 2000.
 Lilly, J.M., and S.C. Olhede. “Higher-Order Properties of Analytic Wavelets.” IEEE Transactions on Signal Processing 57, no. 1 (January 2009): 146–60. https://doi.org/10.1109/TSP.2008.2007607.
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