Documentation

# cwt

Continuous 1-D wavelet transform

See cwt for information on the older version of the cwt. The older version is no longer recommended.

## Description

example

wt = cwt(x) returns the continuous wavelet transform (CWT) of x. The input, x, is a double-precision real- or complex-valued vector, or a single-variable regularly sampled timetable and must have at least four samples. The CWT is obtained using the analytic Morse wavelet with the symmetry parameter (gamma) equal to 3 and the time-bandwidth product equal to 60. cwt uses 10 voices per octave. The minimum and maximum scales are determined automatically based on the energy spread of the wavelet in frequency and time. If x is real-valued, wt is a 2-D matrix where each row corresponds to one scale. The column size of wt is equal to the length of x. If x is complex-valued, wt is a 3-D matrix, where the first page is the CWT for the positive scales (analytic part or counterclockwise component) and the second page is the CWT for the negative scales (anti-analytic part or clockwise component).

The cwt function uses 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 shows a more accurate representation of the signal. See L1 Norm for CWT and Continuous Wavelet Transform of Two Complex Exponentials.

The cwt function supports GPU Code Generation.

wt = cwt(x,wname) uses the analytic wavelet specified by wname to compute the CWT. Valid options for wname are 'morse', 'amor', and 'bump', which specify the Morse, Morlet (Gabor), and bump wavelet, respectively. If you do not specify wname, wname defaults to 'morse'.

example

[wt,f] = cwt(___,fs) specifies the sampling frequency, fs, in Hz as a positive scalar. cwt uses fs to determine the scale-to-frequency conversions and returns the frequencies f in Hz. If you do not specify a sampling frequency, cwt returns f in cycles per sample. If the input x is complex, the scale-to-frequency conversions apply to both pages of wt. If x is a timetable, you cannot specify fs. fs is determined from the RowTimes of the timetable.

[wt,period] = cwt(___,ts) specifies the sampling period, ts, as a positive duration scalar. The duration can be in years, days, hours, minutes, or seconds. cwt uses ts to compute the scale-to-period conversion and returns the time periods in period. The array of durations in period has the same format property as ts. If the input x is complex, the scale-to-period conversions apply to both pages of wt. If x is a timetable, you cannot specify ts. ts is determined from the RowTimes of the timetable when you set the 'PeriodLimits' name-value pair.

example

[wt,f,coi] = cwt(___,fs) returns the cone of influence, coi, which shows where edge effects of the CWT become significant. The cone of influence for the CWT is in Hz. If the input x is complex, the cone of influence applies to both pages of wt.

[wt,period,coi] = cwt(___,ts) returns the cone of influence, coi, which shows where edge effects of the CWT become significant. The cone of influence for the CWT is in periods. If the input x is complex, the cone of influence applies to both pages of wt.

[___] = cwt(___,Name,Value) returns the CWT with additional options specified by one or more Name,Value pair arguments.

[___,coi,fb] = cwt(___) returns the filter bank used in the CWT. See cwtfilterbank.

[___,fb,scalingcfs] = cwt(___) returns the scaling coefficients if the analyzing wavelet is 'morse' or 'amor'. Scaling coefficients are not supported for the bump wavelet.

example

cwt(___) with no output arguments plots the CWT scalogram. The scalogram is the absolute value of the CWT plotted as a function of time and frequency. Frequency is plotted on a logarithmic scale. The cone of influence showing where edge effects become significant is also plotted. Gray regions outside the dashed white line delineate regions where edge effects are significant. If the input signal is complex-valued, the positive (counterclockwise) and negative (clockwise) components are plotted in separate scalograms.

If you do not specify a sampling frequency or sampling period, the frequencies are plotted in cycles per sample. If you specify a sampling frequency, the frequencies are in Hz. If you specify a sampling period, the scalogram is plotted as a function of time and periods. If the input signal is a timetable, the scalogram is plotted as a function of time and frequency in hertz and uses the RowTimes as the basis for the time axis.

To see the time, frequency, and magnitude of a scalogram point, enable data tips in the figure axes toolbar and click the desired point in the scalogram.

### Note

Before plotting, cwt clears (clf) the current figure. To plot the scalogram in a subplot, use a plotting function.

## Examples

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Obtain the continuous wavelet transform of a speech sample using default values.

w = cwt(mtlb);

Obtain the continuous wavelet transform of a speech sample using the bump wavelet instead of the default Morse wavelet.

cwt(mtlb,'bump',Fs)

Compare the result obtained from the CWT using the default Morse wavelet.

cwt(mtlb,Fs)

Obtain the CWT of the Kobe earthquake data. The data are seismograph (vertical acceleration, nm/sq.sec) measurements recorded at Tasmania University, Hobart, Australia on 16 January 1995 beginning at 20:56:51 (GMT) and continuing for 51 minutes. The sampling frequency is 1 Hz.

Plot the earthquake data.

plot((1:numel(kobe))./60,kobe)
xlabel('mins')
ylabel('nm/s^2')
grid on
title('Kobe Earthquake Data')

Obtain the CWT, frequencies, and cone of influence.

[wt,f,coi] = cwt(kobe,1);

Plot the data, including the cone of influence.

cwt(kobe,1)

Obtain the CWT, time periods, and cone of influence by specifying a sampling period instead of a sampling frequency.

[wt,periods,coi] = cwt(kobe,minutes(1/60));

View the same data by specifying a sampling period input instead of a frequency.

cwt(kobe,minutes(1/60))

Create two complex exponentials, of different amplitudes, with frequencies of 32 and 64 Hz. The data is sampled at 1000 Hz. The two complex exponentials have disjoint support in time.

Fs = 1e3;
t = 0:1/Fs:1;
z = exp(1i*2*pi*32*t).*(t>=0.1 & t<0.3)+2*exp(-1i*2*pi*64*t).*(t>0.7);

Add complex white Gaussian noise with a standard deviation of 0.05.

wgnNoise = 0.05/sqrt(2)*randn(size(t))+1i*0.05/sqrt(2)*randn(size(t));
z = z+wgnNoise;

Obtain and plot the cwt using a Morse wavelet.

cwt(z,Fs)

Note the magnitudes of the complex exponential components in the colorbar are essentially their amplitudes even though they are at different scales. This is a direct result of the L1 normalization. You can verify this by executing this script and exploring each subplot with a data cursor.

This example shows that the amplitudes of oscillatory components in a signal agree with the amplitudes of the corresponding wavelet coefficients.

Create a signal composed of two sinusoids with disjoint support in time. One sinusoid has a frequency of 32 Hz and amplitude equal to 1. The other sinusoid has a frequency of 64 Hz and amplitude equal to 2. The signal is sampled for one second at 1000 Hz. Plot the signal.

frq1 = 32;
amp1 = 1;
frq2 = 64;
amp2 = 2;

Fs = 1e3;
t = 0:1/Fs:1;
x = amp1*sin(2*pi*frq1*t).*(t>=0.1 & t<0.3)+amp2*sin(2*pi*frq2*t).*(t>0.6 & t<0.9);

plot(t,x)
grid on
xlabel('Time (sec)')
ylabel('Amplitude')
title('Signal')

Create a CWT filter bank that can be applied to the signal. Since the signal component frequencies are known, set the frequency limits of the filter bank to a narrow range that includes the known frequencies. To confirm the range, plot the magnitude frequency responses for the filter bank.

fb = cwtfilterbank('SignalLength',numel(x),'SamplingFrequency',Fs,...
'FrequencyLimits',[20 100]);
figure
freqz(fb)

Use cwt and the filter bank to plot the scalogram of the signal.

figure
cwt(x,'FilterBank',fb)

Execute this script and use a data cursor to confirm that the amplitudes of the wavelet coefficients are essentially equal to the amplitudes of the sinusoidal components.

This example shows how using a CWT filter bank improves computational efficiency when taking the CWT of multiple time series.

Load the seismograph data recorded during the 1995 Kobe earthquake. The data are seismograph (vertical acceleration, nm/sq.sec) measurements recorded at Tasmania University, Hobart, Australia on 16 January 1995 beginning at 20:56:51 (GMT) and continuing for 51 minutes at 1 second intervals. Create a CWT filter bank that can be applied to the data.

fb = cwtfilterbank('SignalLength',numel(kobe),'SamplingFrequency',1);

Use the cwt function and take the CWT of the data 250 times. Display the elapsed time used.

num = 250;
tic;
for k=1:num
cfs = cwt(kobe);
end
toc
Elapsed time is 6.551628 seconds.

Now use the wt object function of the filter bank to take the CWT of the data. Confirm using the filter bank is faster.

tic;
for k=1:num
cfs = wt(fb,kobe);
end
toc
Elapsed time is 3.782376 seconds.

This example shows how to change the default frequency axis labels for the CWT when you obtain a plot with no output arguments.

Create two sine waves with frequencies of 32 and 64 Hz. The data is sampled at 1000 Hz. The two sine waves have disjoint support in time. Add white Gaussian noise with a standard deviation of 0.05. Obtain and plot the CWT using the default Morse wavelet.

Fs = 1e3;
t = 0:1/Fs:1;
x = cos(2*pi*32*t).*(t>=0.1 & t<0.3)+sin(2*pi*64*t).*(t>0.7);
wgnNoise = 0.05*randn(size(t));
x = x+wgnNoise;
cwt(x,1000)

The plot uses a logarithmic frequency axis because frequencies in the CWT are logarithmic. In MATLAB, logarithmic axes are in powers of 10 (decades). You can use cwtfreqbounds to determine what the minimum and maximum wavelet bandpass frequencies are for a given signal length, sampling frequency, and wavelet.

[minf,maxf] = cwtfreqbounds(numel(x),1000);

You see that by default MATLAB has placed frequency ticks at 10 and 100 because those are the powers of 10 between the minimum and maximum frequencies. If you wish to add more frequency axis ticks, you can obtain a logarithmically spaced set of frequencies between the minimum and maximum frequencies using the following.

numfreq = 10;
freq = logspace(log10(minf),log10(maxf),numfreq);

Next, get the handle to the current axes and replace the frequency axis ticks and labels with the following.

AX = gca;
AX.YTickLabelMode = 'auto';
AX.YTick = freq;

In the CWT, frequencies are computed in powers of two. To create the frequency ticks and tick labels in powers of two, you can do the following.

newplot;
cwt(x,1000);
AX = gca;
freq = 2.^(round(log2(minf)):round(log2(maxf)));
AX.YTickLabelMode = 'auto';
AX.YTick = freq;

This example shows how to scale scalogram values by maximum absolute value at each level for plotting.

Load in a signal and display the default scalogram. Change the colormap to pink(240).

cwt(noisdopp)
colormap(pink(240))

Take the CWT of the signal and obtain the wavelet coefficients and frequencies.

[cfs,frq] = cwt(noisdopp);

To efficiently find the maximum value of the coefficients at each frequency (level), first transpose the absolute value of the coefficients. Find the minimum value at every level. At each level, subtract the level's minimum value.

tmp1 = abs(cfs);
t1 = size(tmp1,2);
tmp1 = tmp1';
minv = min(tmp1);
tmp1 = (tmp1-minv(ones(1,t1),:));

Find the maximum value at every level of tmp1. For each level, divide every value by the maximum value at that level. Multiply the result by the number of colors in the colormap. Set equal to 1 all zero entries. Transpose the result.

maxv = max(tmp1);
maxvArray = maxv(ones(1,t1),:);
indx = maxvArray<eps;
tmp1 = 240*(tmp1./maxvArray);
tmp2 = 1+fix(tmp1);
tmp2(indx) = 1;
tmp2 = tmp2';

Display the result. The scalogram values are now scaled by the maximum absolute value at each level. Frequencies are displayed on a linear scale.

t = 0:length(noisdopp)-1;
pcolor(t,frq,tmp2);
ylabel('Frequency')
title('Scalogram Scaled By Level')
colormap(pink(240))

This example shows that increasing the time-bandwidth product ${P}^{2}$ of the Morse wavelet creates a wavelet with more oscillations under its envelope. Increasing ${P}^{2}$ narrows the wavelet in frequency.

Create two filter banks. One filter bank has the default TimeBandwidth value of 60. The second filter bank has a TimeBandwidth value of 10. The SignalLength for both filter banks is 4096 samples.

sigLen = 4096;
fb60 = cwtfilterbank('SignalLength',sigLen);
fb10 = cwtfilterbank('SignalLength',sigLen,'TimeBandwidth',10);

Obtain the time-domain wavelets for the filter banks.

[psi60,t] = wavelets(fb60);
[psi10,~] = wavelets(fb10);

Use the scales function to find the mother wavelet for each filter bank.

sca60 = scales(fb60);
sca10 = scales(fb10);
[~,idx60] = min(abs(sca60-1));
[~,idx10] = min(abs(sca10-1));
m60 = psi60(idx60,:);
m10 = psi10(idx10,:);

Since the time-bandwidth product is larger for the fb60 filter bank, verify the m60 wavelet has more oscillations under its envelope than the m10 wavelet.

subplot(2,1,1)
plot(t,abs(m60))
grid on
hold on
plot(t,real(m60))
plot(t,imag(m60))
xlim([-30 30])
legend('abs(m60)','real(m60)','imag(m60)')
title('TimeBandwidth = 60')
subplot(2,1,2)
plot(t,abs(m10))
grid on
hold on
plot(t,real(m10))
plot(t,imag(m10))
xlim([-30 30])
legend('abs(m10)','real(m10)','imag(m10)')
title('TimeBandwidth = 10')

Align the peaks of the m60 and m10 magnitude frequency responses. Verify the frequency response of the m60 wavelet is narrower than the frequency response for the m10 wavelet.

cf60 = centerFrequencies(fb60);
cf10 = centerFrequencies(fb10);

m60cFreq = cf60(idx60);
m10cFreq = cf10(idx10);

freqShift = 2*pi*(m60cFreq-m10cFreq);
x10 = m10.*exp(1j*freqShift*(-sigLen/2:sigLen/2-1));

figure
plot([abs(fft(m60)).' abs(fft(x10)).'])
grid on
legend('Time-bandwidth = 60','Time-bandwidth = 10')
title('Magnitude Frequency Responses')

This example shows how to plot the CWT scalogram in a figure subplot.

Load the speech sample. The data is sampled at 7418 Hz. Plot the default CWT scalogram.

cwt(mtlb,Fs)

Obtain the continuous wavelet transform of the signal, and the frequencies of the CWT.

[cfs,frq] = cwt(mtlb,Fs);

The cwt function sets the time and frequency axes in the scalogram. Create a vector representing the sample times.

tms = (0:numel(mtlb)-1)/Fs;

In a new figure, plot the original signal in the upper subplot and the scalogram in the lower subplot. Plot the frequencies on a logarithmic scale.

figure
subplot(2,1,1)
plot(tms,mtlb)
axis tight
title('Signal and Scalogram')
xlabel('Time (s)')
ylabel('Amplitude')
subplot(2,1,2)
surface(tms,frq,abs(cfs))
axis tight
xlabel('Time (s)')
ylabel('Frequency (Hz)')
set(gca,'yscale','log')

## Input Arguments

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Input signal, specified as a double-precision real- or complex-valued vector or a single-variable regularly sampled timetable. x must have at least four samples.

Analytic wavelet used to compute the CWT, specified as 'morse', 'amor', or 'bump'. These character vectors specify the analytic Morse, Morlet (Gabor), and bump wavelet, respectively.

The default Morse wavelet has symmetry parameter ($\gamma$) equal to 3 and time-bandwidth product equal to 60.

Sampling frequency, in Hz, specified as a positive scalar. If you specify fs, then you cannot specify ts.

Sampling period, also known as the time duration, specified as a scalar duration. Valid durations are years, days, hours, minutes, and seconds. You cannot use calendar durations. If you specify ts, then you cannot specify fs.

Example: wt = cwt(x,hours(12))

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is 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 Name1,Value1,...,NameN,ValueN.

Example: 'ExtendSignal',false indicates that the signal is not extended.

Option to extend the input signal symmetrically by reflection, specified as the comma-separated pair consisting of 'ExtendSignal' and either true or false. Extending the signal symmetrically can mitigate boundary effects. If you specify true, then the signal is extended. If you specify false, then the signal is not extended.

Frequency limits to use in the CWT, specified as a two-element vector with positive strictly increasing entries. The first element specifies the lowest peak passband frequency and must be greater than or equal to the product of the wavelet peak frequency in hertz and two time standard deviations divided by the signal length. The second element specifies the highest peak passband frequency and must be less than or equal to the Nyquist frequency. The base 2 logarithm of the ratio of the maximum frequency to the minimum frequency must be greater than or equal to 1/NV where NV is the number of voices per octave.

If you specify frequency limits outside the permissible range, cwt truncates the limits to the minimum and maximum valid values. Use cwtfreqbounds to determine frequency limits for different parameterizations of the CWT. For complex-valued signals, (-1)*flimits is used for the anti-analytic part, where flimits is the vector specified by FrequencyLimits.

Example: 'FrequencyLimits',[0.1 0.3]

Data Types: double

Period limits to use in the CWT, specified as a two-element duration array with strictly increasing positive entries. The first element must be greater than or equal to 2*ts where ts is the sampling period. The base 2 logarithm of the ratio of the minimum period to the maximum period must be less than or equal to -1/NV where NV is the number of voices per octave. The maximum period cannot exceed the signal length divided by the product of two time standard deviations of the wavelet and the wavelet peak frequency.

If you specify period limits outside the permissible range, cwt truncates the limits to the minimum and maximum valid values. Use cwtfreqbounds to determine period limits for different parameterizations of the wavelet transform. For complex-valued signals, (-1)*plimits is used for the anti-analytic part, where plimits is the vector specified by PeriodLimits.

Example: 'PeriodLimits',[seconds(0.2) seconds(1)]

Data Types: duration

Number of voices per octave to use for the CWT, specified as the comma-separated pair consisting of 'VoicesPerOctave' and an even integer from 4 to 48. The CWT scales are discretized using the specified number of voices per octave. The energy spread of the wavelet in frequency and time automatically determines the minimum and maximum scales.

Time-bandwidth product of the Morse wavelet, specified as the comma-separated pair consisting of 'TimeBandwidth' and a scalar greater than 3 and less than or equal to 120. The symmetry parameter, gamma ($\gamma$), is fixed at 3. Wavelets with larger time-bandwidth products have larger spreads in time and narrower spreads in frequency. The standard deviation of the Morse wavelet in time is approximately sqrt(TimeBandwidth/2). The standard deviation of the Morse wavelet in frequency is approximately 1/2*sqrt(2/TimeBandwidth).

If you specify 'TimeBandwidth', you cannot specify 'WaveletParameters'. To specify both the symmetry and time-bandwidth product, use 'WaveletParameters' instead.

In the notation of Morse Wavelets, TimeBandwidth is P2.

Symmetry and time-bandwidth product of the Morse wavelet, specified as the comma-separated pair consisting of 'WaveletParameters' and a two-element vector of scalars.

The first element is the symmetry, $\gamma$, which must be greater than or equal to 1. The second element is the time-bandwidth product, which must be strictly greater than $\gamma$. The ratio of the time-bandwidth product to $\gamma$ cannot exceed 40.

When $\gamma$ is equal to 3, the Morse wavelet is perfectly symmetric in the frequency domain and the skewness is 0. When $\gamma$ is greater than 3, the skewness is positive. When $\gamma$ is less than 3, the skewness is negative.

If you specify 'WaveletParameters', you cannot specify 'TimeBandwidth'.

Number of octaves, specified as the comma-separated pair consisting of 'NumOctaves' and a positive integer. The number of octaves cannot exceed log2(fmax/fmin) where fmax and fmin are the maximum and minimum CWT frequencies (or periods) as determined by the signal length, sampling frequency, and wavelet. See cwtfreqbounds for details.

The 'NumOctaves' name-value pair is not recommended and will be removed in a future release. The recommended way to modify the frequency or period range of the CWT is with the 'FrequencyLimits' or 'PeriodLimits' name-value pairs. You cannot specify both the 'NumOctaves' and 'FrequencyLimits' or 'PeriodLimits' name-value pairs.

CWT filter bank to use to compute the CWT, specified as a CWT filter bank object. If you use the 'FilterBank' name-value pair, you cannot specify any other options. All options for the computation of the CWT are defined as properties of fb.

If x is a timetable, the sampling frequency or sampling period in fb must agree with the sampling frequency or sampling period determined by the RowTimes of the timetable.

Example: wt = cwt(x,'FilterBank',fb)

## Output Arguments

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Continuous wavelet transform, returned as a matrix of complex values. By default, cwt uses the analytic Morse (3,60) wavelet, where 3 is the symmetry and 60 is the time-bandwidth product. cwt uses 10 voices per octave. If x is real-valued, wt is an Na-by-N matrix, where Na is the number of scales, and N is the number of samples in x. If x is complex-valued, wt is a 3-D matrix, where the first page is the CWT for the positive scales (analytic part or counterclockwise component) and the second page is the CWT for the negative scales (anti-analytic part or clockwise component). The minimum and maximum scales are determined automatically based on the energy spread of the wavelet in frequency and time. See Algorithms for information on how the scales are determined.

Data Types: double

Frequencies of the CWT, returned as a vector. If you specify a sampling frequency, fs, then f is in Hz. If you do not specify fs, cwt returns f in cycles per sample.

Time periods, returned as an array of durations. The durations are in the same format as ts. Each row corresponds to a period.

Cone of influence for the CWT, returned as either an array of real numbers or an array of durations. The cone of influence indicates where edge effects occur in the CWT. If you specify a sampling frequency, fs, the cone of influence is in Hz. If you specify a scalar duration, ts, the cone of influence is in periods. Due to the edge effects, give less credence to areas that are outside or overlap the cone of influence.

For additional information, see Boundary Effects and the Cone of Influence.

CWT filter bank used in the CWT, returned as a CWT filter bank object. See cwtfilterbank.

Scaling coefficients for the CWT if the analyzing wavelet is 'morse' or 'amor', returned as a real- or complex-valued vector. The length of scalingcfs is equal to the length of the input x.

Scaling coefficients are not supported for the bump wavelet.

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### Analytic Wavelets

Analytic wavelets are complex-valued wavelets whose Fourier transform vanish for negative frequencies. Analytic wavelets are a good choice when doing time-frequency analysis with the CWT. Because the wavelet coefficients are complex-valued, the coefficients provide phase and amplitude information of the signal being analyzed. Analytic wavelets are well suited for studying how the frequency content in real world nonstationary signals evolves as a function of time.

Analytic wavelets are almost exclusively based on rapidly decreasing functions. If $\psi \left(t\right)$ is an analytic rapidly decreasing function in time, then its Fourier transform $\stackrel{^}{\psi }\left(\omega \right)$ is a rapidly decreasing function in frequency and is small outside of some interval $\alpha <\omega <\beta$ where $0<\alpha <\beta$. Orthogonal and biorthogonal wavelets are typically designed to have compact support in time. Wavelets with compact support in time have relatively poorer energy concentration in frequency than wavelets which rapidly decrease in time. Most orthogonal and biorthogonal wavelets are not symmetric in the Fourier domain.

If your goal is to obtain a joint time-frequency representation of your signal, we recommend you use cwt or cwtfilterbank. Both functions support the following analytic wavelets:

• Morse Wavelet Family (default)

• Analytic Morlet (Gabor) Wavelet

• Bump

If you want to do time-frequency analysis using orthogonal or biorthogonal wavelets, we recommend modwpt.

When using wavelets for time-frequency analysis, you usually convert scales to frequencies or periods to interpret results. cwt and cwtfilterbank do the conversion. You can obtain the corresponding scales associated by using scales on the optional cwt output argument fb.

For more information regarding Morse wavelets, see Morse Wavelets. For guidance on how to choose the wavelet that is right for your application, see Choose a Wavelet.

## Algorithms

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### Minimum Scale

To determine the minimum scale, find the peak frequency ${\omega }_{x}$ of the base wavelet. For Morse wavelets, dilate the wavelet so that the Fourier transform of the wavelet at $\pi$ radians is equal to 10% of the peak frequency. The smallest scale occurs at the largest frequency:

${s}_{0}=\frac{{\omega }_{x}^{\text{'}}}{\pi }$

As a result, the smallest scale is the minimum of (2, ${s}_{0}$). For Morse wavelets, the smallest scale is usually ${s}_{0}$. For the Morlet wavelet, the smallest scale is usually 2.

### Maximum Scale

Both the minimum and maximum scales of the CWT are determined automatically based on the energy spread of the wavelet in frequency and time. To determine the maximum scale, CWT uses the following algorithm.

The standard deviation of the Morse wavelet in time, ${\sigma }_{t}$, is approximately $\sqrt{\frac{{P}^{2}}{2}}$, where ${P}^{2}$ is the time-bandwidth product. The standard deviation in frequency, ${\sigma }_{f}$, is approximately $\frac{1}{2}\sqrt{\frac{2}{{P}^{2}}}$. If you scale the wavelet by some $s>1$, the time duration changes to $2s{\sigma }_{t}=N$, which is the wavelet stretched to equal the full length (N samples) of the input. You cannot translate this wavelet or stretch it further without causing it to wrap, so the largest scale is $floor\left(\frac{N}{2{\sigma }_{t}}\right)$.

Wavelet transform scales are powers of 2 and are denoted by ${s}_{0}{\left({2}^{\frac{1}{NV}}\right)}^{j}$. NV is the number of voices per octave, and j ranges from 0 to the largest scale. For a specific small scale, ${s}_{0}$:

${s}_{0}{\left({2}^{\frac{1}{NV}}\right)}^{j}\le \frac{N}{2{\sigma }_{t}}$

Converting to log2:

$j{\mathrm{log}}_{2}\left({2}^{\frac{1}{NV}}\right)\le {\mathrm{log}}_{2}\left(\frac{N}{2{\sigma }_{t}{s}_{0}}\right)$

$j\le NV{\mathrm{log}}_{2}\left(\frac{N}{2{\sigma }_{t}{s}_{0}}\right)$

Therefore, the maximum scale is

${s}_{0}{\left({2}^{\frac{1}{NV}}\right)}^{floor\left(NV{\mathrm{log}}_{2}\left(\frac{N}{2{\sigma }_{t}{s}_{0}}\right)\right)}$

### L1 Norm for CWT

In integral form, the CWT preserves energy. However, when you implement the CWT numerically, energy is not preserved. In this case, regardless of the normalization you use, the CWT is not an orthonormal transform. The cwt function uses L1 normalization.

Wavelet transforms commonly use L2 normalization of the wavelet. For the L2 norm, dilating a signal by 1/s, where s is greater than 0, is defined as follows:

${‖x\left(\frac{t}{s}\right)‖}_{2}^{2}=s{‖x\left(t\right)‖}_{2}^{2}$

The energy is now s times the original energy. When included in the Fourier transform, multiplying by $1}{\sqrt{s}}$ produces different weights being applied to different scales, so that the peaks at higher frequencies are reduced more than the peaks at lower frequencies.

In many applications, L1 normalization is better. The L1 norm definition does not include squaring the value, so the preserving factor is 1/s instead of $1}{\sqrt{s}}$. Instead of high-frequency amplitudes being reduced as in the L2 norm, for L1 normalization, all frequency amplitudes are normalized to the same value. Therefore, using the L1 norm shows a more accurate representation of the signal. See example Continuous Wavelet Transform of Two Complex Exponentials.

## Compatibility Considerations

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Not recommended starting in R2018a

## References

[1] Lilly, J. M., and S. C. Olhede. “Generalized Morse Wavelets as a Superfamily of Analytic Wavelets.” IEEE Transactions on Signal Processing. Vol. 60, No. 11, 2012, pp. 6036–6041.

[2] Lilly, J. M., and S. C. Olhede. “Higher-Order Properties of Analytic Wavelets.” IEEE Transactions on Signal Processing. Vol. 57, No. 1, 2009, pp. 146–160.

[3] Lilly, J. M. jLab: A data analysis package for Matlab, version 1.6.2. 2016. http://www.jmlilly.net/jmlsoft.html.

[4] Lilly, J. M. “Element analysis: a wavelet-based method for analysing time-localized events in noisy time series.” Proceedings of the Royal Society A. Volume 473: 20160776, 2017, pp. 1–28. dx.doi.org/10.1098/rspa.2016.0776.