MATLAB: Advantages of using the new CWT interface over the old CWT interface.

Wavelet Toolbox

I see that there are two different versions of CWT function in R2018b and Mathworks recommends using the newer interface for the function.

What are the advantages of the newer interface over the old one?

Newer version:

https://www.mathworks.com/help/wavelet/ref/cwt.html

Older version:

https://www.mathworks.com/help/wavelet/ref/cwtold.html

Best Answer

In the newer version you can optionally specify what frequency or period ranges they are interested in. Thinking in terms of periods or frequencies is much more natural for applications than in terms of scales, which are unitless. You are encouraged to look at CWTFILTERBANK for example. As an example of the difficulty in specifying scales, you are using linearly spaced scales in your CWT example. For the CWT, we should prefer logarithmically spaced scales.

Another advantage is that the new CWT uses L1 normalization as opposed to the L2 normalization used in the legacy CWT, The L1 normalization is better for time frequency analysis because it more faithfully represents the magnitude of the components, whereas the L2 normalization is a bandpass filter that has a sqrt(s) term as a multiplier where s is the scale. So as the scale gets bigger, or equivalently as the

frequency becomes lower the wavelet transform is multiplying the frequency content of the signal by a factor of sqrt(s). We find that to be undesirable because it misrepresents the amplitudes.

For example, consider this:

%%New CWT
Fs = 1e3;
t = 0:1/Fs:2;
x = cos(2*pi*32*t).*(t>=0.1 & t <= 0.7)+cos(2*pi*64*t).*(t >= 0.8);
wgnNoise = 0.05*randn(size(t));
x = x+wgnNoise;
cwt(x,1000,'amor','voices',12)
title('New CWT');
%%Old CWT
%
a0 = 2^(1/12);
scales = 2*a0.^(0:8*12);
figure;
[cfs_old,f_old] = cwt(x,scales,'morl','plot',0.001);
colorbar;

The amplitudes of the 32 and 64 Hz sinusoids is 1. BOTH have amplitude 1.

If you look at the new CWT, you see that clearly, it gets it right. Both sine waves are shown to have amplitude 1. Compare that against the legacy CWT, the plot is much more difficult to interpret. Not only that, we have no way of relating the magnitude, or magnitude squared of the wavelet coefficients to the actual signal values. Also, note the new CWT shows clearly the cone of influence where edge effects become significant. The old CWT does not give you this information.

Even if we plot the old CWT as a function of frequency:

surf(t,f_old,abs(cfs_old),'edgecolor','none'); view(0,90); colorbar; axis tight;

The new CWT looks much nicer in terms of representing the time-localized components of constant magnitude.

It is strongly recommended that you use the new interface for the CWT function.

Related Solutions

MATLAB: The difference between the old version and new version of CWT (Continuous Wavelet Transform)

Hi Supatat, Some of this has been discussed here:

https://www.mathworks.com/matlabcentral/answers/474889-why-the-old-version-of-cwt-code-is-no-longer-recommended-continuous-wavelet-transform

You did not show the code that you used in both cases (old CWT api and new CWT api), but obviously we recommend the new API for a number of reasons.

In your specific case, if you are actually looking at the where the maximum values occur in the wavelet transform, then the old version is not recommend because it normalizes the wavelets using L2 normalization. That is discussed here:

See L1 norm for CWT https://www.mathworks.com/help/wavelet/ref/cwt.html

In your case this means that the bandpass filter used has a peak amplitude that depends on the scale, the lower the scale the higher the amplitude. So it biases the results toward larger scales (or lower frequencies). The new CWT does not do that, all filters have the same magnitude in L1 normalization, that is a fairer picture. That is particularly true if you are looking at values. Qualitatively, it looks like the picture of the signal is the same in both othe old and new CWT, but the L2 normalization of the old API is biasing the results toward lower frequencies.

For example: Here we have two oscillations with unambigously unit amplitude. See that the new CWT gets it right.

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);
cwt(x,1000)

Try the above example with the old CWT and you will see that the reported amplitude (magnitude) for the 32-Hz component is larger than the 64-Hz component and neither of them will be close to 1.

Hope that helps,

Wayne

MATLAB: Cwt.m normalization

Hi Simon, that energy normalization should be interpreted here in the correct way. With the CWT, we don't preserve the energy in either case with the L1 or L2 normalization. That energy preservation is only in the integral form of the CWT which is not implemented numerically. The same is true of the spectrogram in the Signal Processing Toolbox. If you look at the integral forms for the CWT with the L2 normalization, then the energy is preserved. However, when you implement the CWT numerically, that is not the case. We will make that clear in the documentation.

Now in the case of the DWT with very specific conditions, i.e. when we implement the classic DWT with a power of two input and the signal length some power of two. You will see the energy preserved. For example:

dwtmode('per')
x = randn(1024,1);
norm(x,2)^2
[C,L] = wavedec(x,10,'sym4');
norm(C,2)^2

But that won't happen with the CWT (by design) and it has nothing to do with the L2 vs L1 normalization. In fact if you look at the legacy CWT, we didn't preserve signal energy there either even though the wavelets were normalized by 1/\sqrt{s}.

If you want a redundant wavelet or wavelet packet transform that does preserve the energy, then MODWT and MODWPT will do that. They are what are referred to as "tight wavelet (and wavelet packet) frames".

Again, the reason for the L1 normalization in the CWT was so that if you have equal amplitude oscillatory components in your data at different scales, they should have equal magnitude in the CWT and NOT be multiplied by a scale factor.

Best Answer

Related Solutions

MATLAB: The difference between the old version and new version of CWT (Continuous Wavelet Transform)

MATLAB: Cwt.m normalization

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