CWT Disapointing Frequency Separation

Question

I'm attempting to perform multi-resolution analysis via Continuous-Wavelet Transform (CWT) using Pywavelets. I've heard that CWT is supposed to be superior to STFT due to varying frequency content as a function of the time-window.

My test signal is two sinusoids of 1Hz and 5Hz, each lasting 10 seconds (see picture): f=np.sin(2.*np.pi*t)*((t>=10)&(t<=20))+np.sin(2*np.pi*5*t)*((t>=30)&(t<=40)). The sampling period is 20Hz.

Using Pywavelets, I perform the CWT as follows with the resulting spectrogram:

scales = np.arange(0.6,65,step=0.2)
coef, freqs=pywt.cwt(f,scales,'cgau1', sampling_period=dT)

As you can see the frequency resolution is quite lousy, and the peak (complex magnitude) doesn't even seem to line up at 5Hz for the second segment.

In contrast, a STFT using the Gaussian window with a standard deviation of 5 results in much sharper frequency resolution (at the expense of time sharpness):

Am I doing something here? I'm willing to sacrifice time resolution but I do need to sharpen up the frequency.

Answer 1

The family of continuous wavelet transforms (CWT) is not, per se, superior to STFT. Due to its variations in scale, the CWT can be better, for instance, when:

• you address natural signals where transients are shorter than more stationary parts,
• you do not know the appropriate scale of observation,
• deterministic or random processes follow scaling laws (like the Brownian motions).

which is not the case from your signal: precise frequency on the same support.

On the one hand, you can work with a continuous wavelet with more oscillations, and better refine the scales and voices.

On the other hand, there are so-called STFT with windows whose width vary with frequency.

Add-ons like hybrid time-scale/time-frequency transformations, reassignement, etc. may be used, but choosing the appropriate tool requires to refine your goal.

Answer 2

The wavelet is too time localized. Also pywt and scipy implems are flawed. On your signal w/ ssqueezepy:

Laurent is correct that CWT isn't always superior.

import numpy as np
from ssqueezepy import cwt
from ssqueezepy.visuals import imshow

t = np.linspace(0, 50, 50*20, 0)
f = np.sin(2.*np.pi*t)*((t>=10)&(t<=20))+np.sin(2*np.pi*5*t)*((t>=30)&(t<=40))

Wx0, _ = cwt(f, ('gmw', {'beta': 10}))
Wx1, _ = cwt(f, ('gmw', {'beta': 90}))

tint = np.round(t).astype(int)
imshow(Wx0, xticks=tint, abs=1, title="abs(CWT) | time-localized")
imshow(Wx1, xticks=tint, abs=1, title="abs(CWT) | freq-localized")