Correct factors for complex Morlet wavelet

[itamwa]

Hi there,

I'm using the continuous complex Morlet transform to compare a local energy spectrum and qualitatively I get what I expect (from Fourier transform, for example). Quantitatively, however, I'm pretty off.

I suspect two things:

1. In the literature I see that for the wavelet I want to use should be in this form - psi1(x)= pi^(-0.25) * e^(iwx) * e^(-x^2/2), but the cwt function only allows me to use psi2(x)= (2pi)^(-0.5) * e^(iw*x) * e^(-x^2/2). Is there a way for me to conduct the transform with psi1(x) rather than psi2(x)
2. In order to compare the amount of energy correctly, do I need to normalize my result by the scale? I believe in the literature it's refereed to as L1 normalization.

Thank you very much,
Itamar

[eulerleibniz]

As far as I know, for a decent visualization, you need a scaling like this code, which still doesn't exactly provide the normalized values but will almost perfectly match the visualization as in MATLAB's cwt function (You will need a constant scaling on top of this I believe). You might also be interested in this https://github.com/neergaard/CWT.git which only works for cmore6.0-0.5, but comes with perfectly correct normalization out of the box :

# IMPORTS
import numpy as np
import matplotlib.pyplot as plt
# import ptwt # FOR GPU CASE WHICH I NEEDED <3
import pywt


# Parameters for the signal generation
WINDOW = 30  # window in seconds
Fs = 512  # sampling rate in Hz, I think pywt has bad implementation and usually requires higher Fs to perform well
# Three distinct frequencies for generating a nice signal, plot the results in Log space ideally
F1 = 10
F2 = 1
F3 = 100
nData = WINDOW * Fs
t = np.linspace(0, WINDOW, nData, endpoint=False)
T1 = 0.33
T2 = 0.66

# Generate the signal with 3 different sin with different time localizations and amplitudes currently set to 1
sig = (1 * np.sin(2 * np.pi * F1 * t) * (t < T1 * WINDOW) +
       1 * np.sin(2 * np.pi * F2 * t) * ((t > T1 * WINDOW) & (t < T2 * WINDOW)) +
       1 * np.sin(2 * np.pi * F3 * t) * (t > T2 * WINDOW))

# sig += 0.1 * np.random.randn(len(sig))
plt.plot(sig)

wavelet = "cmor6.0-0.5"
nv = 10 # Number of Voices
num_scale = int(nv * np.log2(nData // 2))
min_freq = 1 / WINDOW
max_freq = Fs / 2
# Choose the scales wisely. i think the following is a decent one, logarithmic frequencies 
wavescales = pywt.frequency2scale(wavelet, np.geomspace(min_freq, max_freq, num_scale, endpoint=False) / Fs)

coefficients, freqs = pywt.cwt(
    sig, wavescales,  wavelet, sampling_period=1/Fs, method='fft')
# Different visualizations exist .real .imag .... for visualizing phase and so on
coefficients = np.abs(coefficients)

# THE SCALING REQUIRED to make Sin(2 * pi * m * x) and Sin(2 * pi * n * x) appear ALMOST the same
# NOT EXACTLY SCALED
coefficients = coefficients * np.sqrt(freqs[:, np.newaxis])

plt.figure(figsize=(20, 10))
# Use seismic colormap for visualizing the case with negative values, otherwise the default is fine
print(np.max(coefficients))
print(np.min(coefficients))
plt.pcolormesh(t, freqs, coefficients, shading="auto", cmap="seismic") 
plt.yscale("log")
plt.show()