How to define my own continuous wavelet by using Python?

Question

As the title shows, I want to define my own continuous wavelet by Python. However, I don't know how to achieve this exactly.

The formula of my wavelet mother function is below

It looks something like the Mexican hat wavelet, but they are different.

So how can I define such a custom wavelet by Python, then CWT can be performed using this wavelet?

Answer 1

Per this you need a function that takes a number of points and a scale to provide as a wavelet argument

So we define it as such:

import math
import numpy as np
from scipy import signal
import matplotlib.pyplot as plt

mother_wavelet = lambda z : np.exp(-z*z/4)*(2-z*z)/(4*math.sqrt(math.pi))

def mexican_hat_like(n,scale):
    x = np.linspace(-n/2,n/2,n)
    return mother_wavelet(x/scale)

Let's test it. We note that in fact something that looks very similar to yours is available. The difference is in scaling a and also the constant is front looks slightly different. Note math.sqrt(2) scaling for the Ricker wavelet

points = 100
a = 4.0
vec_ours = mexican_hat_like(points, a)
vec_theirs = signal.ricker(points, a*math.sqrt(2))
plt.plot(vec_ours, label = 'ours')
plt.plot(vec_theirs, label = 'ricker')
plt.legend(loc = 'best')
plt.show()

Here is the graph: