spkit.cwt.PoissonWave

spkit.cwt.PoissonWave(t, f=1, n=1, method=3)

Poisson Wavelet

Type 1 (n): Method=1

\[ \begin{align}\begin{aligned}\psi(t) &= \left(\frac{t-n}{n!}\right)t^{n-1} e^{-t}\\\psi(w) &= \frac{-jw}{(1+jw)^{n+1}}\end{aligned}\end{align} \]

where Admiddibility const \(C_{\psi} =\frac{1}{n}\) and \(w = 2\pi f\)

Type 2: Method=2

\[ \begin{align}\begin{aligned}\psi(t) &= \frac{1}{\pi} \frac{1-t^2}{(1+t^2)^2}\\\psi(t) &= p(t) + \frac{d}{dt}p(t)\\\psi(w) &= |w|e^{-|w|}\end{aligned}\end{align} \]

where

\[ \begin{align}\begin{aligned}p(t) &=\frac{1}{\pi}\frac{1}{1+t^2}\\w &= 2\pi f\end{aligned}\end{align} \]

Type 3 (n): Method=3

\[ \begin{align}\begin{aligned}\psi(t) &= \frac{1}{2\pi}(1-jt)^{-(n+1)}\\\psi(w) &= \frac{1}{\Gamma{n+1}}w^{n}e^{-w}u(w)\end{aligned}\end{align} \]

where

Unit step function

\[ \begin{align}\begin{aligned}u(w) &= 1 \quad \text{ if $w>=0$ }\quad \text{else } 0\\w &= 2\pi f\end{aligned}\end{align} \]
Parameters:
t: 1d array,

  • time span array corresponding to signal for analysis,

  • must be centered at 0

f: 1d array,

  • frquency array for wavelet analysis

n: int, array, default=1

  • array of integers

  • np.arange(100), [scale value]

method: {1,2,3}, default=3

  • method to select the type of Poisson wavelet computations method = 1,2,3, different implementation of Poisson funtion, default 3 keep the method=3, other methods are under development and not exactly compatibile with framework yet,

Returns:
wttime-domain wavelet(s)
wffrequency-domain wavelet(s)

See also

ScalogramCWT

Notes

Method 3 produces more stable results It is efficient and easy to use ScalogramCWT with wType==’Poisson’ code:

XW,S = ScalogramCWT(x,t,fs=fs,wType='Poisson',method = 3,PlotPSD=True)

References

Examples

#sp.cwt.PoissonWave
import numpy as np
import matplotlib.pyplot as plt
import spkit as sp
fs = 100                              #sampling frequency
t = np.linspace(-5,5,fs*10+1)         #time
f = np.linspace(-10,10,2*len(t)-1)    #frequency range
#f = np.linspace(-fs//2,fs//2,2*len(t))   #frequency range
n1 = np.arange(5)
n2 = np.arange(5,10)
wt1,wf1 = sp.cwt.PoissonWave(t,f=f,n=n1,method=3)
wt2,wf2 = sp.cwt.PoissonWave(t,f=f,n=n2,method=3)

plt.figure(figsize=(10,5))
plt.subplot(221)
plt.plot(t,wt1.T.real,label='real')
plt.plot(t,wt1.T.imag,'-',label='image')
plt.xlim(t[0],t[-1])
plt.subplot(222)
plt.plot(f,abs(wf1.T))
plt.xlim(f[0],f[-1])
plt.subplot(223)
plt.plot(t,wt2.T.real,label='real')
plt.plot(t,wt2.T.imag,'-',label='image')
plt.xlim(t[0],t[-1])
plt.xlabel('time')
plt.subplot(224)
plt.plot(f,abs(wf2.T))
plt.xlim(f[0],f[-1])
plt.xlabel('Frequency')
plt.suptitle('Poisson Wavelet')
plt.tight_layout()
plt.show()
../../_images/spkit-cwt-PoissonWave-1.png