spkit.cwt.GaussWave

spkit.cwt.GaussWave(t, f, f0, Q, t0=0)

Gaussian Wavelet

Generate time and frequency domain gaussian wavelet functions

\[ \begin{align}\begin{aligned}\psi(t) &= e^{-a(t-t_0)^{2}} \cdot e^{-2\pi jf_0(t-t_0)}\\\psi(f) &= \sqrt{\pi/a}\left( e^{-2\pi jft_0}\cdot e^{-\pi^{2}((f-f_0)^{2})/a}\right)\end{aligned}\end{align} \]

where

\[a = \left( \frac{f_0}{Q} \right)^{2}\]

The main parametres of Gaussian Wavelet are:

• f0 - center frequency

• Q - associated with spread of bandwidth, as a = (f0/Q)^2

Parameters:

t: 1d array,

• time span array corresponding to signal for analysis,

• must be centered at 0

f: 1d array,

• frquency array for gaussian wavelet

f0: array

• array of center frquencies for wavelets, default: np.linspace(0.1,10,100) [scale value]

Q: float or array

• array of q-factor for each wavelet, e.g. 0.5 (default) or np.linspace(0.1,5,100)

• if array, should be of same size as f0

t0: float, default=0,

• time shift of wavelet, or phase shift in frquency, Not suggeestive to change

Returns:

wttime-domain wavelet(s)

wffrequency-domain wavelet(s)

See also

ScalogramCWT

Notes

It is efficient and easy to use ScalogramCWT with wType==’Gauss’

code::

XW,S = ScalogramCWT(x,t,fs=fs,wType=’Gauss’,PlotPSD=True,PlotW=True, f0=f0,Q=Q)

References

• wikipedia -

Examples

#sp.cwt.GaussWave
import numpy as np
import matplotlib.pyplot as plt
import spkit as sp
fs = 128                                 #sampling frequency
t = np.linspace(-5,5,fs*10+1)            #time
f = np.linspace(-fs//2,fs//2,2*len(t))   #frequency range
f0 = 2     #np.linspace(0.1,5,2)[:,None]
Q1  = 2.5   #np.linspace(0.1,5,10)[:,None]
Q2  = 0.5   #np.linspace(0.1,5,10)[:,None]
wt1,wf1 = sp.cwt.GaussWave(t,f,f0=f0,Q=Q1)
wt2,wf2 = sp.cwt.GaussWave(t,f,f0=f0,Q=Q2)
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.xlabel('time')
plt.ylabel('Q=2.5')
plt.legend()
plt.subplot(222)
plt.plot(f,abs(wf1.T), alpha=0.9,label=f'Q={Q1},f0={f0}')
plt.xlim(f[0],f[-1])
plt.xlim(-10,10)
plt.legend()
plt.axvline(f0,color='k',lw=1,ls='--')
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.ylabel('Q=0.5')
plt.legend()
plt.subplot(224)
plt.plot(f,abs(wf2.T), alpha=0.9,label=f'Q={Q2},f0={f0}')
plt.xlim(f[0],f[-1])
plt.xlim(-10,10)
plt.axvline(f0,color='k',lw=1,ls='--')
plt.xlabel('Frequency')
plt.legend()
plt.suptitle('Gauss Wavelet')
plt.tight_layout()
plt.show()

##############################
#sp.cwt.GaussWave
import numpy as np
import matplotlib.pyplot as plt
import spkit as sp
fs = 128                                 #sampling frequency
t = np.linspace(-5,5,fs*10+1)            #time
f = np.linspace(-fs//2,fs//2,2*len(t))   #frequency range
f01  = 2     #np.linspace(0.1,5,2)[:,None]
f02  = 5     #np.linspace(0.1,5,2)[:,None]
Q  = 1   #np.linspace(0.1,5,10)[:,None]
wt1,wf1 = sp.cwt.GaussWave(t,f,f0=f01,Q=Q)
wt2,wf2 = sp.cwt.GaussWave(t,f,f0=f02,Q=Q)
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.xlabel('time')
plt.ylabel('Q=2.5')
plt.legend()
plt.subplot(222)
plt.plot(f,abs(wf1.T), alpha=0.9,label=f'Q={Q},f0={f01}')
plt.xlim(f[0],f[-1])
plt.xlim(-10,10)
plt.legend()
plt.axvline(f01,color='k',lw=1,ls='--')
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.ylabel('Q=0.5')
plt.legend()
plt.subplot(224)
plt.plot(f,abs(wf2.T), alpha=0.9,label=f'Q={Q},f0={f02}')
plt.xlim(f[0],f[-1])
plt.xlim(-10,10)
plt.axvline(f02,color='k',lw=1,ls='--')
plt.xlabel('Frequency')
plt.legend()
plt.suptitle('Gauss Wavelet')
plt.tight_layout()
plt.show()

##############################
#sp.cwt.GaussWave
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(-fs//2,fs//2,2*len(t))   #frequency range
f0 = np.linspace(0.5,10,10)[:,None]
Q  = np.linspace(1,5,10)[:,None]
wt,wf = sp.cwt.GaussWave(t,f,f0=f0,Q=Q)
LABELS = [f'Q={j[0]:,.2f}, f0={i[0]:,.2f}'  for (i, j) in zip(f0, Q)]
plt.figure(figsize=(12,4))
plt.subplot(121)
plt.plot(t,wt.T.real, alpha=0.8)
plt.plot(t,wt.T.imag,'--', alpha=0.6)
plt.xlim(t[0],t[-1])
plt.xlabel('time')
plt.subplot(122)
plt.plot(f,abs(wf.T), alpha=0.6,label=LABELS)
plt.xlim(f[0],f[-1])
plt.xlim(-20,20)
plt.xlabel('Frequency')
plt.legend()
plt.suptitle('Gauss Wavelet')
plt.tight_layout()
plt.show()