Continuous Wavelet Transform (CWT)¶
This section describes functions used to perform single continuous wavelet transforms.
Single level  cwt
¶

pywt.
cwt
(data, scales, wavelet)¶ One dimensional Continuous Wavelet Transform.
Parameters:  data : array_like
Input signal
 scales : array_like
The wavelet scales to use. One can use
f = scale2frequency(scale, wavelet)/sampling_period
to determine what physical frequency,f
. Here,f
is in hertz when thesampling_period
is given in seconds. wavelet : Wavelet object or name
Wavelet to use
 sampling_period : float
Sampling period for the frequencies output (optional). The values computed for
coefs
are independent of the choice ofsampling_period
(i.e.scales
is not scaled by the sampling period).
Returns:  coefs : array_like
Continuous wavelet transform of the input signal for the given scales and wavelet
 frequencies : array_like
If the unit of sampling period are seconds and given, than frequencies are in hertz. Otherwise, a sampling period of 1 is assumed.
Notes
Size of coefficients arrays depends on the length of the input array and the length of given scales.
Examples
>>> import pywt >>> import numpy as np >>> import matplotlib.pyplot as plt >>> x = np.arange(512) >>> y = np.sin(2*np.pi*x/32) >>> coef, freqs=pywt.cwt(y,np.arange(1,129),'gaus1') >>> plt.matshow(coef) # doctest: +SKIP >>> plt.show() # doctest: +SKIP  >>> import pywt >>> import numpy as np >>> import matplotlib.pyplot as plt >>> t = np.linspace(1, 1, 200, endpoint=False) >>> sig = np.cos(2 * np.pi * 7 * t) + np.real(np.exp(7*(t0.4)**2)*np.exp(1j*2*np.pi*2*(t0.4))) >>> widths = np.arange(1, 31) >>> cwtmatr, freqs = pywt.cwt(sig, widths, 'mexh') >>> plt.imshow(cwtmatr, extent=[1, 1, 1, 31], cmap='PRGn', aspect='auto', ... vmax=abs(cwtmatr).max(), vmin=abs(cwtmatr).max()) # doctest: +SKIP >>> plt.show() # doctest: +SKIP
Continuous Wavelet Families¶
A variety of continuous wavelets have been implemented. A list of the available
wavelet names compatible with cwt
can be obtained by:
wavlist = pywt.wavelist(kind='continuous')
Mexican Hat Wavelet¶
The mexican hat wavelet "mexh"
is given by:
where the constant out front is a normalization factor so that the wavelet has unit energy.
Complex Morlet Wavelets¶
The complex Morlet wavelet ("cmorBC"
with floating point values B, C) is
given by:
where \(B\) is the bandwidth and \(C\) is the center frequency.
Gaussian Derivative Wavelets¶
The Gaussian wavelets ("gausP"
where P is an integer between 1 and and 8)
correspond to the Pth order derivatives of the function:
where \(C\) is an orderdependent normalization constant.
Complex Gaussian Derivative Wavelets¶
The complex Gaussian wavelets ("cgauP"
where P is an integer between 1 and
8) correspond to the Pth order derivatives of the function:
where \(C\) is an orderdependent normalization constant.
Shannon Wavelets¶
The Shannon wavelets ("shanBC"
with floating point values B and C)
correspond to the following wavelets:
where \(B\) is the bandwith and \(C\) is the center frequency.
Freuqency BSpline Wavelets¶
The frequency Bspline wavelets ("fpspMBC"
with integer M and floating
point B, C) correspond to the following wavelets:
where \(M\) is the spline order, \(B\) is the bandwidth and \(C\) is the center frequency.
Choosing the scales for cwt
¶
For each of the wavelets described below, the implementation in PyWavelets
evaluates the wavelet function for \(t\) over the range
[wavelet.lower_bound, wavelet.upper_bound]
(with default range
\([8, 8]\)). scale = 1
corresponds to the case where the extent of the
wavelet is (wavelet.upper_bound  wavelet.lower_bound + 1)
samples of the
digital signal being analyzed. Larger scales correspond to stretching of the
wavelet. For example, at scale=10
the wavelet is stretched by a factor of
10, making it sensitive to lower frequencies in the signal.
To relate a given scale to a specific signal frequency, the sampling period
of the signal must be known. pywt.scale2frequency()
can be used to
convert a list of scales to their corresponding frequencies. The proper choice
of scales depends on the chosen wavelet, so pywt.scale2frequency()
should
be used to get an idea of an appropriate range for the signal of interest.
For the cmor
, fbsp
and shan
wavelets, the user can specify a
specific a normalized center frequency. A value of 1.0 corresponds to 1/dt
where dt is the sampling period. In other words, when analyzing a signal
sampled at 100 Hz, a center frequency of 1.0 corresponds to ~100 Hz at
scale = 1
. This is above the Nyquist rate of 50 Hz, so for this
particular wavelet, one would analyze a signal using scales >= 2
.
>>> import numpy as np
>>> import pywt
>>> dt = 0.01 # 100 Hz sampling
>>> frequencies = pywt.scale2frequency('cmor1.51.0', [1, 2, 3, 4]) / dt
>>> frequencies
array([ 100. , 50. , 33.33333333, 25. ])
The CWT in PyWavelets is applied to discrete data by convolution with samples
of the integral of the wavelet. If scale
is too low, this will result in
a discrete filter that is inadequately sampled leading to aliasing as shown
in the example below. Here the wavelet is 'cmor1.51.0'
. The left column of
the figure shows the discrete filters used in the convolution at various
scales. The right column are the corresponding Fourier power spectra of each
filter.. For scales 1 and 2 it can be seen that aliasing due to violation of
the Nyquist limit occurs.
import numpy as np
import pywt
import matplotlib.pyplot as plt
wav = pywt.ContinuousWavelet('cmor1.51.0')
# print the range over which the wavelet will be evaluated
print("Continuous wavelet will be evaluated over the range [{}, {}]".format(
wav.lower_bound, wav.upper_bound))
width = wav.upper_bound  wav.lower_bound
scales = [1, 2, 3, 4, 10, 15]
max_len = int(np.max(scales)*width + 1)
t = np.arange(max_len)
fig, axes = plt.subplots(len(scales), 2, figsize=(12, 6))
for n, scale in enumerate(scales):
# The following code is adapted from the internals of cwt
int_psi, x = pywt.integrate_wavelet(wav, precision=10)
step = x[1]  x[0]
j = np.floor(
np.arange(scale * width + 1) / (scale * step))
if np.max(j) >= np.size(int_psi):
j = np.delete(j, np.where((j >= np.size(int_psi)))[0])
j = j.astype(np.int)
# normalize int_psi for easier plotting
int_psi /= np.abs(int_psi).max()
# discrete samples of the integrated wavelet
filt = int_psi[j][::1]
# The CWT consists of convolution of filt with the signal at this scale
# Here we plot this discrete convolution kernel at each scale.
nt = len(filt)
t = np.linspace(nt//2, nt//2, nt)
axes[n, 0].plot(t, filt.real, t, filt.imag)
axes[n, 0].set_xlim([max_len//2, max_len//2])
axes[n, 0].set_ylim([1, 1])
axes[n, 0].text(50, 0.35, 'scale = {}'.format(scale))
f = np.linspace(np.pi, np.pi, max_len)
filt_fft = np.fft.fftshift(np.fft.fft(filt, n=max_len))
filt_fft /= np.abs(filt_fft).max()
axes[n, 1].plot(f, np.abs(filt_fft)**2)
axes[n, 1].set_xlim([np.pi, np.pi])
axes[n, 1].set_ylim([0, 1])
axes[n, 1].set_xticks([np.pi, 0, np.pi])
axes[n, 1].set_xticklabels([r'$\pi$', '0', r'$\pi$'])
axes[n, 1].grid(True, axis='x')
axes[n, 1].text(np.pi/2, 0.5, 'scale = {}'.format(scale))
axes[n, 0].set_xlabel('time (samples)')
axes[n, 1].set_xlabel('frequency (radians)')
axes[0, 0].legend(['real', 'imaginary'], loc='upper left')
axes[0, 1].legend(['Power'], loc='upper left')
axes[0, 0].set_title('filter')
axes[0, 1].set_title(r'FFT(filter)$^2$')