Plot the 1D wavelet filters

Let us examine the wavelet filters used by kymatio to calculate 1D scattering transforms. Filters are generated using the kymatio.scattering1d.filter_bank.scattering_filter_factory() method, which creates both the first- and second-order filter banks.

Preliminaries

First, we import the scattering_filter_factory method, which we will use to generate the filters.

from kymatio.scattering1d.filter_bank import scattering_filter_factory

We then import numpy and matplotlib to display the filters.

import numpy as np
import matplotlib.pyplot as plt

Filter parameters and generation

The filters are defined for a certain support size T which corresponds to the size of the input signal. The only restriction is that T must be a power of two. Since we are not computing any scattering transforms here, we may pick any power of two for T. Here, we choose 2**13 = 8192.

T = 2**13

The parameter J specifies the maximum scale of the filters as a power of two. In other words, the largest filter will be concentrated in a time interval of size 2**J.

J = 5

The Q parameter controls the number of wavelets per octave in the first-order filter bank. The larger the value, the narrower these filters are in the frequency domain and the wider they are in the time domain (in general, the number of non-negligible oscillations in time is proportional to Q). For audio signals, it is often beneficial to have a large value for Q (between 4 and 16), since these signals are often highly oscillatory and are better localized in frequency than they are in time. We therefore set:

Q = 8

Note that it is currently not possible to control the number of wavelets per octave in the second-order filter bank, which is fixed to one.

We are now ready to create the filters. These are generated by the scattering_filter_factory method, which takes the logarithm of T and the J and Q parameters. It returns the lowpass filter (phi_f), the first-order wavelet filters (psi1_f), and the second-order filters (psi2_f).

phi_f, psi1_f, psi2_f, _ = scattering_filter_factory(np.log2(T), J, Q)

The phi_f output is a dictionary where each integer key corresponds points to the instantiation of the filter at a certain resolution. In other words, phi_f[0] corresponds to the lowpass filter at resolution T, while phi_f[1] corresponds to the filter at resolution T/2, and so on.

While phi_f only contains a single filter (at different resolutions), the psi1_f and psi2_f outputs are lists of filters, one for each wavelet bandpass filter in the filter bank.

Plot the filters

We are now ready to plot the filters. We first display the lowpass filter (at full resolution) in red. We then plot each of the bandpass filters in blue. Since we don’t care about the negative frequencies, we limit the plot to the frequency interval [0, 0.5]. Finally, we add some explanatory labels and title.

plt.figure()
plt.plot(np.arange(T)/T, phi_f[0], 'r')

for psi_f in psi1_f:
    plt.plot(np.arange(T)/T, psi_f[0], 'b')

plt.xlim(0, 0.5)

plt.xlabel(r'$\omega$', fontsize=18)
plt.ylabel(r'$\hat\psi_j(\omega)$', fontsize=18)
plt.title('First-order filters (Q = {})'.format(Q), fontsize=18)
../_images/sphx_glr_plot_filters_001.png

Do the same plot for the second-order filters. Note that since here Q = 1, we obtain wavelets that have higher frequency bandwidth.

plt.figure()
plt.plot(np.arange(T)/T, phi_f[0], 'r')
for psi_f in psi2_f:
    plt.plot(np.arange(T)/T, psi_f[0], 'b')
plt.xlim(0, 0.5)
plt.ylim(0, 1.2)
plt.xlabel(r'$\omega$', fontsize=18)
plt.ylabel(r'$\hat\psi_j(\omega)$', fontsize=18)
plt.title('Second-order filters (Q = 1)', fontsize=18)
../_images/sphx_glr_plot_filters_002.png

Display the plots!

plt.show()

Total running time of the script: ( 0 minutes 0.258 seconds)

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