Synchrosqueezing in Python

Synchrosqueezing is a powerful reassignment method that focuses time-frequency representations, and allows extraction of instantaneous amplitudes and frequencies. Friendly overview.

Features

• Continuous Wavelet Transform (CWT), forward & inverse, and its Synchrosqueezing
• Short-Time Fourier Transform (STFT), forward & inverse, and its Synchrosqueezing
• Clean code with explanations and learning references
• Wavelet visualizations

Coming soon

• Generalized Morse Wavelets

Installation

pip install ssqueezepy. Or, for latest version (most likely stable):

pip install git+https://github.com/OverLordGoldDragon/ssqueezepy

Examples

1. Signal recovery under severe noise

2. Medical: EEG

Introspection

ssqueezepy is equipped with a visualization toolkit, useful for exploring wavelet behavior across scales and configurations. (Also see explanations and code)

Minimal example

import numpy as np
import matplotlib.pyplot as plt
from ssqueezepy import ssq_cwt, ssq_stft

def viz(x, Tx, Wx):
    plt.imshow(np.abs(Wx), aspect='auto', cmap='jet')
    plt.show()
    plt.imshow(np.flipud(np.abs(Tx)), aspect='auto', vmin=0, vmax=.2, cmap='jet')
    plt.show()   

#%%# Define signal ####################################    
N = 2048
t = np.linspace(0, 10, N, endpoint=False)
xo = np.cos(2 * np.pi * 2 * (np.exp(t / 2.2) - 1))
xo += xo[::-1]
x = xo + np.sqrt(2) * np.random.randn(N)

plt.plot(xo); plt.show()
plt.plot(x); plt.show()

#%%# CWT + SSQ CWT ####################################
Twxo, _, Wxo, *_ = ssq_cwt(xo, 'morlet')
viz(xo, Twxo, Wxo)

Twx, _, Wx, *_ = ssq_cwt(x, 'morlet')
viz(x, Twx, Wx)

#%%# STFT + SSQ STFT ##################################
Tsxo, _, Sxo, *_ = ssq_stft(xo)
viz(xo, Tsxo, np.flipud(Sxo))

Tsx, _, Sx, *_ = ssq_stft(x)
viz(x, Tsx, np.flipud(Sx))

Learning resources

1. Continuous Wavelet Transform, & vs STFT
2. Synchrosqueezing's phase transform, intuitively
3. Wavelet time & frequency resolution visuals
4. Why oscillations in SSQ of mixed sines? Separability visuals
5. Zero-padding's effect on spectrum

DSP fundamentals: I recommend starting with 3b1b's Fourier Transform, then proceeding with DSP Guide chapters 7-11. The Discrete Fourier Transform lays the foundation of signal processing with real data. Deeper on DFT coefficients here, also 3b1b.

References

ssqueezepy was originally ported from MATLAB's Synchrosqueezing Toolbox, authored by E. Brevdo and G. Thakur [1]. Synchrosqueezed Wavelet Transform was introduced by I. Daubechies and S. Maes [2], which was followed-up in [3], and adapted to STFT in [4]. Many implementation details draw from [5].

1. G. Thakur, E. Brevdo, N.-S. Fučkar, and H.-T. Wu. "The Synchrosqueezing algorithm for time-varying spectral analysis: robustness properties and new paleoclimate applications", Signal Processing 93:1079-1094, 2013.
2. I. Daubechies, S. Maes. "A Nonlinear squeezing of the Continuous Wavelet Transform Based on Auditory Nerve Models".
3. I. Daubechies, J. Lu, H.T. Wu. "Synchrosqueezed Wavelet Transforms: a Tool for Empirical Mode Decomposition", Applied and Computational Harmonic Analysis 30(2):243-261, 2011.
4. G. Thakur, H.T. Wu. "Synchrosqueezing-based Recovery of Instantaneous Frequency from Nonuniform Samples", SIAM Journal on Mathematical Analysis, 43(5):2078-2095, 2011.
5. Mallat, S. "Wavelet Tour of Signal Processing 3rd ed".

License

ssqueezepy is MIT licensed, as found in the LICENSE file. Some source functions may be under other authorship/licenses; see NOTICE.txt.