ssqueezepy 0.6.6
Synchrosqueezing, wavelet transforms, and time-frequency analysis in Python
License: MIT
Keywords: signal-processing python synchrosqueezing wavelet-transform cwt stft morse-wavelet ridge-extraction time-frequency time-frequency-analysis visualization

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# Synchrosqueezing in Python

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Synchrosqueezing is a powerful _reassignment method_ that focuses time-frequency representations, and allows extraction of instantaneous amplitudes and frequencies. [Friendly overview.](https://dsp.stackexchange.com/a/71399/50076)


## Features
  - Continuous Wavelet Transform (CWT), forward & inverse, and its Synchrosqueezing
  - Short-Time Fourier Transform (STFT), forward & inverse, and its Synchrosqueezing
  - Wavelet visualizations and testing suite
  - Generalized Morse Wavelets
  - Ridge extraction
  - Fastest wavelet transforms in Python<sup>1</sup>, beating MATLAB

<sub>1: feel free to open Issue showing otherwise</sub>


## Installation
`pip install ssqueezepy`. Or, for latest version (most likely stable):

`pip install git+https://github.com/OverLordGoldDragon/ssqueezepy`

## GPU & CPU acceleration

Multi-threaded execution is enabled by default (disable via `os.environ['SSQ_PARALLEL'] = '0'`). GPU requires [CuPy >= 8.0.0](https://docs.cupy.dev/en/stable/install.html)
and [PyTorch >= 1.8.0](https://pytorch.org/get-started/locally/) installed (enable via `os.environ['SSQ_GPU'] = '1'`). `pyfftw` optionally supported for maximum CPU FFT speed.
See [Performance guide](https://github.com/OverLordGoldDragon/ssqueezepy/blob/master/ssqueezepy/README.md#performance-guide).

## Benchmarks

[Code](https://github.com/OverLordGoldDragon/ssqueezepy/blob/master/examples/benchmarks.py). Transforms use padding, `float32` precision (`float64` supported), and output shape
`(300, len(x))`, averaged over 10 runs. `pyfftw` not used, which'd speed 1-thread & parallel further. Benched on author's i7-7700HQ, GTX 1070.

`len(x)`-transform | 1-thread CPU | parallel | gpu | pywavelets | scipy | librosa
:----------------:|:----------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:
10k-cwt       | 0.126 | 0.0462 | 0.00393 | 3.58 | 0.523 | -
10k-stft      | 0.108 | 0.0385 | 0.00534 | -    | 0.118 | 0.0909
10k-ssq_cwt   | 0.372 | 0.148  | 0.00941 | -    | -     | -
10k-ssq_stft  | 0.282 | 0.147  | 0.0278  | -    | -     | -
160k-cwt      | 2.99  | 1.25   | 0.0367  | 12.7 | 10.7  | -
160k-stft     | 1.66  | 0.418  | 0.0643  | -    | 1.93  | 1.38
160k-ssq_cwt  | 8.38  | 3.16   | 0.0856  | -    | -     | -
160k-ssq_stft | 4.65  | 2.48   | 0.159   | -    | -     | -


## Questions?

See [here](#asking-questions).

## Examples

### 1. Signal recovery under severe noise

![image](https://user-images.githubusercontent.com/16495490/99879090-b9f12c00-2c23-11eb-8a40-2011ce84df61.png)

### 2. Reconstruction (differentiable CWT)

<img src="https://media.githubusercontent.com/media/OverLordGoldDragon/ssqueezepy/master/examples/imgs/reconstruction.gif" width="700">

[Code](https://github.com/OverLordGoldDragon/ssqueezepy/blob/master/examples/reconstruction.py)

### 3. Medical: EEG

<img src="https://user-images.githubusercontent.com/16495490/99880110-c88f1180-2c2a-11eb-8932-90bf3406a20d.png">

<img src="https://user-images.githubusercontent.com/16495490/150314341-df5c3092-4bef-4895-99ed-2765504329fd.png">

### 4. Testing suite: CWT vs STFT, reflect-added parallel A.M. linear chirp

<img src="https://github.com/OverLordGoldDragon/ssqueezepy/assets/16495490/c89727db-1bb3-4cf0-ac8c-c524cba75b2d">

### 5. Ridge extraction: cubic polynom. F.M. + pure tone; noiseless & 1.69dB SNR

<img src="https://user-images.githubusercontent.com/16495490/107919540-f4e5d000-6f84-11eb-9f86-dbfd34733084.png">

[More](https://github.com/OverLordGoldDragon/ssqueezepy/tree/master/examples/ridge_extraction)

### 6. Testing suite: GMW vs Morlet, reflect-added hyperbolic chirp (extreme time-loc.)

<img src="https://github.com/OverLordGoldDragon/ssqueezepy/assets/16495490/8c41d5f2-4bdd-4537-8d82-6d5a5c0315d3">

### 7. Higher-order GMW CWT, reflect-added parallel linear chirp, 3.06dB SNR

<img src="https://user-images.githubusercontent.com/16495490/107921072-66bf1900-6f87-11eb-9bf5-afd0a6bbbc4d.png">

[More examples](https://overlordgolddragon.github.io/test-signals/)


## Introspection

`ssqueezepy` is equipped with a visualization toolkit, useful for exploring wavelet behavior across scales and configurations. (Also see [explanations and code](https://dsp.stackexchange.com/a/72044/50076))

<p align="center">
  <img src="https://raw.githubusercontent.com/OverLordGoldDragon/ssqueezepy/master/examples/imgs/anim_tf_morlet20.gif" width="500">
</p>

<img src="https://raw.githubusercontent.com/OverLordGoldDragon/ssqueezepy/master/examples/imgs/morlet_5vs20_tf.png">
<img src="https://user-images.githubusercontent.com/16495490/107297978-e6338080-6a8d-11eb-8a11-60bfd6e4137d.png">

## How's it work?

In a nutshell, synchrosqueezing exploits _redundancy_ of a time-frequency representation to sparsely localize oscillations, by imposing a _prior_. That is, we _assume_ `x` is well-captured by AM-FM components, e.g. based on our knowledge of the underlying process. We surpass Heisenberg's limitations, but only for a _subset_ of all possible signals. It's also akin to an _attention_ mechanism.

Convolve with localized, analytic kernels

<img src="https://raw.githubusercontent.com/OverLordGoldDragon/StackExchangeAnswers/main/SignalProcessing/Q78512%20-%20Wavelet%20Scattering%20explanation/cwt.gif" width="650">

compute phase transform, then combine oscillations with a shared rate

<img src="https://user-images.githubusercontent.com/16495490/150680428-4a651934-85c6-45e8-8a19-c9b4165e5381.png" width="700">

<hr>

## Minimal example

```python
import numpy as np
import matplotlib.pyplot as plt
from ssqueezepy import ssq_cwt, ssq_stft
from ssqueezepy.experimental import scale_to_freq

def viz(x, Tx, Wx):
    plt.imshow(np.abs(Wx), aspect='auto', cmap='turbo')
    plt.show()
    plt.imshow(np.abs(Tx), aspect='auto', vmin=0, vmax=.2, cmap='turbo')
    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]  # add self reflected
x = xo + np.sqrt(2) * np.random.randn(N)  # add noise

plt.plot(xo); plt.show()
plt.plot(x);  plt.show()

#%%# CWT + SSQ CWT ####################################
Twxo, Wxo, *_ = ssq_cwt(xo)
viz(xo, Twxo, Wxo)

Twx, Wx, *_ = ssq_cwt(x)
viz(x, Twx, Wx)

#%%# STFT + SSQ STFT ##################################
Tsxo, Sxo, *_ = ssq_stft(xo)
viz(xo, np.flipud(Tsxo), np.flipud(Sxo))

Tsx, Sx, *_ = ssq_stft(x)
viz(x, np.flipud(Tsx), np.flipud(Sx))

#%%# With units #######################################
from ssqueezepy import Wavelet, cwt, stft, imshow
fs = 400
t = np.linspace(0, N/fs, N)
wavelet = Wavelet()
Wx, scales = cwt(x, wavelet)
Sx = stft(x)[::-1]

freqs_cwt = scale_to_freq(scales, wavelet, len(x), fs=fs)
freqs_stft = np.linspace(1, 0, len(Sx)) * fs/2

ikw = dict(abs=1, xticks=t, xlabel="Time [sec]", ylabel="Frequency [Hz]")
imshow(Wx, **ikw, yticks=freqs_cwt)
imshow(Sx, **ikw, yticks=freqs_stft)
```

Also see ridge extraction [README](https://github.com/OverLordGoldDragon/ssqueezepy/tree/master/examples/ridge_extraction).


## Interesting use cases (with code)

 1. [Identify abrupt changes in audio](https://dsp.stackexchange.com/a/87512/50076) - `ssq_cwt` and `ssq_stft` used together to solve an ML problem without ML
 
Feel free to share yours [here](https://github.com/OverLordGoldDragon/ssqueezepy/issues/9).


## Learning resources

 1. [Continuous Wavelet Transform, & vs STFT](https://ccrma.stanford.edu/~unjung/mylec/WTpart1.html)
 2. [Synchrosqueezing's phase transform, intuitively](https://dsp.stackexchange.com/a/72238/50076)
 3. [Wavelet time & frequency resolution visuals](https://dsp.stackexchange.com/a/72044/50076)
 4. [Why oscillations in SSQ of mixed sines? Separability visuals](https://dsp.stackexchange.com/a/72239/50076)
 5. [Zero-padding's effect on spectrum](https://dsp.stackexchange.com/a/70498/50076)

**DSP fundamentals**: I recommend starting with 3b1b's [Fourier Transform](https://youtu.be/spUNpyF58BY), then proceeding with [DSP Guide](https://www.dspguide.com/CH7.PDF) chapters 7-11.
The Discrete Fourier Transform lays the foundation of signal processing with real data. Deeper on DFT coefficients [here](https://dsp.stackexchange.com/a/70395/50076), also [3b1b](https://youtu.be/g8RkArhtCc4).


## Contributors (noteworthy)

 - [David Bondesson](https://github.com/DavidBondesson): ridge extraction (`ridge_extraction.py`; `examples/`: `extracting_ridges.py`, `ridge_extraction/README.md`)

## Asking questions

Open an Issue, and follow the [Issues Template](https://github.com/OverLordGoldDragon/ssqueezepy/issues/new/choose). Mainly code-related questions go to [Stack Overflow](https://stackoverflow.com/) (SO). Applications, theory questions, etc go elsewhere, e.g. [DSP.SE](https://dsp.stackexchange.com/). I may or may not respond, but others may (or may not) help. I don't follow SO.

**Do not** send e-mail, LinkedIn messages, etc - they will be ignored.

## How to cite

Short form:

> John Muradeli, ssqueezepy, 2020. GitHub repository, https://github.com/OverLordGoldDragon/ssqueezepy/. DOI: 10.5281/zenodo.5080508

BibTeX:

```bibtex
@article{OverLordGoldDragon2020ssqueezepy,
  title={ssqueezepy},
  author={John Muradeli},
  journal={GitHub. Note: https://github.com/OverLordGoldDragon/ssqueezepy/},
  year={2020},
  doi={10.5281/zenodo.5080508},
}
```

## References

`ssqueezepy` was originally ported from MATLAB's [Synchrosqueezing Toolbox](https://github.com/ebrevdo/synchrosqueezing), 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]. Ridge extraction based on [6].

  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"](https://arxiv.org/abs/1105.0010), Signal Processing 93:1079-1094, 2013.
  2. I. Daubechies, S. Maes. ["A Nonlinear squeezing of the Continuous Wavelet Transform Based on Auditory Nerve Models"](https://services.math.duke.edu/%7Eingrid/publications/DM96.pdf).
  3. I. Daubechies, J. Lu, H.T. Wu. ["Synchrosqueezed Wavelet Transforms: a Tool for Empirical Mode Decomposition"](https://arxiv.org/pdf/0912.2437.pdf), 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"](https://arxiv.org/abs/1006.2533), SIAM Journal on Mathematical Analysis, 43(5):2078-2095, 2011.
  5. Mallat, S. ["Wavelet Tour of Signal Processing 3rd ed"](https://www.di.ens.fr/~mallat/papiers/WaveletTourChap1-2-3.pdf).
  6. D. Iatsenko, P. V. E. McClintock, A. Stefanovska. ["On the extraction of instantaneous frequencies from ridges in time-frequency representations of signals"](https://arxiv.org/pdf/1310.7276.pdf).


## License

ssqueezepy is MIT licensed, as found in the [LICENSE](https://github.com/OverLordGoldDragon/ssqueezepy/blob/master/LICENSE) file. Some source functions may be under other authorship/licenses; see [NOTICE.txt](https://github.com/OverLordGoldDragon/ssqueezepy/blob/master/NOTICE.txt).