<p align="center"><img src="https://user-images.githubusercontent.com/16495490/99882586-faa86f80-2c3a-11eb-899c-b3984e98b1c7.png" width="300"></p>
# Synchrosqueezing in Python
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[](https://doi.org/10.5281/zenodo.5080508)
[](https://opensource.org/licenses/MIT)
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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
### 2. 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">
### 3. 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">
### 4. 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)
### 5. 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">
### 6. 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).
Raw data
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"keywords": "signal-processing python synchrosqueezing wavelet-transform cwt stft morse-wavelet ridge-extraction time-frequency time-frequency-analysis visualization",
"author": "John Muradeli",
"author_email": "john.muradeli@gmail.com",
"download_url": "https://files.pythonhosted.org/packages/7e/0e/1cfccf7c4cb73e6f3c344597aa91b7e8768d3ca1fcc19c2e26b789b18ae2/ssqueezepy-0.6.5.tar.gz",
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"description": "<p align=\"center\"><img src=\"https://user-images.githubusercontent.com/16495490/99882586-faa86f80-2c3a-11eb-899c-b3984e98b1c7.png\" width=\"300\"></p>\r\n\r\n\r\n# Synchrosqueezing in Python\r\n\r\n[](https://github.com/OverLordGoldDragon/ssqueezepy/actions/workflows/conda.yml)\r\n[](https://codecov.io/gh/OverLordGoldDragon/ssqueezepy)\r\n[](https://badge.fury.io/py/ssqueezepy)\r\n[](https://www.codacy.com/gh/OverLordGoldDragon/ssqueezepy/dashboard?utm_source=github.com&utm_medium=referral&utm_content=OverLordGoldDragon/ssqueezepy&utm_campaign=Badge_Grade)\r\n[](https://doi.org/10.5281/zenodo.5080508)\r\n[](https://opensource.org/licenses/MIT)\r\n<!-- [](https://travis-ci.com/OverLordGoldDragon/ssqueezepy)\r\n[](https://coveralls.io/github/OverLordGoldDragon/ssqueezepy) -->\r\n\r\nSynchrosqueezing 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)\r\n\r\n\r\n## Features\r\n - Continuous Wavelet Transform (CWT), forward & inverse, and its Synchrosqueezing\r\n - Short-Time Fourier Transform (STFT), forward & inverse, and its Synchrosqueezing\r\n - Wavelet visualizations and testing suite\r\n - Generalized Morse Wavelets\r\n - Ridge extraction\r\n - Fastest wavelet transforms in Python<sup>1</sup>, beating MATLAB\r\n\r\n<sub>1: feel free to open Issue showing otherwise</sub>\r\n\r\n\r\n## Installation\r\n`pip install ssqueezepy`. Or, for latest version (most likely stable):\r\n\r\n`pip install git+https://github.com/OverLordGoldDragon/ssqueezepy`\r\n\r\n## GPU & CPU acceleration\r\n\r\nMulti-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)\r\nand [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.\r\nSee [Performance guide](https://github.com/OverLordGoldDragon/ssqueezepy/blob/master/ssqueezepy/README.md#performance-guide).\r\n\r\n## Benchmarks\r\n\r\n[Code](https://github.com/OverLordGoldDragon/ssqueezepy/blob/master/examples/benchmarks.py). Transforms use padding, `float32` precision (`float64` supported), and output shape\r\n`(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.\r\n\r\n`len(x)`-transform | 1-thread CPU | parallel | gpu | pywavelets | scipy | librosa\r\n:----------------:|:----------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:\r\n10k-cwt | 0.126 | 0.0462 | 0.00393 | 3.58 | 0.523 | -\r\n10k-stft | 0.108 | 0.0385 | 0.00534 | - | 0.118 | 0.0909\r\n10k-ssq_cwt | 0.372 | 0.148 | 0.00941 | - | - | -\r\n10k-ssq_stft | 0.282 | 0.147 | 0.0278 | - | - | -\r\n160k-cwt | 2.99 | 1.25 | 0.0367 | 12.7 | 10.7 | -\r\n160k-stft | 1.66 | 0.418 | 0.0643 | - | 1.93 | 1.38\r\n160k-ssq_cwt | 8.38 | 3.16 | 0.0856 | - | - | -\r\n160k-ssq_stft | 4.65 | 2.48 | 0.159 | - | - | -\r\n\r\n\r\n## Questions?\r\n\r\nSee [here](#asking-questions).\r\n\r\n## Examples\r\n\r\n### 1. Signal recovery under severe noise\r\n\r\n\r\n\r\n### 2. Medical: EEG\r\n\r\n<img src=\"https://user-images.githubusercontent.com/16495490/99880110-c88f1180-2c2a-11eb-8932-90bf3406a20d.png\">\r\n\r\n<img src=\"https://user-images.githubusercontent.com/16495490/150314341-df5c3092-4bef-4895-99ed-2765504329fd.png\">\r\n\r\n### 3. Testing suite: CWT vs STFT, reflect-added parallel A.M. linear chirp\r\n\r\n<img src=\"https://github.com/OverLordGoldDragon/ssqueezepy/assets/16495490/c89727db-1bb3-4cf0-ac8c-c524cba75b2d\">\r\n\r\n### 4. Ridge extraction: cubic polynom. F.M. + pure tone; noiseless & 1.69dB SNR\r\n\r\n<img src=\"https://user-images.githubusercontent.com/16495490/107919540-f4e5d000-6f84-11eb-9f86-dbfd34733084.png\">\r\n\r\n[More](https://github.com/OverLordGoldDragon/ssqueezepy/tree/master/examples/ridge_extraction)\r\n\r\n### 5. Testing suite: GMW vs Morlet, reflect-added hyperbolic chirp (extreme time-loc.)\r\n\r\n<img src=\"https://github.com/OverLordGoldDragon/ssqueezepy/assets/16495490/8c41d5f2-4bdd-4537-8d82-6d5a5c0315d3\">\r\n\r\n### 6. Higher-order GMW CWT, reflect-added parallel linear chirp, 3.06dB SNR\r\n\r\n<img src=\"https://user-images.githubusercontent.com/16495490/107921072-66bf1900-6f87-11eb-9bf5-afd0a6bbbc4d.png\">\r\n\r\n[More examples](https://overlordgolddragon.github.io/test-signals/)\r\n\r\n\r\n## Introspection\r\n\r\n`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))\r\n\r\n<p align=\"center\">\r\n <img src=\"https://raw.githubusercontent.com/OverLordGoldDragon/ssqueezepy/master/examples/imgs/anim_tf_morlet20.gif\" width=\"500\">\r\n</p>\r\n\r\n<img src=\"https://raw.githubusercontent.com/OverLordGoldDragon/ssqueezepy/master/examples/imgs/morlet_5vs20_tf.png\">\r\n<img src=\"https://user-images.githubusercontent.com/16495490/107297978-e6338080-6a8d-11eb-8a11-60bfd6e4137d.png\">\r\n\r\n## How's it work?\r\n\r\nIn 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.\r\n\r\nConvolve with localized, analytic kernels\r\n\r\n<img src=\"https://raw.githubusercontent.com/OverLordGoldDragon/StackExchangeAnswers/main/SignalProcessing/Q78512%20-%20Wavelet%20Scattering%20explanation/cwt.gif\" width=\"650\">\r\n\r\ncompute phase transform, then combine oscillations with a shared rate\r\n\r\n<img src=\"https://user-images.githubusercontent.com/16495490/150680428-4a651934-85c6-45e8-8a19-c9b4165e5381.png\" width=\"700\">\r\n\r\n<hr>\r\n\r\n## Minimal example\r\n\r\n```python\r\nimport numpy as np\r\nimport matplotlib.pyplot as plt\r\nfrom ssqueezepy import ssq_cwt, ssq_stft\r\nfrom ssqueezepy.experimental import scale_to_freq\r\n\r\ndef viz(x, Tx, Wx):\r\n plt.imshow(np.abs(Wx), aspect='auto', cmap='turbo')\r\n plt.show()\r\n plt.imshow(np.abs(Tx), aspect='auto', vmin=0, vmax=.2, cmap='turbo')\r\n plt.show()\r\n\r\n#%%# Define signal ####################################\r\nN = 2048\r\nt = np.linspace(0, 10, N, endpoint=False)\r\nxo = np.cos(2 * np.pi * 2 * (np.exp(t / 2.2) - 1))\r\nxo += xo[::-1] # add self reflected\r\nx = xo + np.sqrt(2) * np.random.randn(N) # add noise\r\n\r\nplt.plot(xo); plt.show()\r\nplt.plot(x); plt.show()\r\n\r\n#%%# CWT + SSQ CWT ####################################\r\nTwxo, Wxo, *_ = ssq_cwt(xo)\r\nviz(xo, Twxo, Wxo)\r\n\r\nTwx, Wx, *_ = ssq_cwt(x)\r\nviz(x, Twx, Wx)\r\n\r\n#%%# STFT + SSQ STFT ##################################\r\nTsxo, Sxo, *_ = ssq_stft(xo)\r\nviz(xo, np.flipud(Tsxo), np.flipud(Sxo))\r\n\r\nTsx, Sx, *_ = ssq_stft(x)\r\nviz(x, np.flipud(Tsx), np.flipud(Sx))\r\n\r\n#%%# With units #######################################\r\nfrom ssqueezepy import Wavelet, cwt, stft, imshow\r\nfs = 400\r\nt = np.linspace(0, N/fs, N)\r\nwavelet = Wavelet()\r\nWx, scales = cwt(x, wavelet)\r\nSx = stft(x)[::-1]\r\n\r\nfreqs_cwt = scale_to_freq(scales, wavelet, len(x), fs=fs)\r\nfreqs_stft = np.linspace(1, 0, len(Sx)) * fs/2\r\n\r\nikw = dict(abs=1, xticks=t, xlabel=\"Time [sec]\", ylabel=\"Frequency [Hz]\")\r\nimshow(Wx, **ikw, yticks=freqs_cwt)\r\nimshow(Sx, **ikw, yticks=freqs_stft)\r\n```\r\n\r\nAlso see ridge extraction [README](https://github.com/OverLordGoldDragon/ssqueezepy/tree/master/examples/ridge_extraction).\r\n\r\n\r\n## Interesting use cases (with code)\r\n\r\n 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\r\n \r\nFeel free to share yours [here](https://github.com/OverLordGoldDragon/ssqueezepy/issues/9).\r\n\r\n\r\n## Learning resources\r\n\r\n 1. [Continuous Wavelet Transform, & vs STFT](https://ccrma.stanford.edu/~unjung/mylec/WTpart1.html)\r\n 2. [Synchrosqueezing's phase transform, intuitively](https://dsp.stackexchange.com/a/72238/50076)\r\n 3. [Wavelet time & frequency resolution visuals](https://dsp.stackexchange.com/a/72044/50076)\r\n 4. [Why oscillations in SSQ of mixed sines? Separability visuals](https://dsp.stackexchange.com/a/72239/50076)\r\n 5. [Zero-padding's effect on spectrum](https://dsp.stackexchange.com/a/70498/50076)\r\n\r\n**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.\r\nThe 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).\r\n\r\n\r\n## Contributors (noteworthy)\r\n\r\n - [David Bondesson](https://github.com/DavidBondesson): ridge extraction (`ridge_extraction.py`; `examples/`: `extracting_ridges.py`, `ridge_extraction/README.md`)\r\n\r\n## Asking questions\r\n\r\nOpen 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.\r\n\r\n**Do not** send e-mail, LinkedIn messages, etc - they will be ignored.\r\n\r\n## How to cite\r\n\r\nShort form:\r\n\r\n> John Muradeli, ssqueezepy, 2020. GitHub repository, https://github.com/OverLordGoldDragon/ssqueezepy/. DOI: 10.5281/zenodo.5080508\r\n\r\nBibTeX:\r\n\r\n```bibtex\r\n@article{OverLordGoldDragon2020ssqueezepy,\r\n title={ssqueezepy},\r\n author={John Muradeli},\r\n journal={GitHub. Note: https://github.com/OverLordGoldDragon/ssqueezepy/},\r\n year={2020},\r\n doi={10.5281/zenodo.5080508},\r\n}\r\n```\r\n\r\n## References\r\n\r\n`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].\r\n\r\n 1. G. Thakur, E. Brevdo, N.-S. Fu\u010dkar, 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.\r\n 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).\r\n 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.\r\n 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.\r\n 5. Mallat, S. [\"Wavelet Tour of Signal Processing 3rd ed\"](https://www.di.ens.fr/~mallat/papiers/WaveletTourChap1-2-3.pdf).\r\n 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).\r\n\r\n\r\n## License\r\n\r\nssqueezepy 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).\r\n",
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