VQT: Variable Q-Transform

Contributions are welcome! Feel free to open an issue or a pull request.

Variable Q-Transform

This is a novel python implementation of the variable Q-transform that was developed due to the need for a more accurate and flexible VQT for use in research. It is battle-tested and has been used in a number of research projects.

• Accuracy: The approach is different in that it is a direct implementation of a spectrogram via a Hilbert transformation at each desired frequency. This results in an exact computation of the spectrogram and is appropriate for research applications where accuracy is critical. The implementation seen in librosa and nnAudio uses recursive downsampling, which can introduce artifacts in the spectrogram under certain conditions.
• Flexibility: The parameters and codebase are less complex than in other libraries, and the filter bank is fully customizable and exposed to the user. Built in plotting of the filter bank makes tuning the parameters easy and intuitive. The main class is a PyTorch Module and the gradient function is maintained, so backpropagation is possible.
• Speed: The backend is written using PyTorch, and allows for GPU acceleration. It is faster than the librosa implementation under most cases. Though it is typically a bit slower (1X-8X) than the nnAudio implementation, however under some conditions (low hop_length), it is as fast or faster. See below section 'What to improve on?' for more details on how to speed it up further.

Installation

Using pip:

pip install vqt

From source:

git clone https://github.com/RichieHakim/vqt.git
cd vqt
pip install -e .

Requirements: torch, numpy, scipy, matplotlib, tqdm
These will be installed automatically if you install from PyPI.

Usage

import vqt

signal = torch.as_tensor(X)  ## torch Tensor of shape (n_channels, n_samples)

my_vqt = vqt.VQT(
    Fs_sample=1000,  ## In Hz
    Q_lowF=3,  ## In periods per octave
    Q_highF=20,  ## In periods per octave
    F_min=10,  ## In Hz
    F_max=400,  ## In Hz
    n_freq_bins=55,  ## Number of frequency bins
    window_type='hann',
    downsample_factor=8,  ## Reduce the output sample rate
    fft_conv=True,  ## Use FFT convolution for speed
    plot_pref=False,  ## Can show the filter bank
)

spectrograms = my_vqt(signal)
x_axis = my_vqt.get_xAxis(n_samples=signal.shape[1])
frequencies = my_vqt.get_freqs()

What is the Variable Q-Transform?

The Variable Q-Transform (VQT) is a time-frequency analysis tool that generates spectrograms, similar to the Short-time Fourier Transform (STFT). It can also be defined as a special case of a wavelet transform (complex Morlet), as well as the generalization of the Constant Q-Transform (CQT). In fact, the VQT subsumes the CQT and the STFT since both can be recreated using specific parameters of the VQT.

In brief, the VQT generates a spectrogram where the frequencies are spaced logarithmically, and the bandwidth of the filters are tuned using two parameters: Q_low and Q_high, where Q describes the number of periods of the oscillatory wavelet at a particular frequency (aka the 'bandwidth'); 'low' refers to the lowest frequency bin, and 'high' refers to the highest frequency bin.

Why use the VQT?

It provides enough knobs to tune the time-frequency resolution trade-off to suit your needs. It is especially useful when time resolution is needed at lower frequencies.

How exactly does this implementation differ from others?

This function works differently than the VQT from librosa or nnAudio in that it does not use the recursive downsampling algorithm from this paper. Instead, it computes the power at each frequency using either direct- or FFT-convolution with a filter bank of complex oscillations, followed by a Hilbert transform. This results in a more accurate computation of the same spectrogram without any artifacts. The direct computation approach also results in code that is more flexible, easier to understand, and it has fewer constraints on the input parameters compared to librosa and nnAudio.

What to improve on?

Contributions are welcome! Feel free to open an issue or a pull request.

• Speed / Memory usage:
  • Lossless approaches:
    • For the conv1d approach: I think it would be much faster if we cropped the filters to remove the blank space from the higher frequency filters. This would be pretty easy to implement and could give a >10x speedup.
  • Lossy approaches:
    • For the fft_conv approach: I believe a large (5-50x) speedup is possible. The lower frequency filters use only a small portion of the spectrum, therefore most of the compute is spent multiplying zeros.
      • Idea 1: Separate out filters in the filter bank whose spectra are all zeros above n_samples_downsampled, crop the spectra above that level, then use ifft with n=n_samples_downsampled to downsample the filter. This would allow for a much faster convolution. For filters that can't be cropped, downsampling would have to be done after the iFFT.
      • Idea 2: using an efficient sparse or non-uniform FFT. An approach where only the non-zero frequencies are computed in the fft, product, and ifft. There is an implmentation of the NUFFT in PyTorch here.
      • Idea 3: Similar to above, a log-frequency iFFT could be used to allow for only the non-zero segment of the filter's spectrum to be used in the convolution.
      • Idea 4: Try using the overlap-add method.
    • Recursive downsampling: Under many circumstances (like when Q_high is not much greater than Q_low), recursive downsampling is fine. Implementing it would be nice just for completeness (from this paper)
    • For conv1d approach: Use a strided convolution.
    • For fftconv approach: Downsample using n=n_samples_downsampled in ifft function.
  • Non-trivial ideas that theoretically could speed things up:
    • An FFT implementation that allows for a reduced set of frequencies to be computed.
• Flexibility:
  • librosa parameter mode: It would be nice to have a mode that allows for the same parameters as librosa to be used.

Demo:

import vqt
import numpy as np
import torch
import matplotlib.pyplot as plt
import scipy

data_ecg = torch.as_tensor(scipy.datasets.electrocardiogram()[:10000])
sample_rate = 360

my_vqt = vqt.VQT(
    Fs_sample=sample_rate,
    Q_lowF=2,
    Q_highF=8,
    F_min=1,
    F_max=120,
    n_freq_bins=150,
    win_size=1501,
    window_type='gaussian',
    downsample_factor=8,
    padding='same',
    fft_conv=True,
    take_abs=True,
    plot_pref=False,
)

specs = my_vqt(data_ecg)
xaxis = my_vqt.get_xAxis(n_samples=data_ecg.shape[0])
freqs = my_vqt.get_freqs()

fig, axs = plt.subplots(nrows=2, ncols=1, sharex=True, )
axs[0].plot(np.arange(data_ecg.shape[0]) / sample_rate, data_ecg)
axs[0].title.set_text('Electrocardiogram')
axs[1].pcolor(
    xaxis / sample_rate, 
    np.arange(specs[0].shape[0]), specs[0] * (freqs)[:, None], 
    vmin=0, 
    vmax=30,
    cmap='hot',
)
axs[1].set_yticks(np.arange(specs.numpy()[0].shape[0])[::10], np.round(freqs.numpy()[::10], 1));
axs[1].set_xlim([13, 22])
axs[0].set_ylabel('mV')
axs[1].set_ylabel('frequency (Hz)')
axs[1].set_xlabel('time (s)')
plt.show()