audio_resampling_tutorial.py

# -*- coding: utf-8 -*-
"""
Audio Resampling
================

Here, we will walk through resampling audio waveforms using ``torchaudio``.

"""

# When running this tutorial in Google Colab, install the required packages
# with the following.
# !pip install torchaudio librosa

import torch
import torchaudio
import torchaudio.functional as F
import torchaudio.transforms as T

print(torch.__version__)
print(torchaudio.__version__)

######################################################################
# Preparing data and utility functions (skip this section)
# --------------------------------------------------------
#

# @title Prepare data and utility functions. {display-mode: "form"}
# @markdown
# @markdown You do not need to look into this cell.
# @markdown Just execute once and you are good to go.

# -------------------------------------------------------------------------------
# Preparation of data and helper functions.
# -------------------------------------------------------------------------------

import math
import time

import librosa
import matplotlib.pyplot as plt
from IPython.display import Audio, display
import pandas as pd


DEFAULT_OFFSET = 201
SWEEP_MAX_SAMPLE_RATE = 48000
DEFAULT_LOWPASS_FILTER_WIDTH = 6
DEFAULT_ROLLOFF = 0.99
DEFAULT_RESAMPLING_METHOD = "sinc_interpolation"


def _get_log_freq(sample_rate, max_sweep_rate, offset):
    """Get freqs evenly spaced out in log-scale, between [0, max_sweep_rate // 2]

    offset is used to avoid negative infinity `log(offset + x)`.

    """
    start, stop = math.log(offset), math.log(offset + max_sweep_rate // 2)
    return (
        torch.exp(torch.linspace(start, stop, sample_rate, dtype=torch.double)) - offset
    )


def _get_inverse_log_freq(freq, sample_rate, offset):
    """Find the time where the given frequency is given by _get_log_freq"""
    half = sample_rate // 2
    return sample_rate * (math.log(1 + freq / offset) / math.log(1 + half / offset))


def _get_freq_ticks(sample_rate, offset, f_max):
    # Given the original sample rate used for generating the sweep,
    # find the x-axis value where the log-scale major frequency values fall in
    time, freq = [], []
    for exp in range(2, 5):
        for v in range(1, 10):
            f = v * 10 ** exp
            if f < sample_rate // 2:
                t = _get_inverse_log_freq(f, sample_rate, offset) / sample_rate
                time.append(t)
                freq.append(f)
    t_max = _get_inverse_log_freq(f_max, sample_rate, offset) / sample_rate
    time.append(t_max)
    freq.append(f_max)
    return time, freq


def get_sine_sweep(sample_rate, offset=DEFAULT_OFFSET):
    max_sweep_rate = sample_rate
    freq = _get_log_freq(sample_rate, max_sweep_rate, offset)
    delta = 2 * math.pi * freq / sample_rate
    cummulative = torch.cumsum(delta, dim=0)
    signal = torch.sin(cummulative).unsqueeze(dim=0)
    return signal


def plot_sweep(
    waveform,
    sample_rate,
    title,
    max_sweep_rate=SWEEP_MAX_SAMPLE_RATE,
    offset=DEFAULT_OFFSET,
):
    x_ticks = [100, 500, 1000, 5000, 10000, 20000, max_sweep_rate // 2]
    y_ticks = [1000, 5000, 10000, 20000, sample_rate // 2]

    time, freq = _get_freq_ticks(max_sweep_rate, offset, sample_rate // 2)
    freq_x = [f if f in x_ticks and f <= max_sweep_rate // 2 else None for f in freq]
    freq_y = [f for f in freq if f >= 1000 and f in y_ticks and f <= sample_rate // 2]

    figure, axis = plt.subplots(1, 1)
    axis.specgram(waveform[0].numpy(), Fs=sample_rate)
    plt.xticks(time, freq_x)
    plt.yticks(freq_y, freq_y)
    axis.set_xlabel("Original Signal Frequency (Hz, log scale)")
    axis.set_ylabel("Waveform Frequency (Hz)")
    axis.xaxis.grid(True, alpha=0.67)
    axis.yaxis.grid(True, alpha=0.67)
    figure.suptitle(f"{title} (sample rate: {sample_rate} Hz)")
    plt.show(block=True)


def play_audio(waveform, sample_rate):
    waveform = waveform.numpy()

    num_channels, num_frames = waveform.shape
    if num_channels == 1:
        display(Audio(waveform[0], rate=sample_rate))
    elif num_channels == 2:
        display(Audio((waveform[0], waveform[1]), rate=sample_rate))
    else:
        raise ValueError("Waveform with more than 2 channels are not supported.")


def plot_specgram(waveform, sample_rate, title="Spectrogram", xlim=None):
    waveform = waveform.numpy()

    num_channels, num_frames = waveform.shape

    figure, axes = plt.subplots(num_channels, 1)
    if num_channels == 1:
        axes = [axes]
    for c in range(num_channels):
        axes[c].specgram(waveform[c], Fs=sample_rate)
        if num_channels > 1:
            axes[c].set_ylabel(f"Channel {c+1}")
        if xlim:
            axes[c].set_xlim(xlim)
    figure.suptitle(title)
    plt.show(block=False)


def benchmark_resample(
    method,
    waveform,
    sample_rate,
    resample_rate,
    lowpass_filter_width=DEFAULT_LOWPASS_FILTER_WIDTH,
    rolloff=DEFAULT_ROLLOFF,
    resampling_method=DEFAULT_RESAMPLING_METHOD,
    beta=None,
    librosa_type=None,
    iters=5,
):
    if method == "functional":
        begin = time.time()
        for _ in range(iters):
            F.resample(
                waveform,
                sample_rate,
                resample_rate,
                lowpass_filter_width=lowpass_filter_width,
                rolloff=rolloff,
                resampling_method=resampling_method,
            )
        elapsed = time.time() - begin
        return elapsed / iters
    elif method == "transforms":
        resampler = T.Resample(
            sample_rate,
            resample_rate,
            lowpass_filter_width=lowpass_filter_width,
            rolloff=rolloff,
            resampling_method=resampling_method,
            dtype=waveform.dtype,
        )
        begin = time.time()
        for _ in range(iters):
            resampler(waveform)
        elapsed = time.time() - begin
        return elapsed / iters
    elif method == "librosa":
        waveform_np = waveform.squeeze().numpy()
        begin = time.time()
        for _ in range(iters):
            librosa.resample(
                waveform_np, sample_rate, resample_rate, res_type=librosa_type
            )
        elapsed = time.time() - begin
        return elapsed / iters


######################################################################
# To resample an audio waveform from one freqeuncy to another, you can use
# ``transforms.Resample`` or ``functional.resample``.
# ``transforms.Resample`` precomputes and caches the kernel used for
# resampling, while ``functional.resample`` computes it on the fly, so
# using ``transforms.Resample`` will result in a speedup when resampling
# multiple waveforms using the same parameters (see Benchmarking section).
#
# Both resampling methods use `bandlimited sinc
# interpolation <https://ccrma.stanford.edu/~jos/resample/>`__ to compute
# signal values at arbitrary time steps. The implementation involves
# convolution, so we can take advantage of GPU / multithreading for
# performance improvements. When using resampling in multiple
# subprocesses, such as data loading with multiple worker processes, your
# application might create more threads than your system can handle
# efficiently. Setting ``torch.set_num_threads(1)`` might help in this
# case.
#
# Because a finite number of samples can only represent a finite number of
# frequencies, resampling does not produce perfect results, and a variety
# of parameters can be used to control for its quality and computational
# speed. We demonstrate these properties through resampling a logarithmic
# sine sweep, which is a sine wave that increases exponentially in
# frequency over time.
#
# The spectrograms below show the frequency representation of the signal,
# where the x-axis corresponds to the frequency of the original
# waveform (in log scale), y-axis the frequency of the
# plotted waveform, and color intensity the amplitude.
#


sample_rate = 48000
resample_rate = 32000

waveform = get_sine_sweep(sample_rate)
plot_sweep(waveform, sample_rate, title="Original Waveform")
play_audio(waveform, sample_rate)

resampler = T.Resample(sample_rate, resample_rate, dtype=waveform.dtype)
resampled_waveform = resampler(waveform)
plot_sweep(resampled_waveform, resample_rate, title="Resampled Waveform")
play_audio(waveform, sample_rate)


######################################################################
# Controling resampling quality with parameters
# ---------------------------------------------
#
# Lowpass filter width
# ~~~~~~~~~~~~~~~~~~~~
#
# Because the filter used for interpolation extends infinitely, the
# ``lowpass_filter_width`` parameter is used to control for the width of
# the filter to use to window the interpolation. It is also referred to as
# the number of zero crossings, since the interpolation passes through
# zero at every time unit. Using a larger ``lowpass_filter_width``
# provides a sharper, more precise filter, but is more computationally
# expensive.
#


sample_rate = 48000
resample_rate = 32000

resampled_waveform = F.resample(
    waveform, sample_rate, resample_rate, lowpass_filter_width=6
)
plot_sweep(resampled_waveform, resample_rate, title="lowpass_filter_width=6")

resampled_waveform = F.resample(
    waveform, sample_rate, resample_rate, lowpass_filter_width=128
)
plot_sweep(resampled_waveform, resample_rate, title="lowpass_filter_width=128")


######################################################################
# Rolloff
# ~~~~~~~
#
# The ``rolloff`` parameter is represented as a fraction of the Nyquist
# frequency, which is the maximal frequency representable by a given
# finite sample rate. ``rolloff`` determines the lowpass filter cutoff and
# controls the degree of aliasing, which takes place when frequencies
# higher than the Nyquist are mapped to lower frequencies. A lower rolloff
# will therefore reduce the amount of aliasing, but it will also reduce
# some of the higher frequencies.
#


sample_rate = 48000
resample_rate = 32000

resampled_waveform = F.resample(waveform, sample_rate, resample_rate, rolloff=0.99)
plot_sweep(resampled_waveform, resample_rate, title="rolloff=0.99")

resampled_waveform = F.resample(waveform, sample_rate, resample_rate, rolloff=0.8)
plot_sweep(resampled_waveform, resample_rate, title="rolloff=0.8")


######################################################################
# Window function
# ~~~~~~~~~~~~~~~
#
# By default, ``torchaudio``’s resample uses the Hann window filter, which is
# a weighted cosine function. It additionally supports the Kaiser window,
# which is a near optimal window function that contains an additional
# ``beta`` parameter that allows for the design of the smoothness of the
# filter and width of impulse. This can be controlled using the
# ``resampling_method`` parameter.
#


sample_rate = 48000
resample_rate = 32000

resampled_waveform = F.resample(
    waveform, sample_rate, resample_rate, resampling_method="sinc_interpolation"
)
plot_sweep(resampled_waveform, resample_rate, title="Hann Window Default")

resampled_waveform = F.resample(
    waveform, sample_rate, resample_rate, resampling_method="kaiser_window"
)
plot_sweep(resampled_waveform, resample_rate, title="Kaiser Window Default")


######################################################################
# Comparison against librosa
# --------------------------
#
# ``torchaudio``’s resample function can be used to produce results similar to
# that of librosa (resampy)’s kaiser window resampling, with some noise
#


sample_rate = 48000
resample_rate = 32000

# kaiser_best
resampled_waveform = F.resample(
    waveform,
    sample_rate,
    resample_rate,
    lowpass_filter_width=64,
    rolloff=0.9475937167399596,
    resampling_method="kaiser_window",
    beta=14.769656459379492,
)
plot_sweep(resampled_waveform, resample_rate, title="Kaiser Window Best (torchaudio)")

librosa_resampled_waveform = torch.from_numpy(
    librosa.resample(
        waveform.squeeze().numpy(), sample_rate, resample_rate, res_type="kaiser_best"
    )
).unsqueeze(0)
plot_sweep(
    librosa_resampled_waveform, resample_rate, title="Kaiser Window Best (librosa)"
)

mse = torch.square(resampled_waveform - librosa_resampled_waveform).mean().item()
print("torchaudio and librosa kaiser best MSE:", mse)

# kaiser_fast
resampled_waveform = F.resample(
    waveform,
    sample_rate,
    resample_rate,
    lowpass_filter_width=16,
    rolloff=0.85,
    resampling_method="kaiser_window",
    beta=8.555504641634386,
)
plot_specgram(
    resampled_waveform, resample_rate, title="Kaiser Window Fast (torchaudio)"
)

librosa_resampled_waveform = torch.from_numpy(
    librosa.resample(
        waveform.squeeze().numpy(), sample_rate, resample_rate, res_type="kaiser_fast"
    )
).unsqueeze(0)
plot_sweep(
    librosa_resampled_waveform, resample_rate, title="Kaiser Window Fast (librosa)"
)

mse = torch.square(resampled_waveform - librosa_resampled_waveform).mean().item()
print("torchaudio and librosa kaiser fast MSE:", mse)


######################################################################
# Performance Benchmarking
# ------------------------
#
# Below are benchmarks for downsampling and upsampling waveforms between
# two pairs of sampling rates. We demonstrate the performance implications
# that the ``lowpass_filter_wdith``, window type, and sample rates can
# have. Additionally, we provide a comparison against ``librosa``\ ’s
# ``kaiser_best`` and ``kaiser_fast`` using their corresponding parameters
# in ``torchaudio``.
#
# To elaborate on the results:
#
# - a larger ``lowpass_filter_width`` results in a larger resampling kernel,
#   and therefore increases computation time for both the kernel computation
#   and convolution
# - using ``kaiser_window`` results in longer computation times than the default
#   ``sinc_interpolation`` because it is more complex to compute the intermediate
#   window values - a large GCD between the sample and resample rate will result
#   in a simplification that allows for a smaller kernel and faster kernel computation.
#


configs = {
    "downsample (48 -> 44.1 kHz)": [48000, 44100],
    "downsample (16 -> 8 kHz)": [16000, 8000],
    "upsample (44.1 -> 48 kHz)": [44100, 48000],
    "upsample (8 -> 16 kHz)": [8000, 16000],
}

for label in configs:
    times, rows = [], []
    sample_rate = configs[label][0]
    resample_rate = configs[label][1]
    waveform = get_sine_sweep(sample_rate)

    # sinc 64 zero-crossings
    f_time = benchmark_resample(
        "functional", waveform, sample_rate, resample_rate, lowpass_filter_width=64
    )
    t_time = benchmark_resample(
        "transforms", waveform, sample_rate, resample_rate, lowpass_filter_width=64
    )
    times.append([None, 1000 * f_time, 1000 * t_time])
    rows.append("sinc (width 64)")

    # sinc 6 zero-crossings
    f_time = benchmark_resample(
        "functional", waveform, sample_rate, resample_rate, lowpass_filter_width=16
    )
    t_time = benchmark_resample(
        "transforms", waveform, sample_rate, resample_rate, lowpass_filter_width=16
    )
    times.append([None, 1000 * f_time, 1000 * t_time])
    rows.append("sinc (width 16)")

    # kaiser best
    lib_time = benchmark_resample(
        "librosa", waveform, sample_rate, resample_rate, librosa_type="kaiser_best"
    )
    f_time = benchmark_resample(
        "functional",
        waveform,
        sample_rate,
        resample_rate,
        lowpass_filter_width=64,
        rolloff=0.9475937167399596,
        resampling_method="kaiser_window",
        beta=14.769656459379492,
    )
    t_time = benchmark_resample(
        "transforms",
        waveform,
        sample_rate,
        resample_rate,
        lowpass_filter_width=64,
        rolloff=0.9475937167399596,
        resampling_method="kaiser_window",
        beta=14.769656459379492,
    )
    times.append([1000 * lib_time, 1000 * f_time, 1000 * t_time])
    rows.append("kaiser_best")

    # kaiser fast
    lib_time = benchmark_resample(
        "librosa", waveform, sample_rate, resample_rate, librosa_type="kaiser_fast"
    )
    f_time = benchmark_resample(
        "functional",
        waveform,
        sample_rate,
        resample_rate,
        lowpass_filter_width=16,
        rolloff=0.85,
        resampling_method="kaiser_window",
        beta=8.555504641634386,
    )
    t_time = benchmark_resample(
        "transforms",
        waveform,
        sample_rate,
        resample_rate,
        lowpass_filter_width=16,
        rolloff=0.85,
        resampling_method="kaiser_window",
        beta=8.555504641634386,
    )
    times.append([1000 * lib_time, 1000 * f_time, 1000 * t_time])
    rows.append("kaiser_fast")

    df = pd.DataFrame(
        times, columns=["librosa", "functional", "transforms"], index=rows
    )
    df.columns = pd.MultiIndex.from_product([[f"{label} time (ms)"], df.columns])
    display(df.round(2))