#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""Constant-Q transforms"""
import warnings
import numpy as np
from numba import jit
from . import audio
from .intervals import interval_frequencies
from .fft import get_fftlib
from .convert import cqt_frequencies, note_to_hz
from .spectrum import stft, istft
from .pitch import estimate_tuning
from .._cache import cache
from .. import filters
from .. import util
from ..util.exceptions import ParameterError
from numpy.typing import DTypeLike
from typing import Optional, Union, Collection, List
from .._typing import _WindowSpec, _PadMode, _FloatLike_co, _ensure_not_reachable
__all__ = ["cqt", "hybrid_cqt", "pseudo_cqt", "icqt", "griffinlim_cqt", "vqt"]
# TODO: ivqt, griffinlim_vqt
[docs]@cache(level=20)
def cqt(
y: np.ndarray,
*,
sr: float = 22050,
hop_length: int = 512,
fmin: Optional[_FloatLike_co] = None,
n_bins: int = 84,
bins_per_octave: int = 12,
tuning: Optional[float] = 0.0,
filter_scale: float = 1,
norm: Optional[float] = 1,
sparsity: float = 0.01,
window: _WindowSpec = "hann",
scale: bool = True,
pad_mode: _PadMode = "constant",
res_type: Optional[str] = "soxr_hq",
dtype: Optional[DTypeLike] = None,
) -> np.ndarray:
"""Compute the constant-Q transform of an audio signal.
This implementation is based on the recursive sub-sampling method
described by [#]_.
.. [#] Schoerkhuber, Christian, and Anssi Klapuri.
"Constant-Q transform toolbox for music processing."
7th Sound and Music Computing Conference, Barcelona, Spain. 2010.
Parameters
----------
y : np.ndarray [shape=(..., n)]
audio time series. Multi-channel is supported.
sr : number > 0 [scalar]
sampling rate of ``y``
hop_length : int > 0 [scalar]
number of samples between successive CQT columns.
fmin : float > 0 [scalar]
Minimum frequency. Defaults to `C1 ~= 32.70 Hz`
n_bins : int > 0 [scalar]
Number of frequency bins, starting at ``fmin``
bins_per_octave : int > 0 [scalar]
Number of bins per octave
tuning : None or float
Tuning offset in fractions of a bin.
If ``None``, tuning will be automatically estimated from the signal.
The minimum frequency of the resulting CQT will be modified to
``fmin * 2**(tuning / bins_per_octave)``.
filter_scale : float > 0
Filter scale factor. Small values (<1) use shorter windows
for improved time resolution.
norm : {inf, -inf, 0, float > 0}
Type of norm to use for basis function normalization.
See `librosa.util.normalize`.
sparsity : float in [0, 1)
Sparsify the CQT basis by discarding up to ``sparsity``
fraction of the energy in each basis.
Set ``sparsity=0`` to disable sparsification.
window : str, tuple, number, or function
Window specification for the basis filters.
See `filters.get_window` for details.
scale : bool
If ``True``, scale the CQT response by square-root the length of
each channel's filter. This is analogous to ``norm='ortho'`` in FFT.
If ``False``, do not scale the CQT. This is analogous to
``norm=None`` in FFT.
pad_mode : string
Padding mode for centered frame analysis.
See also: `librosa.stft` and `numpy.pad`.
res_type : string
The resampling mode for recursive downsampling.
dtype : np.dtype
The (complex) data type of the output array. By default, this is inferred to match
the numerical precision of the input signal.
Returns
-------
CQT : np.ndarray [shape=(..., n_bins, t)]
Constant-Q value each frequency at each time.
See Also
--------
vqt
librosa.resample
librosa.util.normalize
Notes
-----
This function caches at level 20.
Examples
--------
Generate and plot a constant-Q power spectrum
>>> import matplotlib.pyplot as plt
>>> y, sr = librosa.load(librosa.ex('trumpet'))
>>> C = np.abs(librosa.cqt(y, sr=sr))
>>> fig, ax = plt.subplots()
>>> img = librosa.display.specshow(librosa.amplitude_to_db(C, ref=np.max),
... sr=sr, x_axis='time', y_axis='cqt_note', ax=ax)
>>> ax.set_title('Constant-Q power spectrum')
>>> fig.colorbar(img, ax=ax, format="%+2.0f dB")
Limit the frequency range
>>> C = np.abs(librosa.cqt(y, sr=sr, fmin=librosa.note_to_hz('C2'),
... n_bins=60))
>>> C
array([[6.830e-04, 6.361e-04, ..., 7.362e-09, 9.102e-09],
[5.366e-04, 4.818e-04, ..., 8.953e-09, 1.067e-08],
...,
[4.288e-02, 4.580e-01, ..., 1.529e-05, 5.572e-06],
[2.965e-03, 1.508e-01, ..., 8.965e-06, 1.455e-05]])
Using a higher frequency resolution
>>> C = np.abs(librosa.cqt(y, sr=sr, fmin=librosa.note_to_hz('C2'),
... n_bins=60 * 2, bins_per_octave=12 * 2))
>>> C
array([[5.468e-04, 5.382e-04, ..., 5.911e-09, 6.105e-09],
[4.118e-04, 4.014e-04, ..., 7.788e-09, 8.160e-09],
...,
[2.780e-03, 1.424e-01, ..., 4.225e-06, 2.388e-05],
[5.147e-02, 6.959e-02, ..., 1.694e-05, 5.811e-06]])
"""
# CQT is the special case of VQT with gamma=0
return vqt(
y=y,
sr=sr,
hop_length=hop_length,
fmin=fmin,
n_bins=n_bins,
intervals="equal",
gamma=0,
bins_per_octave=bins_per_octave,
tuning=tuning,
filter_scale=filter_scale,
norm=norm,
sparsity=sparsity,
window=window,
scale=scale,
pad_mode=pad_mode,
res_type=res_type,
dtype=dtype,
)
[docs]@cache(level=20)
def hybrid_cqt(
y: np.ndarray,
*,
sr: float = 22050,
hop_length: int = 512,
fmin: Optional[_FloatLike_co] = None,
n_bins: int = 84,
bins_per_octave: int = 12,
tuning: Optional[float] = 0.0,
filter_scale: float = 1,
norm: Optional[float] = 1,
sparsity: float = 0.01,
window: _WindowSpec = "hann",
scale: bool = True,
pad_mode: _PadMode = "constant",
res_type: str = "soxr_hq",
dtype: Optional[DTypeLike] = None,
) -> np.ndarray:
"""Compute the hybrid constant-Q transform of an audio signal.
Here, the hybrid CQT uses the pseudo CQT for higher frequencies where
the hop_length is longer than half the filter length and the full CQT
for lower frequencies.
Parameters
----------
y : np.ndarray [shape=(..., n)]
audio time series. Multi-channel is supported.
sr : number > 0 [scalar]
sampling rate of ``y``
hop_length : int > 0 [scalar]
number of samples between successive CQT columns.
fmin : float > 0 [scalar]
Minimum frequency. Defaults to `C1 ~= 32.70 Hz`
n_bins : int > 0 [scalar]
Number of frequency bins, starting at ``fmin``
bins_per_octave : int > 0 [scalar]
Number of bins per octave
tuning : None or float
Tuning offset in fractions of a bin.
If ``None``, tuning will be automatically estimated from the signal.
The minimum frequency of the resulting CQT will be modified to
``fmin * 2**(tuning / bins_per_octave)``.
filter_scale : float > 0
Filter filter_scale factor. Larger values use longer windows.
norm : {inf, -inf, 0, float > 0}
Type of norm to use for basis function normalization.
See `librosa.util.normalize`.
sparsity : float in [0, 1)
Sparsify the CQT basis by discarding up to ``sparsity``
fraction of the energy in each basis.
Set ``sparsity=0`` to disable sparsification.
window : str, tuple, number, or function
Window specification for the basis filters.
See `filters.get_window` for details.
scale : bool
If ``True``, scale the CQT response by square-root the length of
each channel's filter. This is analogous to ``norm='ortho'`` in FFT.
If ``False``, do not scale the CQT. This is analogous to
``norm=None`` in FFT.
pad_mode : string
Padding mode for centered frame analysis.
See also: `librosa.stft` and `numpy.pad`.
res_type : string
Resampling mode. See `librosa.cqt` for details.
dtype : np.dtype, optional
The complex dtype to use for computing the CQT.
By default, this is inferred to match the precision of
the input signal.
Returns
-------
CQT : np.ndarray [shape=(..., n_bins, t), dtype=np.float]
Constant-Q energy for each frequency at each time.
See Also
--------
cqt
pseudo_cqt
Notes
-----
This function caches at level 20.
"""
if fmin is None:
# C1 by default
fmin = note_to_hz("C1")
if tuning is None:
tuning = estimate_tuning(y=y, sr=sr, bins_per_octave=bins_per_octave)
# Apply tuning correction
fmin = fmin * 2.0 ** (tuning / bins_per_octave)
# Get all CQT frequencies
freqs = cqt_frequencies(n_bins, fmin=fmin, bins_per_octave=bins_per_octave)
# Pre-compute alpha
if n_bins == 1:
alpha = __et_relative_bw(bins_per_octave)
else:
alpha = filters._relative_bandwidth(freqs=freqs)
# Compute the length of each constant-Q basis function
lengths, _ = filters.wavelet_lengths(
freqs=freqs, sr=sr, filter_scale=filter_scale, window=window, alpha=alpha
)
# Determine which filters to use with Pseudo CQT
# These are the ones that fit within 2 hop lengths after padding
pseudo_filters = 2.0 ** np.ceil(np.log2(lengths)) < 2 * hop_length
n_bins_pseudo = int(np.sum(pseudo_filters))
n_bins_full = n_bins - n_bins_pseudo
cqt_resp = []
if n_bins_pseudo > 0:
fmin_pseudo = np.min(freqs[pseudo_filters])
cqt_resp.append(
pseudo_cqt(
y,
sr=sr,
hop_length=hop_length,
fmin=fmin_pseudo,
n_bins=n_bins_pseudo,
bins_per_octave=bins_per_octave,
filter_scale=filter_scale,
norm=norm,
sparsity=sparsity,
window=window,
scale=scale,
pad_mode=pad_mode,
dtype=dtype,
)
)
if n_bins_full > 0:
cqt_resp.append(
np.abs(
cqt(
y,
sr=sr,
hop_length=hop_length,
fmin=fmin,
n_bins=n_bins_full,
bins_per_octave=bins_per_octave,
filter_scale=filter_scale,
norm=norm,
sparsity=sparsity,
window=window,
scale=scale,
pad_mode=pad_mode,
res_type=res_type,
dtype=dtype,
)
)
)
# Propagate dtype from the last component
return __trim_stack(cqt_resp, n_bins, cqt_resp[-1].dtype)
[docs]@cache(level=20)
def pseudo_cqt(
y: np.ndarray,
*,
sr: float = 22050,
hop_length: int = 512,
fmin: Optional[_FloatLike_co] = None,
n_bins: int = 84,
bins_per_octave: int = 12,
tuning: Optional[float] = 0.0,
filter_scale: float = 1,
norm: Optional[float] = 1,
sparsity: float = 0.01,
window: _WindowSpec = "hann",
scale: bool = True,
pad_mode: _PadMode = "constant",
dtype: Optional[DTypeLike] = None,
) -> np.ndarray:
"""Compute the pseudo constant-Q transform of an audio signal.
This uses a single fft size that is the smallest power of 2 that is greater
than or equal to the max of:
1. The longest CQT filter
2. 2x the hop_length
Parameters
----------
y : np.ndarray [shape=(..., n)]
audio time series. Multi-channel is supported.
sr : number > 0 [scalar]
sampling rate of ``y``
hop_length : int > 0 [scalar]
number of samples between successive CQT columns.
fmin : float > 0 [scalar]
Minimum frequency. Defaults to `C1 ~= 32.70 Hz`
n_bins : int > 0 [scalar]
Number of frequency bins, starting at ``fmin``
bins_per_octave : int > 0 [scalar]
Number of bins per octave
tuning : None or float
Tuning offset in fractions of a bin.
If ``None``, tuning will be automatically estimated from the signal.
The minimum frequency of the resulting CQT will be modified to
``fmin * 2**(tuning / bins_per_octave)``.
filter_scale : float > 0
Filter filter_scale factor. Larger values use longer windows.
norm : {inf, -inf, 0, float > 0}
Type of norm to use for basis function normalization.
See `librosa.util.normalize`.
sparsity : float in [0, 1)
Sparsify the CQT basis by discarding up to ``sparsity``
fraction of the energy in each basis.
Set ``sparsity=0`` to disable sparsification.
window : str, tuple, number, or function
Window specification for the basis filters.
See `filters.get_window` for details.
scale : bool
If ``True``, scale the CQT response by square-root the length of
each channel's filter. This is analogous to ``norm='ortho'`` in FFT.
If ``False``, do not scale the CQT. This is analogous to
``norm=None`` in FFT.
pad_mode : string
Padding mode for centered frame analysis.
See also: `librosa.stft` and `numpy.pad`.
dtype : np.dtype, optional
The complex data type for CQT calculations.
By default, this is inferred to match the precision of the input signal.
Returns
-------
CQT : np.ndarray [shape=(..., n_bins, t), dtype=np.float]
Pseudo Constant-Q energy for each frequency at each time.
Notes
-----
This function caches at level 20.
"""
if fmin is None:
# C1 by default
fmin = note_to_hz("C1")
if tuning is None:
tuning = estimate_tuning(y=y, sr=sr, bins_per_octave=bins_per_octave)
if dtype is None:
dtype = util.dtype_r2c(y.dtype)
# Apply tuning correction
fmin = fmin * 2.0 ** (tuning / bins_per_octave)
freqs = cqt_frequencies(fmin=fmin, n_bins=n_bins, bins_per_octave=bins_per_octave)
if n_bins == 1:
alpha = __et_relative_bw(bins_per_octave)
else:
alpha = filters._relative_bandwidth(freqs=freqs)
lengths, _ = filters.wavelet_lengths(
freqs=freqs, sr=sr, window=window, filter_scale=filter_scale, alpha=alpha
)
fft_basis, n_fft, _ = __vqt_filter_fft(
sr,
freqs,
filter_scale,
norm,
sparsity,
hop_length=hop_length,
window=window,
dtype=dtype,
alpha=alpha,
)
fft_basis = np.abs(fft_basis)
# Compute the magnitude-only CQT response
C: np.ndarray = __cqt_response(
y,
n_fft,
hop_length,
fft_basis,
pad_mode,
window="hann",
dtype=dtype,
phase=False,
)
if scale:
C /= np.sqrt(n_fft)
else:
# reshape lengths to match dimension properly
lengths = util.expand_to(lengths, ndim=C.ndim, axes=-2)
C *= np.sqrt(lengths / n_fft)
return C
[docs]@cache(level=40)
def icqt(
C: np.ndarray,
*,
sr: float = 22050,
hop_length: int = 512,
fmin: Optional[_FloatLike_co] = None,
bins_per_octave: int = 12,
tuning: float = 0.0,
filter_scale: float = 1,
norm: Optional[float] = 1,
sparsity: float = 0.01,
window: _WindowSpec = "hann",
scale: bool = True,
length: Optional[int] = None,
res_type: str = "soxr_hq",
dtype: Optional[DTypeLike] = None,
) -> np.ndarray:
"""Compute the inverse constant-Q transform.
Given a constant-Q transform representation ``C`` of an audio signal ``y``,
this function produces an approximation ``y_hat``.
Parameters
----------
C : np.ndarray, [shape=(..., n_bins, n_frames)]
Constant-Q representation as produced by `cqt`
sr : number > 0 [scalar]
sampling rate of the signal
hop_length : int > 0 [scalar]
number of samples between successive frames
fmin : float > 0 [scalar]
Minimum frequency. Defaults to `C1 ~= 32.70 Hz`
bins_per_octave : int > 0 [scalar]
Number of bins per octave
tuning : float [scalar]
Tuning offset in fractions of a bin.
The minimum frequency of the CQT will be modified to
``fmin * 2**(tuning / bins_per_octave)``.
filter_scale : float > 0 [scalar]
Filter scale factor. Small values (<1) use shorter windows
for improved time resolution.
norm : {inf, -inf, 0, float > 0}
Type of norm to use for basis function normalization.
See `librosa.util.normalize`.
sparsity : float in [0, 1)
Sparsify the CQT basis by discarding up to ``sparsity``
fraction of the energy in each basis.
Set ``sparsity=0`` to disable sparsification.
window : str, tuple, number, or function
Window specification for the basis filters.
See `filters.get_window` for details.
scale : bool
If ``True``, scale the CQT response by square-root the length
of each channel's filter. This is analogous to ``norm='ortho'`` in FFT.
If ``False``, do not scale the CQT. This is analogous to ``norm=None``
in FFT.
length : int > 0, optional
If provided, the output ``y`` is zero-padded or clipped to exactly
``length`` samples.
res_type : string
Resampling mode.
See `librosa.resample` for supported modes.
dtype : numeric type
Real numeric type for ``y``. Default is inferred to match the numerical
precision of the input CQT.
Returns
-------
y : np.ndarray, [shape=(..., n_samples), dtype=np.float]
Audio time-series reconstructed from the CQT representation.
See Also
--------
cqt
librosa.resample
Notes
-----
This function caches at level 40.
Examples
--------
Using default parameters
>>> y, sr = librosa.load(librosa.ex('trumpet'))
>>> C = librosa.cqt(y=y, sr=sr)
>>> y_hat = librosa.icqt(C=C, sr=sr)
Or with a different hop length and frequency resolution:
>>> hop_length = 256
>>> bins_per_octave = 12 * 3
>>> C = librosa.cqt(y=y, sr=sr, hop_length=256, n_bins=7*bins_per_octave,
... bins_per_octave=bins_per_octave)
>>> y_hat = librosa.icqt(C=C, sr=sr, hop_length=hop_length,
... bins_per_octave=bins_per_octave)
"""
if fmin is None:
fmin = note_to_hz("C1")
# Apply tuning correction
fmin = fmin * 2.0 ** (tuning / bins_per_octave)
# Get the top octave of frequencies
n_bins = C.shape[-2]
n_octaves = int(np.ceil(float(n_bins) / bins_per_octave))
freqs = cqt_frequencies(fmin=fmin, n_bins=n_bins, bins_per_octave=bins_per_octave)
if n_bins == 1:
alpha = __et_relative_bw(bins_per_octave)
else:
alpha = filters._relative_bandwidth(freqs=freqs)
lengths, f_cutoff = filters.wavelet_lengths(
freqs=freqs, sr=sr, window=window, filter_scale=filter_scale, alpha=alpha
)
# Trim the CQT to only what's necessary for reconstruction
if length is not None:
n_frames = int(np.ceil((length + max(lengths)) / hop_length))
C = C[..., :n_frames]
C_scale = np.sqrt(lengths)
# This shape array will be used for broadcasting the basis scale
# we'll have to adapt this per octave within the loop
y: Optional[np.ndarray] = None
# Assume the top octave is at the full rate
srs = [sr]
hops = [hop_length]
for i in range(n_octaves - 1):
if hops[0] % 2 == 0:
# We can downsample:
srs.insert(0, srs[0] * 0.5)
hops.insert(0, hops[0] // 2)
else:
# We're out of downsamplings, carry forward
srs.insert(0, srs[0])
hops.insert(0, hops[0])
for i, (my_sr, my_hop) in enumerate(zip(srs, hops)):
# How many filters are in this octave?
n_filters = min(bins_per_octave, n_bins - bins_per_octave * i)
# Slice out the current octave
sl = slice(bins_per_octave * i, bins_per_octave * i + n_filters)
fft_basis, n_fft, _ = __vqt_filter_fft(
my_sr,
freqs[sl],
filter_scale,
norm,
sparsity,
window=window,
alpha=alpha[sl],
)
# Transpose the basis
inv_basis = fft_basis.H.todense()
# Compute each filter's frequency-domain power
freq_power = 1 / np.sum(util.abs2(np.asarray(inv_basis)), axis=0)
# Compensate for length normalization in the forward transform
freq_power *= n_fft / lengths[sl]
# Inverse-project the basis for each octave
if scale:
# scale=True ==> re-scale by sqrt(lengths)
D_oct = np.einsum(
"fc,c,c,...ct->...ft",
inv_basis,
C_scale[sl],
freq_power,
C[..., sl, :],
optimize=True,
)
else:
D_oct = np.einsum(
"fc,c,...ct->...ft", inv_basis, freq_power, C[..., sl, :], optimize=True
)
y_oct = istft(D_oct, window="ones", hop_length=my_hop, dtype=dtype)
y_oct = audio.resample(
y_oct,
orig_sr=1,
target_sr=sr // my_sr,
res_type=res_type,
scale=False,
fix=False,
)
if y is None:
y = y_oct
else:
y[..., : y_oct.shape[-1]] += y_oct
# make mypy happy
assert y is not None
if length:
y = util.fix_length(y, size=length)
return y
[docs]@cache(level=20)
def vqt(
y: np.ndarray,
*,
sr: float = 22050,
hop_length: int = 512,
fmin: Optional[_FloatLike_co] = None,
n_bins: int = 84,
intervals: Union[str, Collection[float]] = "equal",
gamma: Optional[float] = None,
bins_per_octave: int = 12,
tuning: Optional[float] = 0.0,
filter_scale: float = 1,
norm: Optional[float] = 1,
sparsity: float = 0.01,
window: _WindowSpec = "hann",
scale: bool = True,
pad_mode: _PadMode = "constant",
res_type: Optional[str] = "soxr_hq",
dtype: Optional[DTypeLike] = None,
) -> np.ndarray:
"""Compute the variable-Q transform of an audio signal.
This implementation is based on the recursive sub-sampling method
described by [#]_.
.. [#] Schörkhuber, Christian, Anssi Klapuri, Nicki Holighaus, and Monika Dörfler.
"A Matlab toolbox for efficient perfect reconstruction time-frequency
transforms with log-frequency resolution."
In Audio Engineering Society Conference: 53rd International Conference: Semantic Audio.
Audio Engineering Society, 2014.
Parameters
----------
y : np.ndarray [shape=(..., n)]
audio time series. Multi-channel is supported.
sr : number > 0 [scalar]
sampling rate of ``y``
hop_length : int > 0 [scalar]
number of samples between successive VQT columns.
fmin : float > 0 [scalar]
Minimum frequency. Defaults to `C1 ~= 32.70 Hz`
n_bins : int > 0 [scalar]
Number of frequency bins, starting at ``fmin``
intervals : str or array of floats in [1, 2)
Either a string specification for an interval set, e.g.,
`'equal'`, `'pythagorean'`, `'ji3'`, etc. or an array of
intervals expressed as numbers between 1 and 2.
.. see also:: librosa.interval_frequencies
gamma : number > 0 [scalar]
Bandwidth offset for determining filter lengths.
If ``gamma=0``, produces the constant-Q transform.
If 'gamma=None', gamma will be calculated such that filter bandwidths are equal to a
constant fraction of the equivalent rectangular bandwidths (ERB). This is accomplished
by solving for the gamma which gives::
B_k = alpha * f_k + gamma = C * ERB(f_k),
where ``B_k`` is the bandwidth of filter ``k`` with center frequency ``f_k``, alpha
is the inverse of what would be the constant Q-factor, and ``C = alpha / 0.108`` is the
constant fraction across all filters.
Here we use ``ERB(f_k) = 24.7 + 0.108 * f_k``, the best-fit curve derived
from experimental data in [#]_.
.. [#] Glasberg, Brian R., and Brian CJ Moore.
"Derivation of auditory filter shapes from notched-noise data."
Hearing research 47.1-2 (1990): 103-138.
bins_per_octave : int > 0 [scalar]
Number of bins per octave
tuning : None or float
Tuning offset in fractions of a bin.
If ``None``, tuning will be automatically estimated from the signal.
The minimum frequency of the resulting VQT will be modified to
``fmin * 2**(tuning / bins_per_octave)``.
filter_scale : float > 0
Filter scale factor. Small values (<1) use shorter windows
for improved time resolution.
norm : {inf, -inf, 0, float > 0}
Type of norm to use for basis function normalization.
See `librosa.util.normalize`.
sparsity : float in [0, 1)
Sparsify the VQT basis by discarding up to ``sparsity``
fraction of the energy in each basis.
Set ``sparsity=0`` to disable sparsification.
window : str, tuple, number, or function
Window specification for the basis filters.
See `filters.get_window` for details.
scale : bool
If ``True``, scale the VQT response by square-root the length of
each channel's filter. This is analogous to ``norm='ortho'`` in FFT.
If ``False``, do not scale the VQT. This is analogous to
``norm=None`` in FFT.
pad_mode : string
Padding mode for centered frame analysis.
See also: `librosa.stft` and `numpy.pad`.
res_type : string
The resampling mode for recursive downsampling.
dtype : np.dtype
The dtype of the output array. By default, this is inferred to match the
numerical precision of the input signal.
Returns
-------
VQT : np.ndarray [shape=(..., n_bins, t), dtype=np.complex]
Variable-Q value each frequency at each time.
See Also
--------
cqt
Notes
-----
This function caches at level 20.
Examples
--------
Generate and plot a variable-Q power spectrum
>>> import matplotlib.pyplot as plt
>>> y, sr = librosa.load(librosa.ex('choice'), duration=5)
>>> C = np.abs(librosa.cqt(y, sr=sr))
>>> V = np.abs(librosa.vqt(y, sr=sr))
>>> fig, ax = plt.subplots(nrows=2, sharex=True, sharey=True)
>>> librosa.display.specshow(librosa.amplitude_to_db(C, ref=np.max),
... sr=sr, x_axis='time', y_axis='cqt_note', ax=ax[0])
>>> ax[0].set(title='Constant-Q power spectrum', xlabel=None)
>>> ax[0].label_outer()
>>> img = librosa.display.specshow(librosa.amplitude_to_db(V, ref=np.max),
... sr=sr, x_axis='time', y_axis='cqt_note', ax=ax[1])
>>> ax[1].set_title('Variable-Q power spectrum')
>>> fig.colorbar(img, ax=ax, format="%+2.0f dB")
"""
# If intervals are provided as an array, override BPO
if not isinstance(intervals, str):
bins_per_octave = len(intervals)
# How many octaves are we dealing with?
n_octaves = int(np.ceil(float(n_bins) / bins_per_octave))
n_filters = min(bins_per_octave, n_bins)
if fmin is None:
# C1 by default
fmin = note_to_hz("C1")
if tuning is None:
tuning = estimate_tuning(y=y, sr=sr, bins_per_octave=bins_per_octave)
if dtype is None:
dtype = util.dtype_r2c(y.dtype)
# Apply tuning correction
fmin = fmin * 2.0 ** (tuning / bins_per_octave)
# First thing, get the freqs of the top octave
freqs = interval_frequencies(
n_bins=n_bins,
fmin=fmin,
intervals=intervals,
bins_per_octave=bins_per_octave,
sort=True,
)
freqs_top = freqs[-bins_per_octave:]
fmax_t: float = np.max(freqs_top)
if n_bins == 1:
alpha = __et_relative_bw(bins_per_octave)
else:
alpha = filters._relative_bandwidth(freqs=freqs)
lengths, filter_cutoff = filters.wavelet_lengths(
freqs=freqs,
sr=sr,
window=window,
filter_scale=filter_scale,
gamma=gamma,
alpha=alpha,
)
# Determine required resampling quality
nyquist = sr / 2.0
if filter_cutoff > nyquist:
raise ParameterError(
f"Wavelet basis with max frequency={fmax_t} would exceed the Nyquist frequency={nyquist}. "
"Try reducing the number of frequency bins."
)
if res_type is None:
warnings.warn(
"Support for VQT with res_type=None is deprecated in librosa 0.10\n"
"and will be removed in version 1.0.",
category=FutureWarning,
stacklevel=2,
)
res_type = "soxr_hq"
y, sr, hop_length = __early_downsample(
y, sr, hop_length, res_type, n_octaves, nyquist, filter_cutoff, scale
)
vqt_resp = []
# Iterate down the octaves
my_y, my_sr, my_hop = y, sr, hop_length
for i in range(n_octaves):
# Slice out the current octave of filters
if i == 0:
sl = slice(-n_filters, None)
else:
sl = slice(-n_filters * (i + 1), -n_filters * i)
# This may be incorrect with early downsampling
freqs_oct = freqs[sl]
alpha_oct = alpha[sl]
fft_basis, n_fft, _ = __vqt_filter_fft(
my_sr,
freqs_oct,
filter_scale,
norm,
sparsity,
window=window,
gamma=gamma,
dtype=dtype,
alpha=alpha_oct,
)
# Re-scale the filters to compensate for downsampling
fft_basis[:] *= np.sqrt(sr / my_sr)
# Compute the vqt filter response and append to the stack
vqt_resp.append(
__cqt_response(my_y, n_fft, my_hop, fft_basis, pad_mode, dtype=dtype)
)
if my_hop % 2 == 0:
my_hop //= 2
my_sr /= 2.0
my_y = audio.resample(
my_y, orig_sr=2, target_sr=1, res_type=res_type, scale=True
)
V = __trim_stack(vqt_resp, n_bins, dtype)
if scale:
# Recompute lengths here because early downsampling may have changed
# our sampling rate
lengths, _ = filters.wavelet_lengths(
freqs=freqs,
sr=sr,
window=window,
filter_scale=filter_scale,
gamma=gamma,
alpha=alpha,
)
# reshape lengths to match V shape
lengths = util.expand_to(lengths, ndim=V.ndim, axes=-2)
V /= np.sqrt(lengths)
return V
@cache(level=10)
def __vqt_filter_fft(
sr,
freqs,
filter_scale,
norm,
sparsity,
hop_length=None,
window="hann",
gamma=0.0,
dtype=np.complex64,
alpha=None,
):
"""Generate the frequency domain variable-Q filter basis."""
basis, lengths = filters.wavelet(
freqs=freqs,
sr=sr,
filter_scale=filter_scale,
norm=norm,
pad_fft=True,
window=window,
gamma=gamma,
alpha=alpha,
)
# Filters are padded up to the nearest integral power of 2
n_fft = basis.shape[1]
if hop_length is not None and n_fft < 2.0 ** (1 + np.ceil(np.log2(hop_length))):
n_fft = int(2.0 ** (1 + np.ceil(np.log2(hop_length))))
# re-normalize bases with respect to the FFT window length
basis *= lengths[:, np.newaxis] / float(n_fft)
# FFT and retain only the non-negative frequencies
fft = get_fftlib()
fft_basis = fft.fft(basis, n=n_fft, axis=1)[:, : (n_fft // 2) + 1]
# sparsify the basis
fft_basis = util.sparsify_rows(fft_basis, quantile=sparsity, dtype=dtype)
return fft_basis, n_fft, lengths
def __trim_stack(
cqt_resp: List[np.ndarray], n_bins: int, dtype: DTypeLike
) -> np.ndarray:
"""Trim and stack a collection of CQT responses"""
max_col = min(c_i.shape[-1] for c_i in cqt_resp)
# Grab any leading dimensions
shape = list(cqt_resp[0].shape)
shape[-2] = n_bins
shape[-1] = max_col
cqt_out = np.empty(shape, dtype=dtype, order="F")
# Copy per-octave data into output array
end = n_bins
for c_i in cqt_resp:
# By default, take the whole octave
n_oct = c_i.shape[-2]
# If the whole octave is more than we can fit,
# take the highest bins from c_i
if end < n_oct:
cqt_out[..., :end, :] = c_i[..., -end:, :max_col]
else:
cqt_out[..., end - n_oct : end, :] = c_i[..., :max_col]
end -= n_oct
return cqt_out
def __cqt_response(
y, n_fft, hop_length, fft_basis, mode, window="ones", phase=True, dtype=None
):
"""Compute the filter response with a target STFT hop."""
# Compute the STFT matrix
D = stft(
y, n_fft=n_fft, hop_length=hop_length, window=window, pad_mode=mode, dtype=dtype
)
if not phase:
D = np.abs(D)
# Reshape D to Dr
Dr = D.reshape((-1, D.shape[-2], D.shape[-1]))
output_flat = np.empty(
(Dr.shape[0], fft_basis.shape[0], Dr.shape[-1]), dtype=D.dtype
)
# iterate over channels
# project fft_basis.dot(Dr[i])
for i in range(Dr.shape[0]):
output_flat[i] = fft_basis.dot(Dr[i])
# reshape Dr to match D's leading dimensions again
shape = list(D.shape)
shape[-2] = fft_basis.shape[0]
return output_flat.reshape(shape)
def __early_downsample_count(nyquist, filter_cutoff, hop_length, n_octaves):
"""Compute the number of early downsampling operations"""
downsample_count1 = max(0, int(np.ceil(np.log2(nyquist / filter_cutoff)) - 1) - 1)
num_twos = __num_two_factors(hop_length)
downsample_count2 = max(0, num_twos - n_octaves + 1)
return min(downsample_count1, downsample_count2)
def __early_downsample(
y, sr, hop_length, res_type, n_octaves, nyquist, filter_cutoff, scale
):
"""Perform early downsampling on an audio signal, if it applies."""
downsample_count = __early_downsample_count(
nyquist, filter_cutoff, hop_length, n_octaves
)
if downsample_count > 0:
downsample_factor = 2 ** (downsample_count)
hop_length //= downsample_factor
if y.shape[-1] < downsample_factor:
raise ParameterError(
f"Input signal length={len(y):d} is too short for "
f"{n_octaves:d}-octave CQT"
)
new_sr = sr / float(downsample_factor)
y = audio.resample(
y, orig_sr=downsample_factor, target_sr=1, res_type=res_type, scale=True
)
# If we're not going to length-scale after CQT, we
# need to compensate for the downsampling factor here
if not scale:
y *= np.sqrt(downsample_factor)
sr = new_sr
return y, sr, hop_length
@jit(nopython=True, cache=True)
def __num_two_factors(x):
"""Return how many times integer x can be evenly divided by 2.
Returns 0 for non-positive integers.
"""
if x <= 0:
return 0
num_twos = 0
while x % 2 == 0:
num_twos += 1
x //= 2
return num_twos
[docs]def griffinlim_cqt(
C: np.ndarray,
*,
n_iter: int = 32,
sr: float = 22050,
hop_length: int = 512,
fmin: Optional[_FloatLike_co] = None,
bins_per_octave: int = 12,
tuning: float = 0.0,
filter_scale: float = 1,
norm: Optional[float] = 1,
sparsity: float = 0.01,
window: _WindowSpec = "hann",
scale: bool = True,
pad_mode: _PadMode = "constant",
res_type: str = "soxr_hq",
dtype: Optional[DTypeLike] = None,
length: Optional[int] = None,
momentum: float = 0.99,
init: Optional[str] = "random",
random_state: Optional[
Union[int, np.random.RandomState, np.random.Generator]
] = None,
) -> np.ndarray:
"""Approximate constant-Q magnitude spectrogram inversion using the "fast" Griffin-Lim
algorithm.
Given the magnitude of a constant-Q spectrogram (``C``), the algorithm randomly initializes
phase estimates, and then alternates forward- and inverse-CQT operations. [#]_
This implementation is based on the (fast) Griffin-Lim method for Short-time Fourier Transforms, [#]_
but adapted for use with constant-Q spectrograms.
.. [#] D. W. Griffin and J. S. Lim,
"Signal estimation from modified short-time Fourier transform,"
IEEE Trans. ASSP, vol.32, no.2, pp.236–243, Apr. 1984.
.. [#] Perraudin, N., Balazs, P., & Søndergaard, P. L.
"A fast Griffin-Lim algorithm,"
IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (pp. 1-4),
Oct. 2013.
Parameters
----------
C : np.ndarray [shape=(..., n_bins, n_frames)]
The constant-Q magnitude spectrogram
n_iter : int > 0
The number of iterations to run
sr : number > 0
Audio sampling rate
hop_length : int > 0
The hop length of the CQT
fmin : number > 0
Minimum frequency for the CQT.
If not provided, it defaults to `C1`.
bins_per_octave : int > 0
Number of bins per octave
tuning : float
Tuning deviation from A440, in fractions of a bin
filter_scale : float > 0
Filter scale factor. Small values (<1) use shorter windows
for improved time resolution.
norm : {inf, -inf, 0, float > 0}
Type of norm to use for basis function normalization.
See `librosa.util.normalize`.
sparsity : float in [0, 1)
Sparsify the CQT basis by discarding up to ``sparsity``
fraction of the energy in each basis.
Set ``sparsity=0`` to disable sparsification.
window : str, tuple, or function
Window specification for the basis filters.
See `filters.get_window` for details.
scale : bool
If ``True``, scale the CQT response by square-root the length
of each channel's filter. This is analogous to ``norm='ortho'``
in FFT.
If ``False``, do not scale the CQT. This is analogous to ``norm=None``
in FFT.
pad_mode : string
Padding mode for centered frame analysis.
See also: `librosa.stft` and `numpy.pad`.
res_type : string
The resampling mode for recursive downsampling.
See ``librosa.resample`` for a list of available options.
dtype : numeric type
Real numeric type for ``y``. Default is inferred to match the precision
of the input CQT.
length : int > 0, optional
If provided, the output ``y`` is zero-padded or clipped to exactly
``length`` samples.
momentum : float > 0
The momentum parameter for fast Griffin-Lim.
Setting this to 0 recovers the original Griffin-Lim method.
Values near 1 can lead to faster convergence, but above 1 may not converge.
init : None or 'random' [default]
If 'random' (the default), then phase values are initialized randomly
according to ``random_state``. This is recommended when the input ``C`` is
a magnitude spectrogram with no initial phase estimates.
If ``None``, then the phase is initialized from ``C``. This is useful when
an initial guess for phase can be provided, or when you want to resume
Griffin-Lim from a previous output.
random_state : None, int, np.random.RandomState, or np.random.Generator
If int, random_state is the seed used by the random number generator
for phase initialization.
If `np.random.RandomState` or `np.random.Generator` instance, the random number generator itself.
If ``None``, defaults to the `np.random.default_rng()` object.
Returns
-------
y : np.ndarray [shape=(..., n)]
time-domain signal reconstructed from ``C``
See Also
--------
cqt
icqt
griffinlim
filters.get_window
resample
Examples
--------
A basis CQT inverse example
>>> y, sr = librosa.load(librosa.ex('trumpet', hq=True), sr=None)
>>> # Get the CQT magnitude, 7 octaves at 36 bins per octave
>>> C = np.abs(librosa.cqt(y=y, sr=sr, bins_per_octave=36, n_bins=7*36))
>>> # Invert using Griffin-Lim
>>> y_inv = librosa.griffinlim_cqt(C, sr=sr, bins_per_octave=36)
>>> # And invert without estimating phase
>>> y_icqt = librosa.icqt(C, sr=sr, bins_per_octave=36)
Wave-plot the results
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(nrows=3, sharex=True, sharey=True)
>>> librosa.display.waveshow(y, sr=sr, color='b', ax=ax[0])
>>> ax[0].set(title='Original', xlabel=None)
>>> ax[0].label_outer()
>>> librosa.display.waveshow(y_inv, sr=sr, color='g', ax=ax[1])
>>> ax[1].set(title='Griffin-Lim reconstruction', xlabel=None)
>>> ax[1].label_outer()
>>> librosa.display.waveshow(y_icqt, sr=sr, color='r', ax=ax[2])
>>> ax[2].set(title='Magnitude-only icqt reconstruction')
"""
if fmin is None:
fmin = note_to_hz("C1")
if random_state is None:
rng = np.random.default_rng()
elif isinstance(random_state, int):
rng = np.random.RandomState(seed=random_state) # type: ignore
elif isinstance(random_state, (np.random.RandomState, np.random.Generator)):
rng = random_state # type: ignore
else:
_ensure_not_reachable(random_state)
raise ParameterError(f"Unsupported random_state={random_state!r}")
if momentum > 1:
warnings.warn(
f"Griffin-Lim with momentum={momentum} > 1 can be unstable. "
"Proceed with caution!",
stacklevel=2,
)
elif momentum < 0:
raise ParameterError(f"griffinlim_cqt() called with momentum={momentum} < 0")
# using complex64 will keep the result to minimal necessary precision
angles = np.empty(C.shape, dtype=np.complex64)
eps = util.tiny(angles)
if init == "random":
# randomly initialize the phase
angles[:] = util.phasor(2 * np.pi * rng.random(size=C.shape))
elif init is None:
# Initialize an all ones complex matrix
angles[:] = 1.0
else:
raise ParameterError(f"init={init} must either None or 'random'")
# And initialize the previous iterate to 0
rebuilt: np.ndarray = np.array(0.0)
for _ in range(n_iter):
# Store the previous iterate
tprev = rebuilt
# Invert with our current estimate of the phases
inverse = icqt(
C * angles,
sr=sr,
hop_length=hop_length,
bins_per_octave=bins_per_octave,
fmin=fmin,
tuning=tuning,
filter_scale=filter_scale,
window=window,
length=length,
res_type=res_type,
norm=norm,
scale=scale,
sparsity=sparsity,
dtype=dtype,
)
# Rebuild the spectrogram
rebuilt = cqt(
inverse,
sr=sr,
bins_per_octave=bins_per_octave,
n_bins=C.shape[-2],
hop_length=hop_length,
fmin=fmin,
tuning=tuning,
filter_scale=filter_scale,
window=window,
norm=norm,
scale=scale,
sparsity=sparsity,
pad_mode=pad_mode,
res_type=res_type,
)
# Update our phase estimates
angles[:] = rebuilt - (momentum / (1 + momentum)) * tprev
angles[:] /= np.abs(angles) + eps
# Return the final phase estimates
return icqt(
C * angles,
sr=sr,
hop_length=hop_length,
bins_per_octave=bins_per_octave,
tuning=tuning,
filter_scale=filter_scale,
fmin=fmin,
window=window,
length=length,
res_type=res_type,
norm=norm,
scale=scale,
sparsity=sparsity,
dtype=dtype,
)
def __et_relative_bw(bins_per_octave: int) -> np.ndarray:
"""Compute the relative bandwidth coefficient for equal
(geometric) freuqency spacing and a give number of bins
per octave.
This is a special case of the more general `relative_bandwidth`
calculation that can be used when only a single basis frequency
is used.
Parameters
----------
bins_per_octave : int
Returns
-------
alpha : np.ndarray > 0
Value is cast up to a 1d array to allow slicing
"""
r = 2 ** (1 / bins_per_octave)
return np.atleast_1d((r**2 - 1) / (r**2 + 1))