code: #!/usr/bin/env python
  # -*- coding: utf-8 -*-
  '''Constant-Q transforms'''
  from __future__ import division
  
  import numpy as np
  import scipy.fftpack as fft
  
  from . import audio
  from .time_frequency import cqt_frequencies, note_to_hz
  from .spectrum import stft
  from .pitch import estimate_tuning
  from .. import cache
  from .. import filters
  from .. import util
  from ..util.exceptions import ParameterError
  
  __all__ = ['cqt', 'hybrid_cqt', 'pseudo_cqt']
  
  
  @cache(level=20)
  def cqt(y, sr=22050, hop_length=512, fmin=None, n_bins=84,
          bins_per_octave=12, tuning=0.0, filter_scale=1,
          norm=1, sparsity=0.01, window='hann',
          scale=True,
          real=util.Deprecated()):
      '''Compute the constant-Q transform of an audio signal.
  
      This implementation is based on the recursive sub-sampling method
      described by [1]_.
  
      .. [1] 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
  
      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 in `[-0.5, 0.5)`
          Tuning offset in fractions of a bin (cents).
  
          If `None`, tuning will be automatically estimated from the signal.
  
      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.
  
      real : [DEPRECATED]
          .. warning:: This parameter name deprecated in librosa 0.5.0
              It will be removed in librosa 0.6.0.
  
  
      Returns
      -------
      CQT : np.ndarray [shape=(n_bins, t), dtype=np.complex or np.float]
          Constant-Q value each frequency at each time.
  
      Raises
      ------
      ParameterError
          If `hop_length` is not an integer multiple of
          `2**(n_bins / bins_per_octave)`
  
          Or if `y` is too short to support the frequency range of the CQT.
  
      See Also
      --------
      librosa.core.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.util.example_audio_file())
      >>> C = librosa.cqt(y, sr=sr)
      >>> librosa.display.specshow(librosa.logamplitude(C**2, ref_power=np.max),
      ...                          sr=sr, x_axis='time', y_axis='cqt_note')
      >>> plt.colorbar(format='%+2.0f dB')
      >>> plt.title('Constant-Q power spectrum')
      >>> plt.tight_layout()
  
  
      Limit the frequency range
  
      >>> C = librosa.cqt(y, sr=sr, fmin=librosa.note_to_hz('C2'),
      ...                 n_bins=60)
      >>> C
      array([[  8.827e-04,   9.293e-04, ...,   3.133e-07,   2.942e-07],
             [  1.076e-03,   1.068e-03, ...,   1.153e-06,   1.148e-06],
             ...,
             [  1.042e-07,   4.087e-07, ...,   1.612e-07,   1.928e-07],
             [  2.363e-07,   5.329e-07, ...,   1.294e-07,   1.611e-07]])
  
  
      Using a higher frequency resolution
  
      >>> C = librosa.cqt(y, sr=sr, fmin=librosa.note_to_hz('C2'),
      ...                 n_bins=60 * 2, bins_per_octave=12 * 2)
      >>> C
      array([[  1.536e-05,   5.848e-05, ...,   3.241e-07,   2.453e-07],
             [  1.856e-03,   1.854e-03, ...,   2.397e-08,   3.549e-08],
             ...,
             [  2.034e-07,   4.245e-07, ...,   6.213e-08,   1.463e-07],
             [  4.896e-08,   5.407e-07, ...,   9.176e-08,   1.051e-07]])
      '''
  
      # 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)
  
      len_orig = len(y)
  
      if fmin is None:
          # C1 by default
          fmin = note_to_hz('C1')
  
      if tuning is None:
          tuning = estimate_tuning(y=y, sr=sr)
  
      # First thing, get the freqs of the top octave
      freqs = cqt_frequencies(n_bins, fmin,
                              bins_per_octave=bins_per_octave)[-bins_per_octave:]
  
      fmin_t = np.min(freqs)
      fmax_t = np.max(freqs)
  
      # Determine required resampling quality
      Q = float(filter_scale) / (2.0**(1. / bins_per_octave) - 1)
      filter_cutoff = fmax_t * (1 + 0.5 * filters.window_bandwidth(window) / Q)
      nyquist = sr / 2.0
      if filter_cutoff < audio.BW_FASTEST * nyquist:
          res_type = 'kaiser_fast'
      else:
          res_type = 'kaiser_best'
  
      y, sr, hop_length = __early_downsample(y, sr, hop_length,
                                             res_type,
                                             n_octaves,
                                             nyquist, filter_cutoff, scale)
  
      cqt_resp = []
  
      if res_type != 'kaiser_fast':
  
          # Do the top octave before resampling to allow for fast resampling
          fft_basis, n_fft, _ = __cqt_filter_fft(sr, fmin_t,
                                                 n_filters,
                                                 bins_per_octave,
                                                 tuning,
                                                 filter_scale,
                                                 norm,
                                                 sparsity,
                                                 window=window)
  
          # Compute the CQT filter response and append it to the stack
          cqt_resp.append(__cqt_response(y, n_fft, hop_length, fft_basis))
  
          fmin_t /= 2
          fmax_t /= 2
          n_octaves -= 1
  
          filter_cutoff = fmax_t * (1 + 0.5 * filters.window_bandwidth(window) / Q)
  
          res_type = 'kaiser_fast'
  
      # Make sure our hop is long enough to support the bottom octave
      num_twos = __num_two_factors(hop_length)
      if num_twos < n_octaves - 1:
          raise ParameterError('hop_length must be a positive integer '
                               'multiple of 2^{0:d} for {1:d}-octave CQT'
                               .format(n_octaves - 1, n_octaves))
  
      # Now do the recursive bit
      fft_basis, n_fft, _ = __cqt_filter_fft(sr, fmin_t,
                                             n_filters,
                                             bins_per_octave,
                                             tuning,
                                             filter_scale,
                                             norm,
                                             sparsity,
                                             window=window)
  
      my_y, my_sr, my_hop = y, sr, hop_length
  
      # Iterate down the octaves
      for i in range(n_octaves):
  
          # Resample (except first time)
          if i > 0:
              if len(my_y) < 2:
                  raise ParameterError('Input signal length={} is too short for '
                                       '{:d}-octave CQT'.format(len_orig,
                                                                n_octaves))
  
              # The additional scaling of sqrt(2) here is to implicitly rescale
              # the filters
              my_y = np.sqrt(2) * audio.resample(my_y, my_sr, my_sr/2.0,
                                                 res_type=res_type,
                                                 scale=True)
              my_sr /= 2.0
              my_hop //= 2
  
          # Compute the cqt filter response and append to the stack
          cqt_resp.append(__cqt_response(my_y, n_fft, my_hop, fft_basis))
  
      C = __trim_stack(cqt_resp, n_bins)
  
      if scale:
          lengths = filters.constant_q_lengths(sr, fmin,
                                               n_bins=n_bins,
                                               bins_per_octave=bins_per_octave,
                                               tuning=tuning,
                                               window=window,
                                               filter_scale=filter_scale)
          C /= np.sqrt(lengths[:, np.newaxis])
  
      return C
  
  
  @cache(level=20)
  def hybrid_cqt(y, sr=22050, hop_length=512, fmin=None, n_bins=84,
                 bins_per_octave=12, tuning=0.0, filter_scale=1,
                 norm=1, sparsity=0.01, window='hann', scale=True):
      '''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
  
      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 in `[-0.5, 0.5)`
          Tuning offset in fractions of a bin (cents).
  
          If `None`, tuning will be automatically estimated from the signal.
  
      filter_scale : float > 0
          Filter filter_scale factor. Larger values use longer windows.
  
      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.
  
  
      Returns
      -------
      CQT : np.ndarray [shape=(n_bins, t), dtype=np.float]
          Constant-Q energy for each frequency at each time.
  
      Raises
      ------
      ParameterError
          If `hop_length` is not an integer multiple of
          `2**(n_bins / bins_per_octave)`
  
          Or if `y` is too short to support the frequency range of the CQT.
  
      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)
  
      # Get all CQT frequencies
      freqs = cqt_frequencies(n_bins, fmin,
                              bins_per_octave=bins_per_octave,
                              tuning=tuning)
  
      # Compute the length of each constant-Q basis function
      lengths = filters.constant_q_lengths(sr, fmin,
                                           n_bins=n_bins,
                                           bins_per_octave=bins_per_octave,
                                           tuning=tuning,
                                           filter_scale=filter_scale,
                                           window=window)
  
      # 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,
                                     hop_length=hop_length,
                                     fmin=fmin_pseudo,
                                     n_bins=n_bins_pseudo,
                                     bins_per_octave=bins_per_octave,
                                     tuning=tuning,
                                     filter_scale=filter_scale,
                                     norm=norm,
                                     sparsity=sparsity,
                                     window=window,
                                     scale=scale))
  
      if n_bins_full > 0:
          cqt_resp.append(np.abs(cqt(y, sr,
                                     hop_length=hop_length,
                                     fmin=fmin,
                                     n_bins=n_bins_full,
                                     bins_per_octave=bins_per_octave,
                                     tuning=tuning,
                                     filter_scale=filter_scale,
                                     norm=norm,
                                     sparsity=sparsity,
                                     window=window,
                                     scale=scale)))
  
      return __trim_stack(cqt_resp, n_bins)
  
  
  @cache(level=20)
  def pseudo_cqt(y, sr=22050, hop_length=512, fmin=None, n_bins=84,
                 bins_per_octave=12, tuning=0.0, filter_scale=1,
                 norm=1, sparsity=0.01, window='hann', scale=True):
      '''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
  
      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 in `[-0.5, 0.5)`
          Tuning offset in fractions of a bin (cents).
  
          If `None`, tuning will be automatically estimated from the signal.
  
      filter_scale : float > 0
          Filter filter_scale factor. Larger values use longer windows.
  
      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.
  
  
      Returns
      -------
      CQT : np.ndarray [shape=(n_bins, t), dtype=np.float]
          Pseudo Constant-Q energy for each frequency at each time.
  
      Raises
      ------
      ParameterError
          If `hop_length` is not an integer multiple of
          `2**(n_bins / bins_per_octave)`
  
          Or if `y` is too short to support the frequency range of the 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)
  
      fft_basis, n_fft, _ = __cqt_filter_fft(sr, fmin, n_bins,
                                             bins_per_octave,
                                             tuning, filter_scale,
                                             norm, sparsity,
                                             hop_length=hop_length,
                                             window=window)
  
      fft_basis = np.abs(fft_basis)
  
      # Compute the magnitude STFT with Hann window
      D = np.abs(stft(y, n_fft=n_fft, hop_length=hop_length))
  
      # Project onto the pseudo-cqt basis
      C = fft_basis.dot(D)
  
      if scale:
          C /= np.sqrt(n_fft)
      else:
          lengths = filters.constant_q_lengths(sr, fmin,
                                               n_bins=n_bins,
                                               bins_per_octave=bins_per_octave,
                                               tuning=tuning,
                                               window=window,
                                               filter_scale=filter_scale)
  
          C *= np.sqrt(lengths[:, np.newaxis] / n_fft)
  
      return C
  
  
  @cache(level=10)
  def __cqt_filter_fft(sr, fmin, n_bins, bins_per_octave, tuning,
                       filter_scale, norm, sparsity, hop_length=None,
                       window='hann'):
      '''Generate the frequency domain constant-Q filter basis.'''
  
      basis, lengths = filters.constant_q(sr,
                                          fmin=fmin,
                                          n_bins=n_bins,
                                          bins_per_octave=bins_per_octave,
                                          tuning=tuning,
                                          filter_scale=filter_scale,
                                          norm=norm,
                                          pad_fft=True,
                                          window=window)
  
      # 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_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)
  
      return fft_basis, n_fft, lengths
  
  
  def __trim_stack(cqt_resp, n_bins):
      '''Helper function to trim and stack a collection of CQT responses'''
  
      # cleanup any framing errors at the boundaries
      max_col = min([x.shape[1] for x in cqt_resp])
  
      cqt_resp = np.vstack([x[:, :max_col] for x in cqt_resp][::-1])
  
      # Finally, clip out any bottom frequencies that we don't really want
      # Transpose magic here to ensure column-contiguity
      return np.ascontiguousarray(cqt_resp[-n_bins:].T).T
  
  
  def __cqt_response(y, n_fft, hop_length, fft_basis):
      '''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=np.ones)
  
      # And filter response energy
      return fft_basis.dot(D)
  
  
  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(audio.BW_FASTEST * 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 and res_type == 'kaiser_fast':
          downsample_factor = 2**(downsample_count)
  
          hop_length //= downsample_factor
  
          if len(y) < downsample_factor:
              raise ParameterError('Input signal length={:d} is too short for '
                                   '{:d}-octave CQT'.format(len(y), n_octaves))
  
          new_sr = sr / float(downsample_factor)
          y = audio.resample(y, sr, new_sr,
                             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
  
  
  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
  
apis: numpy.vstack numpy.sum numpy.log2 scipy.fftpack.fft numpy.abs numpy.max numpy.min numpy.sqrt numpy.ascontiguousarray