Return a matrix that can post-multiply spectrogram rows to make mel.

  Returns a np.array matrix A that can be used to post-multiply a matrix S of
  spectrogram values (STFT magnitudes) arranged as frames x bins to generate a
  "mel spectrogram" M of frames x num_mel_bins.  M = S A.

  The classic HTK algorithm exploits the complementarity of adjacent mel bands
  to multiply each FFT bin by only one mel weight, then add it, with positive
  and negative signs, to the two adjacent mel bands to which that bin
  contributes.  Here, by expressing this operation as a matrix multiply, we go
  from num_fft multiplies per frame (plus around 2*num_fft adds) to around
  num_fft^2 multiplies and adds.  However, because these are all presumably
  accomplished in a single call to np.dot(), it's not clear which approach is
  faster in Python.  The matrix multiplication has the attraction of being more
  general and flexible, and much easier to read.

  Args:
    num_mel_bins: How many bands in the resulting mel spectrum.  This is
      the number of columns in the output matrix.
    num_spectrogram_bins: How many bins there are in the source spectrogram
      data, which is understood to be fft_size/2 + 1, i.e. the spectrogram
      only contains the nonredundant FFT bins.
    audio_sample_rate: Samples per second of the audio at the input to the
      spectrogram. We need this to figure out the actual frequencies for
      each spectrogram bin, which dictates how they are mapped into mel.
    lower_edge_hertz: Lower bound on the frequencies to be included in the mel
      spectrum.  This corresponds to the lower edge of the lowest triangular
      band.
    upper_edge_hertz: The desired top edge of the highest frequency band.

  Returns:
    An np.array with shape (num_spectrogram_bins, num_mel_bins).

  Raises:
    ValueError: if frequency edges are incorrectly ordered or out of range.

def spectrogram_to_mel_matrix(num_mel_bins=20,
                              num_spectrogram_bins=129,
                              audio_sample_rate=8000,
                              lower_edge_hertz=125.0,
                              upper_edge_hertz=3800.0):
  """Return a matrix that can post-multiply spectrogram rows to make mel.

  Returns a np.array matrix A that can be used to post-multiply a matrix S of
  spectrogram values (STFT magnitudes) arranged as frames x bins to generate a
  "mel spectrogram" M of frames x num_mel_bins.  M = S A.

  The classic HTK algorithm exploits the complementarity of adjacent mel bands
  to multiply each FFT bin by only one mel weight, then add it, with positive
  and negative signs, to the two adjacent mel bands to which that bin
  contributes.  Here, by expressing this operation as a matrix multiply, we go
  from num_fft multiplies per frame (plus around 2*num_fft adds) to around
  num_fft^2 multiplies and adds.  However, because these are all presumably
  accomplished in a single call to np.dot(), it's not clear which approach is
  faster in Python.  The matrix multiplication has the attraction of being more
  general and flexible, and much easier to read.

  Args:
    num_mel_bins: How many bands in the resulting mel spectrum.  This is
      the number of columns in the output matrix.
    num_spectrogram_bins: How many bins there are in the source spectrogram
      data, which is understood to be fft_size/2 + 1, i.e. the spectrogram
      only contains the nonredundant FFT bins.
    audio_sample_rate: Samples per second of the audio at the input to the
      spectrogram. We need this to figure out the actual frequencies for
      each spectrogram bin, which dictates how they are mapped into mel.
    lower_edge_hertz: Lower bound on the frequencies to be included in the mel
      spectrum.  This corresponds to the lower edge of the lowest triangular
      band.
    upper_edge_hertz: The desired top edge of the highest frequency band.

  Returns:
    An np.array with shape (num_spectrogram_bins, num_mel_bins).

  Raises:
    ValueError: if frequency edges are incorrectly ordered or out of range.
  """
  nyquist_hertz = audio_sample_rate / 2.
  if lower_edge_hertz < 0.0:
    raise ValueError("lower_edge_hertz %.1f must be >= 0" % lower_edge_hertz)
  if lower_edge_hertz >= upper_edge_hertz:
    raise ValueError("lower_edge_hertz %.1f >= upper_edge_hertz %.1f" %
                     (lower_edge_hertz, upper_edge_hertz))
  if upper_edge_hertz > nyquist_hertz:
    raise ValueError("upper_edge_hertz %.1f is greater than Nyquist %.1f" %
                     (upper_edge_hertz, nyquist_hertz))
  spectrogram_bins_hertz = np.linspace(0.0, nyquist_hertz, num_spectrogram_bins)
  spectrogram_bins_mel = hertz_to_mel(spectrogram_bins_hertz)
  # The i'th mel band (starting from i=1) has center frequency
  # band_edges_mel[i], lower edge band_edges_mel[i-1], and higher edge
  # band_edges_mel[i+1].  Thus, we need num_mel_bins + 2 values in
  # the band_edges_mel arrays.
  band_edges_mel = np.linspace(hertz_to_mel(lower_edge_hertz),
                               hertz_to_mel(upper_edge_hertz), num_mel_bins + 2)
  # Matrix to post-multiply feature arrays whose rows are num_spectrogram_bins
  # of spectrogram values.
  mel_weights_matrix = np.empty((num_spectrogram_bins, num_mel_bins))
  for i in range(num_mel_bins):
    lower_edge_mel, center_mel, upper_edge_mel = band_edges_mel[i:i + 3]
    # Calculate lower and upper slopes for every spectrogram bin.
    # Line segments are linear in the *mel* domain, not hertz.
    lower_slope = ((spectrogram_bins_mel - lower_edge_mel) /
                   (center_mel - lower_edge_mel))
    upper_slope = ((upper_edge_mel - spectrogram_bins_mel) /
                   (upper_edge_mel - center_mel))
    # .. then intersect them with each other and zero.
    mel_weights_matrix[:, i] = np.maximum(0.0, np.minimum(lower_slope,
                                                          upper_slope))
  # HTK excludes the spectrogram DC bin; make sure it always gets a zero
  # coefficient.
  mel_weights_matrix[0, :] = 0.0
  return mel_weights_matrix