在语音分析，合成，转换中，第一步往往是提取语音特征参数。
利用机器学习方法进行上述语音任务，常用到梅尔频谱。
本文介绍从音频文件提取梅尔频谱，和从梅尔频谱变成音频波形。

从音频波形提取Mel频谱：

1. 对音频信号预加重、分帧和加窗
2. 对每帧信号进行短时傅立叶变换STFT，得到短时幅度谱
3. 短时幅度谱通过Mel滤波器组得到Mel频谱

从Mel频谱重建音频波形

1. Mel频谱转换成幅度谱
2. griffin_lim声码器算法重建波形
3. 去加重

声码器有很多种，比如world,straight等，但是griffin_lim是特殊的，它不需要相位信息就可以重频谱重建波形，实际上它根据帧之间的关系估计相位信息。和成的音频质量也较高，代码也比较简单。

音频波形 到 mel-spectrogram

sr = 24000 # Sample rate.
n_fft = 2048 # fft points (samples)
frame_shift = 0.0125 # seconds
frame_length = 0.05 # seconds
hop_length = int(sr*frame_shift) # samples.
win_length = int(sr*frame_length) # samples.
n_mels = 512 # Number of Mel banks to generate
power = 1.2 # Exponent for amplifying the predicted magnitude
n_iter = 100 # Number of inversion iterations
preemphasis = .97 # or None
max_db = 100
ref_db = 20
top_db = 15

def get_spectrograms(fpath):
    '''Returns normalized log(melspectrogram) and log(magnitude) from `sound_file`.
    Args:
      sound_file: A string. The full path of a sound file.

    Returns:
      mel: A 2d array of shape (T, n_mels) <- Transposed
      mag: A 2d array of shape (T, 1+n_fft/2) <- Transposed
 '''
    # Loading sound file
    y, sr = librosa.load(fpath, sr=sr)

    # Trimming
    y, _ = librosa.effects.trim(y, top_db=top_db)

    # Preemphasis
    y = np.append(y[0], y[1:] - preemphasis * y[:-1])
    
    # stft
    linear = librosa.stft(y=y,
                          n_fft=n_fft,
                          hop_length=hop_length,
                          win_length=win_length)

    # magnitude spectrogram
    mag = np.abs(linear)  # (1+n_fft//2, T)

    # mel spectrogram
    mel_basis = librosa.filters.mel(sr, n_fft, n_mels)  # (n_mels, 1+n_fft//2)
    mel = np.dot(mel_basis, mag)  # (n_mels, t)

    # to decibel
    mel = 20 * np.log10(np.maximum(1e-5, mel))
    mag = 20 * np.log10(np.maximum(1e-5, mag))

    # normalize
    mel = np.clip((mel - ref_db + max_db) / max_db, 1e-8, 1)
    mag = np.clip((mag - ref_db + max_db) / max_db, 1e-8, 1)

    # Transpose
    mel = mel.T.astype(np.float32)  # (T, n_mels)
    mag = mag.T.astype(np.float32)  # (T, 1+n_fft//2)

    return mel, mag

mel-spectrogram 到 音频波形

def melspectrogram2wav(mel):
    '''# Generate wave file from spectrogram'''
    # transpose
    mel = mel.T

    # de-noramlize
    mel = (np.clip(mel, 0, 1) * max_db) - max_db + ref_db

    # to amplitude
    mel = np.power(10.0, mel * 0.05)
    m = _mel_to_linear_matrix(sr, n_fft, n_mels)
    mag = np.dot(m, mel)

    # wav reconstruction
    wav = griffin_lim(mag)

    # de-preemphasis
    wav = signal.lfilter([1], [1, -preemphasis], wav)

    # trim
    wav, _ = librosa.effects.trim(wav)

    return wav.astype(np.float32)

def spectrogram2wav(mag):
    '''# Generate wave file from spectrogram'''
    # transpose
    mag = mag.T

    # de-noramlize
    mag = (np.clip(mag, 0, 1) * max_db) - max_db + ref_db

    # to amplitude
    mag = np.power(10.0, mag * 0.05)

    # wav reconstruction
    wav = griffin_lim(mag)

    # de-preemphasis
    wav = signal.lfilter([1], [1, -preemphasis], wav)

    # trim
    wav, _ = librosa.effects.trim(wav)

    return wav.astype(np.float32)

几个辅助函数：

def _mel_to_linear_matrix(sr, n_fft, n_mels):
    m = librosa.filters.mel(sr, n_fft, n_mels)
    m_t = np.transpose(m)
    p = np.matmul(m, m_t)
    d = [1.0 / x if np.abs(x) > 1.0e-8 else x for x in np.sum(p, axis=0)]
    return np.matmul(m_t, np.diag(d))

def griffin_lim(spectrogram):
    '''Applies Griffin-Lim's raw.
    '''
    X_best = copy.deepcopy(spectrogram)
    for i in range(n_iter):
        X_t = invert_spectrogram(X_best)
        est = librosa.stft(X_t, n_fft, hop_length, win_length=win_length)
        phase = est / np.maximum(1e-8, np.abs(est))
        X_best = spectrogram * phase
    X_t = invert_spectrogram(X_best)
    y = np.real(X_t)

    return y


def invert_spectrogram(spectrogram):
    '''
    spectrogram: [f, t]
    '''
    return librosa.istft(spectrogram, hop_length, win_length=win_length, window="hann")

预加重：
语音信号的平均功率谱受声门激励和口鼻辐射影响，高频端约在800HZ以上按6dB/倍频程衰落，预加重的目的是提升高频成分，使信号频谱平坦化，以便于频谱分析或声道参数分析．