短时傅里叶变换(Short Time Fourier Transform)原理及 Python 实现

原理

　　短时傅里叶变换(Short Time Fourier Transform, STFT) 是一个用于语音信号处理的通用工具.它定义了一个非常有用的时间和频率分布类, 其指定了任意信号随时间和频率变化的复数幅度. 实际上,计算短时傅里叶变换的过程是把一个较长的时间信号分成相同长度的更短的段, 在每个更短的段上计算傅里叶变换, 即傅里叶频谱.

短时傅里叶变换通常的数学定义如下:

其中,

DTFT (Decrete Time Fourier Transform) 为离散时间傅里叶变换.  其数学公式, 如下所示:

　　其中,  x(n) 为在采样数 n 处的信号幅度. ω^~ 的定义如下:

　　实现时, 短时傅里叶变换被计算为一系列加窗数据帧的快速傅里叶变换 (Fast Fourier Transform, FFT)，其中窗口随时间 “滑动” (slide) 或“跳跃” (hop) 。

 

Python 实现

　　在程序中, frame_size 为将信号分为较短的帧的大小, 在语音处理中, 通常帧大小在 20ms 到 40ms 之间. 这里设置为 25ms, 即 frame_size = 0.025;

　　frame_stride 为相邻帧的滑动尺寸或跳跃尺寸, 通常帧的滑动尺寸在 10ms 到 20ms 之间, 这里设置为 10ms, 即 frame_stride = 0.01. 此时, 相邻帧的交叠大小为 15ms;

　　窗函数采用汉明窗函数 (Hamming Function) ;

　　在每一帧, 进行 512 点快速傅里叶变换, 即 NFFT = 512. 具体程序如下: 

# -*- coding: utf8 -*-
import numpy as np

def calc_stft(signal, sample_rate=16000, frame_size=0.025, frame_stride=0.01, winfunc=np.hamming, NFFT=512):

    # Calculate the number of frames from the signal
    frame_length = frame_size * sample_rate
    frame_step = frame_stride * sample_rate
    signal_length = len(signal)
    frame_length = int(round(frame_length))
    frame_step = int(round(frame_step))
    num_frames = 1 + int(np.ceil(float(np.abs(signal_length - frame_length)) / frame_step))
    # zero padding
    pad_signal_length = num_frames * frame_step + frame_length
    z = np.zeros((pad_signal_length - signal_length))
    # Pad signal to make sure that all frames have equal number of samples 
    # without truncating any samples from the original signal
    pad_signal = np.append(signal, z)

    # Slice the signal into frames from indices
    indices = np.tile(np.arange(0, frame_length), (num_frames, 1)) + 
            np.tile(np.arange(0, num_frames * frame_step, frame_step), (frame_length, 1)).T
    frames = pad_signal[indices.astype(np.int32, copy=False)]
    # Get windowed frames
    frames *= winfunc(frame_length)
    # Compute the one-dimensional n-point discrete Fourier Transform(DFT) of
    # a real-valued array by means of an efficient algorithm called Fast Fourier Transform (FFT)
    mag_frames = np.absolute(np.fft.rfft(frames, NFFT))
    # Compute power spectrum
    pow_frames = (1.0 / NFFT) * ((mag_frames) ** 2)

    return pow_frames

if __name__ == '__main__':
    import scipy.io.wavfile
    import matplotlib.pyplot as plt

    # Read wav file
    # "OSR_us_000_0010_8k.wav" is downloaded from http://www.voiptroubleshooter.com/open_speech/american.html
    sample_rate, signal = scipy.io.wavfile.read("OSR_us_000_0010_8k.wav")
    # Get speech data in the first 2 seconds
    signal = signal[0:int(2. * sample_rate)]

    # Calculate the short time fourier transform
    pow_spec = calc_stft(signal, sample_rate)

    plt.imshow(pow_spec)
    plt.tight_layout()
    plt.show()

参考资料

1. DISCRETE TIME FOURIER TRANSFORM (DTFT). https://www.dsprelated.com/freebooks/mdft/Discrete_Time_Fourier_Transform.html

2. THE SHORT-TIME FOURIER TRANSFORM. https://www.dsprelated.com/freebooks/sasp/Short_Time_Fourier_Transform.html

3. Short-time Fourier transform. https://en.wikipedia.org/wiki/Short-time_Fourier_transform

4. Speech Processing for Machine Learning: Filter banks, Mel-Frequency Cepstral Coefficients (MFCCs) and What's In-Between. https://haythamfayek.com/2016/04/21/speech-processing-for-machine-learning.html