def stft(y, n_fft=2048, hop_length=None, win_length=None, window='hann',
center=True, dtype=np.complex64, pad_mode='reflect'):
"""Short-time Fourier transform (STFT)
Returns a complex-valued matrix D such that
`np.abs(D[f, t])` is the magnitude of frequency bin `f`
at frame `t`
`np.angle(D[f, t])` is the phase of frequency bin `f`
at frame `t`
Parameters
----------
y : np.ndarray [shape=(n,)], real-valued
the input signal (audio time series)
n_fft : int > 0 [scalar]
FFT window size，做N点FFT
hop_length : int > 0 [scalar]
number audio of frames between STFT columns.
If unspecified, defaults `win_length / 4`.
也就是帧之间的overlap，默认为窗口长度的1/4
win_length : int <= n_fft [scalar]
Each frame of audio is windowed by `window()`.
The window will be of length `win_length` and then padded
with zeros to match `n_fft`.
窗口长度必须小于等于N_FFT的长度，如果未指定，则默认窗口长度为FFT点数
If unspecified, defaults to ``win_length = n_fft``.
window : string, tuple, number, function, or np.ndarray [shape=(n_fft,)]
- a window specification (string, tuple, or number);
see `scipy.signal.get_window`
- a window function, such as `scipy.signal.hanning`
- a vector or array of length `n_fft`
.. see also:: `filters.get_window`
center : boolean
- If `True`, the signal `y` is padded so that frame
`D[:, t]` is centered at `y[t * hop_length]`.
- If `False`, then `D[:, t]` begins at `y[t * hop_length]`
对于此处的描述需要仔细理解：
反过来理解也就是说，是以窗的中心时刻作为时间标示，还是以窗的开始位置作为时间标示
dtype : numeric type
Complex numeric type for `D`. Default is 64-bit complex.
此处对信号做FFT，得到的数值为复数，默认为64位复数
复数具有幅值和相位属性，但是在一般的应用当中，较少考虑相位因素
pad_mode : string
If `center=True`, the padding mode to use at the edges of the signal.
By default, STFT uses reflection padding.（镜面反射padding）
Returns
-------
D : np.ndarray [shape=(1 + n_fft/2, t), dtype=dtype]
需要进一步对FFT进行理解，只取信号FFT的一半进行分析，是因为实数信号的FFT具有对称性，
完整的FFT结果具有信息冗余。
STFT matrix
See Also
--------
istft : Inverse STFT
ifgram : Instantaneous（即时的，同时发生的） frequency spectrogram
np.pad : array padding
Notes
-----
This function caches(cache memory) at level 20.
Examples
--------
>>> y, sr = librosa.load(librosa.util.example_audio_file())
>>> D = np.abs(librosa.stft(y))
>>> D
array([[2.58028018e-03, 4.32422794e-02, 6.61255598e-01, ...,
6.82710262e-04, 2.51654536e-04, 7.23036574e-05],
[2.49403086e-03, 5.15930466e-02, 6.00107312e-01, ...,
3.48026224e-04, 2.35853557e-04, 7.54836728e-05],
[7.82410789e-04, 1.05394892e-01, 4.37517226e-01, ...,
6.29352580e-04, 3.38571583e-04, 8.38094638e-05],
...,
[9.48568513e-08, 4.74725084e-07, 1.50052492e-05, ...,
1.85637656e-08, 2.89708542e-08, 5.74304337e-09],
[1.25165826e-07, 8.58259284e-07, 1.11157215e-05, ...,
3.49099771e-08, 3.11740926e-08, 5.29926236e-09],
[1.70630571e-07, 8.92518756e-07, 1.23656537e-05, ...,
5.33256745e-08, 3.33264900e-08, 5.13272980e-09]], dtype=float32)
Use left-aligned frames, instead of centered frames
>>> D_left = np.abs(librosa.stft(y, center=False))
Use a shorter hop length
>>> D_short = np.abs(librosa.stft(y, hop_length=64))
Display a spectrogram
>>> import matplotlib.pyplot as plt
>>> librosa.display.specshow(librosa.amplitude_to_db(D,
... ref=np.max),
... y_axis='log', x_axis='time')
>>> plt.title('Power spectrogram')
>>> plt.colorbar(format='%+2.0f dB')
>>> plt.tight_layout()
"""
# By default, use the entire frame
if win_length is None:
win_length = n_fft
# Set the default hop, if it's not already specified
if hop_length is None:
hop_length = int(win_length // 4)
fft_window = get_window(window, win_length, fftbins=True)
# Pad the window out to n_fft size 将窗口大小扩展到与N_FFT大小相同
fft_window = util.pad_center(fft_window, n_fft)
# Reshape so that the window can be broadcast
fft_window = fft_window.reshape((-1, 1))
# Check audio is valid
util.valid_audio(y)
# Pad the time series so that frames are centered
if center:
y = np.pad(y, int(n_fft // 2), mode=pad_mode)
# 在y的两侧，分别镜像填充n_fft//2个数据
# eg:[3, 2, 1, 2, 3, 4, 5, 4, 3]（填充两个数据）
# Window the time series.将信号进行分帧
y_frames = util.frame(y, frame_length=n_fft, hop_length=hop_length)
# Pre-allocate the STFT matrix 对输出矩阵进行内存分配，应该有助于计算速度的提升
stft_matrix = np.empty((int(1 + n_fft // 2), y_frames.shape[1]),
dtype=dtype,
order='F')
# how many columns can we fit within MAX_MEM_BLOCK? 在librosa当中，设定为256KB
# 计算在最大内存的限制下，最多能够存储多少帧（列）数据的FFT变换
n_columns = int(util.MAX_MEM_BLOCK / (stft_matrix.shape[0] *
stft_matrix.itemsize))
for bl_s in range(0, stft_matrix.shape[1], n_columns):
bl_t = min(bl_s + n_columns, stft_matrix.shape[1])
# 当n_columns大于分帧数的时候，for循环当中就只有一次bl_s=0
# 反之，每次进行n_columns帧数据的运算，然后进行循环拼接
stft_matrix[:, bl_s:bl_t] = fft.fft(fft_window *
y_frames[:, bl_s:bl_t],
axis=0)[:stft_matrix.shape[0]]
return stft_matrix
