librosa的短时傅里叶实现librosa.stft()
目录
1. API参数解析
2. 分帧机制
2.1 C语言实现
2.1.1 测试代码
2.2 librosa的短时傅里叶变正换实现
2.2.1 libsora.stft()的分帧机制
2.2.2 测试代码
2.2.3 运行结果分析
2.3 librosa的短时傅里叶逆变换实现
3. 附录：librosa官网
1. API参数解析
|
函数原型 |
librosa.stft((y, n_fft=2048, hop_length=None, win_length=None, \ window='hann', center=True, pad_mode='reflect')) |
|
函数功能 |
对语音进行分帧，加窗，及短时傅里叶变换的计算(FFT) |
|
参数 |
(1) y: 输入的音频时间序列(一般是通过librosa.load()得到) (2) n_fft : FFT的点数，fft的点数跟窗长可以是不同的，但是一般设置成相同 (3) hop_length : 帧移 (4) win_length : 指的是滑动窗[截断窗]的窗长，一般来说，让这两个相等win_length=n_fft (5) window : 指定窗函数，字符串的形式，如果不指定，则默认是汉宁窗 (6) center : bool类型，默认值是True，一般使用默认的值即可。指”中心对齐”。这和分帧的机制相关，是我们重点需要研究的参数。 (7) pad_mode : 填充模式设置，使用默认的参数即可，表示使用数据0填充帧数据。 |
|
返回值 |
返回复数矩阵（实数序列经过分帧、加窗以及傅里叶变换后得到的是复数序列，a+bi的形式）。 |
|
定义处（源文件） |
|
|
声明处（头文件） |
2. 分帧机制
Libsora的短时傅里叶变换，分帧有点“不常规”。以256个采样点的傅里叶变换为例。
实验数据：为了更直观、简洁、方便地进行试验和理解API的参数以及实现机制，预先从音频文件中读出256个采样点，用数组或者列表进行存储。同样，汉宁窗也可以实现计算出来，并存储，使用时直接查表，可避免运算量，这是算法处理中常见的“以牺牲空间换取时间效率”的做法。
2.1 C语言实现
直接对这256个点进行傅里叶变换，傅里叶变换的点数就是256，阶数就是8。采用复数的FFT计算算法，虚部全为0。
2.1.1 测试代码
#define FRAME_LEN 256
#define FFT_ORDER ((int)log2(FRAME_LEN))
float dat_i[FRAME_LEN], dat_a[FRAME_LEN];
float dat_r[FRAME_LEN]={0.000214,0.000122,-0.000061,0.000275,0.000092,0.000336,0.000153,-0.000092,0.000031,0.000305,0.000641,0.000458,0.000305,0.000153,0.000397,0.000732,0.000885,0.000580,0.000427,0.000397,0.000641,0.001190,0.001282,0.000977,0.000336,0.000458,0.001007,0.001617,0.001465,0.000824,0.000275,0.000366,0.001007,0.001282,0.000885,0.000122,-0.000336,-0.000122,0.000336,0.000275,-0.000153,-0.000824,-0.000916,-0.000580,-0.000519,-0.000763,-0.001099,-0.001221,-0.001007,-0.000580,-0.000793,-0.000885,-0.001038,-0.000580,-0.000275,-0.000214,-0.000549,-0.000610,-0.000061,0.000702,0.001190,0.000763,0.000305,0.000275,0.001068,0.001923,0.001953,0.001190,0.000488,0.000519,0.001251,0.001740,0.001282,0.000397,-0.000183,0.000000,0.000214,0.000305,-0.000092,-0.000458,-0.000732,-0.000610,-0.000671,-0.000916,-0.000946,-0.000824,-0.000702,-0.001068,-0.000977,-0.001190,-0.000671,-0.000244,-0.000122,-0.000580,-0.000854,-0.000519,0.000153,0.000641,0.000671,0.000214,-0.000122,0.000336,0.000732,0.000702,0.000397,0.000122,0.000122,0.000275,0.000000,-0.000458,-0.000580,-0.000275,0.000092,-0.000244,-0.001373,-0.001892,-0.001465,-0.000153,0.000214,-0.000641,-0.002533,-0.002808,-0.001526,0.000549,0.000946,-0.000824,-0.002594,-0.002625,-0.000427,0.001678,0.001587,-0.000397,-0.002075,-0.001526,0.000549,0.002014,0.001740,-0.000122,-0.001678,-0.000916,0.000610,0.001556,0.000977,-0.000580,-0.001251,-0.000854,-0.000183,0.000214,-0.000031,-0.000671,-0.001007,-0.001221,-0.000916,-0.000824,-0.000427,-0.000549,-0.000732,-0.000610,-0.000854,-0.000671,-0.000458,-0.000183,0.000031,0.000275,0.000427,-0.000092,-0.000397,0.000000,0.000793,0.001526,0.001465,0.000122,-0.001068,-0.000854,0.000946,0.002472,0.002014,-0.000488,-0.002350,-0.002075,0.000702,0.002686,0.001801,-0.001526,-0.003662,-0.002472,0.000793,0.002747,0.001740,-0.001251,-0.002991,-0.001648,0.001160,0.003143,0.002289,0.000244,-0.000916,-0.000305,0.001526,0.002960,0.002808,0.002289,0.001221,0.000092,0.000427,0.001404,0.003021,0.003479,0.001709,-0.001068,-0.002502,-0.001007,0.001831,0.002960,0.000427,-0.003510,-0.005157,-0.002930,0.000580,0.001648,-0.001038,-0.004486,-0.005463,-0.002991,-0.000122,0.000885,-0.000488,-0.002045,-0.002289,-0.001129,0.000214,0.001343,0.002441,0.003448,0.002991,0.001282,0.000153,0.002014,0.006134,0.008820,0.006439,0.000793,-0.002625,0.000488,0.007111,0.010132,0.004669,-0.004791,-0.009033,-0.004028,0.004089,0.006104,-0.002228,-0.013214,-0.016327,-0.009583,
};int main(void)
{for (int i=0; i<FRAME_LEN; i++) {dat_i[i]=0.0;dat_r[i]=dat_r[i]*hanning[i];}FFT(dat_r, dat_i, dat_a, FRAME_LEN, FFT_ORDER);
// for (int i=0; i<FRAME_LEN; i++) {
// printf("[%d] (%f+%fj)\n", i, dat_r[i], dat_i[i]);
// printf("%d %f\n", i, dat_r[i]);
// }
#if 1FILE* fp_dbg=NULL;fp_dbg = fopen("res_fft.txt", "wb");for (int i=0; i<FRAME_LEN; i++) {fprintf(fp_dbg, "[%d]:%f\n", i, dat_r[i]);}fclose(fp_dbg);printf("data output done\n");
#endifreturn 0;
}2.1.2 运行结果分析
|
[0] (-0.016482+0.000000j) //直流分量 [1] (0.023277+0.005782j) [2] (-0.022870+-0.002839j) [3] (0.008954+-0.001403j) [4] (0.009356+-0.005058j) [5] (0.003737+0.002048j) [6] (-0.016804+0.000179j) [7] (-0.005669+0.038893j) [8] (0.003740+-0.042136j) [9] (0.001611+0.004286j) [10] (0.001527+0.002011j) [11] (0.000186+0.000361j) [12] (0.000990+-0.000215j) [13] (0.000373+-0.000256j) [14] (-0.001273+0.000464j) [15] (0.001159+0.001058j) [16] (0.000621+-0.001287j) [17] (-0.000673+0.001001j) [18] (0.000025+-0.000228j) [19] (-0.000244+-0.000012j) [20] (0.000698+0.000222j) [21] (-0.000256+-0.000116j) [22] (-0.000189+-0.000112j) [23] (0.000449+0.000111j) [24] (-0.000790+0.000218j) [25] (0.000185+0.000019j) [26] (0.001250+-0.000179j) [27] (-0.000985+-0.000131j) [28] (-0.000077+0.000429j) [29] (0.000429+-0.000058j) [30] (-0.000620+-0.000187j) [31] (0.000907+0.000340j) [32] (-0.000126+-0.000314j) [33] (0.000024+-0.000142j) [34] (-0.000503+0.000532j) [35] (0.001116+0.000584j) [36] (0.000258+-0.000051j) [37] (0.000807+0.000773j) [38] (0.003857+0.003912j) [39] (-0.022648+0.023224j) [40] (-0.000071+-0.056523j) [41] (0.021825+0.018984j) [42] (0.006530+0.003042j) [43] (-0.019067+0.025972j) [44] (0.002945+-0.045785j) [45] (0.010391+0.040381j) [46] (-0.018414+-0.019886j) [47] (0.010679+0.007567j) [48] (0.014649+-0.022715j) [49] (-0.007632+0.028023j) [50] (-0.004731+-0.005551j) [51] (0.000074+-0.000718j) [52] (-0.000791+-0.000904j) [53] (0.000014+-0.000019j) [54] (0.000344+0.000090j) [55] (0.000134+-0.000584j) [56] (-0.000264+0.000585j) [57] (-0.000724+-0.000490j) [58] (0.001307+-0.000154j) [59] (-0.001079+0.000291j) [60] (0.000548+-0.000334j) [61] (-0.000573+0.000041j) [62] (0.000840+-0.000249j) [63] (-0.000619+0.000665j) [64] (-0.000148+-0.000633j) [65] (0.000158+0.000457j) [66] (0.000138+-0.000554j) [67] (0.000081+0.000580j) [68] (-0.000484+-0.000325j) [69] (0.000552+-0.000122j) [70] (-0.000070+0.000219j) [71] (-0.000362+-0.000100j) [72] (0.000310+-0.000062j) [73] (0.000635+0.000270j) [74] (-0.001663+-0.000394j) [75] (0.001318+0.000141j) [76] (-0.000177+0.000009j) [77] (-0.000646+-0.000014j) [78] (0.000624+0.000356j) [79] (-0.000049+-0.000515j) [80] (-0.000366+-0.000084j) [81] (0.000828+0.000682j) [82] (-0.001138+-0.000252j) [83] (0.000686+-0.000317j) [84] (-0.000067+0.000474j) [85] (-0.000097+-0.000849j) [86] (0.000413+0.001005j) [87] (-0.000761+-0.000557j) [88] (0.000237+0.000295j) [89] (0.000096+-0.000673j) [90] (0.000482+0.000729j) [91] (-0.000311+-0.000147j) [92] (-0.000463+-0.000385j) [93] (0.000049+0.000497j) [94] (0.000472+-0.000785j) [95] (0.000007+0.001097j) [96] (0.000009+-0.000468j) [97] (-0.000771+0.000036j) [98] (0.000657+-0.000847j) [99] (0.000456+0.001364j) [100] (-0.001066+-0.000980j) [101] (0.000792+0.000747j) [102] (-0.000536+-0.000571j) [103] (0.000696+-0.000112j) [104] (-0.000424+0.000539j) [105] (-0.000023+-0.000251j) [106] (0.000105+-0.000071j) [107] (-0.000341+0.000353j) [108] (0.000462+-0.001015j) [109] (-0.000282+0.001091j) [110] (0.000073+-0.000221j) [111] (0.000250+-0.000085j) [112] (-0.000258+-0.000122j) [113] (-0.000355+-0.000028j) [114] (0.001097+0.000032j) [115] (-0.001402+0.000154j) [116] (0.001382+0.000325j) [117] (-0.001009+-0.001084j) [118] (0.000305+0.000141j) [119] (-0.000007+0.001139j) [120] (0.000040+-0.000427j) [121] (-0.000253+-0.000659j) [122] (0.000403+0.000886j) [123] (-0.000065+-0.000793j) [124] (-0.000628+0.000475j) [125] (0.001202+-0.000259j) [126] (-0.000712+0.000480j) [127] (0.000087+-0.000369j) [128] (-0.000109+0.000000j) //直流分量 [129] (0.000087+0.000369j) [130] (-0.000712+-0.000480j) [131] (0.001202+0.000259j) [132] (-0.000628+-0.000475j) [133] (-0.000065+0.000793j) [134] (0.000403+-0.000886j) [135] (-0.000253+0.000659j) [136] (0.000040+0.000427j) [137] (-0.000007+-0.001139j) [138] (0.000305+-0.000141j) [139] (-0.001009+0.001084j) [140] (0.001382+-0.000325j) [141] (-0.001402+-0.000154j) [142] (0.001097+-0.000032j) [143] (-0.000355+0.000028j) [144] (-0.000258+0.000122j) [145] (0.000251+0.000085j) [146] (0.000073+0.000221j) [147] (-0.000282+-0.001091j) [148] (0.000462+0.001015j) [149] (-0.000341+-0.000353j) [150] (0.000105+0.000071j) [151] (-0.000023+0.000251j) [152] (-0.000424+-0.000539j) [153] (0.000696+0.000112j) [154] (-0.000536+0.000571j) [155] (0.000792+-0.000747j) [156] (-0.001066+0.000980j) [157] (0.000456+-0.001364j) [158] (0.000657+0.000847j) [159] (-0.000771+-0.000036j) [160] (0.000009+0.000468j) [161] (0.000007+-0.001097j) [162] (0.000472+0.000785j) [163] (0.000049+-0.000497j) [164] (-0.000463+0.000385j) [165] (-0.000311+0.000147j) [166] (0.000482+-0.000729j) [167] (0.000096+0.000673j) [168] (0.000237+-0.000295j) [169] (-0.000761+0.000557j) [170] (0.000413+-0.001005j) [171] (-0.000097+0.000849j) [172] (-0.000067+-0.000474j) [173] (0.000686+0.000317j) [174] (-0.001138+0.000252j) [175] (0.000828+-0.000682j) [176] (-0.000366+0.000084j) [177] (-0.000049+0.000515j) [178] (0.000624+-0.000356j) [179] (-0.000646+0.000014j) [180] (-0.000177+-0.000009j) [181] (0.001318+-0.000141j) [182] (-0.001663+0.000394j) [183] (0.000635+-0.000270j) [184] (0.000310+0.000062j) [185] (-0.000362+0.000100j) [186] (-0.000070+-0.000219j) [187] (0.000552+0.000122j) [188] (-0.000484+0.000325j) [189] (0.000081+-0.000580j) [190] (0.000138+0.000554j) [191] (0.000158+-0.000457j) [192] (-0.000148+0.000633j) [193] (-0.000619+-0.000665j) [194] (0.000840+0.000249j) [195] (-0.000573+-0.000041j) [196] (0.000548+0.000334j) [197] (-0.001079+-0.000291j) [198] (0.001307+0.000154j) [199] (-0.000724+0.000490j) [200] (-0.000264+-0.000585j) [201] (0.000134+0.000584j) [202] (0.000344+-0.000090j) [203] (0.000014+0.000019j) [204] (-0.000791+0.000904j) [205] (0.000074+0.000718j) [206] (-0.004731+0.005551j) [207] (-0.007631+-0.028023j) [208] (0.014648+0.022715j) [209] (0.010679+-0.007567j) [210] (-0.018414+0.019886j) [211] (0.010392+-0.040381j) [212] (0.002945+0.045785j) [213] (-0.019067+-0.025972j) [214] (0.006530+-0.003042j) [215] (0.021826+-0.018984j) [216] (-0.000071+0.056523j) [217] (-0.022648+-0.023224j) [218] (0.003857+-0.003912j) [219] (0.000807+-0.000773j) [220] (0.000258+0.000051j) [221] (0.001116+-0.000584j) [222] (-0.000503+-0.000532j) [223] (0.000024+0.000142j) [224] (-0.000126+0.000314j) [225] (0.000907+-0.000340j) [226] (-0.000620+0.000187j) [227] (0.000429+0.000058j) [228] (-0.000077+-0.000429j) [229] (-0.000985+0.000131j) [230] (0.001250+0.000179j) [231] (0.000185+-0.000019j) [232] (-0.000790+-0.000218j) [233] (0.000449+-0.000111j) [234] (-0.000189+0.000112j) [235] (-0.000256+0.000116j) [236] (0.000698+-0.000222j) [237] (-0.000244+0.000012j) [238] (0.000025+0.000228j) [239] (-0.000673+-0.001001j) [240] (0.000621+0.001286j) [241] (0.001159+-0.001058j) [242] (-0.001273+-0.000464j) [243] (0.000373+0.000256j) [244] (0.000990+0.000215j) [245] (0.000186+-0.000361j) [246] (0.001527+-0.002011j) [247] (0.001611+-0.004286j) [248] (0.003740+0.042136j) [249] (-0.005668+-0.038893j) [250] (-0.016804+-0.000179j) [251] (0.003737+-0.002048j) [252] (0.009356+0.005058j) [253] (0.008954+0.001403j) [254] (-0.022870+0.002839j) [255] (0.023277+-0.005782j) |
注：
（1）实数FFT的运算结果得到复数序列
（2）第0个和第FRAME_LEN/2个是直流分量，其余的结果具有“共轭对称性”
（3）注意运算结果的前129个数据，要与下文的libsora.stft()的结果做对比
根据实数的FFT结果具有“共轭对称性”的性质，可以优化FFT计算算法，将大大减小运算量和存储空间，一般来说像Matlab或者Python都有实数FFT计算的API，比如做256个点的FFT，只输出129个点的数据即可(没必要返回冗余数据)，。为方便理解，这里采用的是复数的FFT算法。测试代码中，个人实现的API简单说明如下：
|
/* float dataR[]：时域上，输入序列的实部；计算结束，变成频域上频点的实部 float dataR[]：时域上，输入序列的虚部(都是0)；计算结束，变成频域上频点的虚部 float dataA[]：运算结果序列的幅值。 int N：FFT序列的长度 int M：FFT序列的阶数 */ void FFT(float dataR[], float dataI[], float dataA[], int N, int M); |
2.2 librosa的短时傅里叶变正换实现
用相同的数据源，由于librosa.stft()默认加的是汉宁窗，这里就没必要先进行计算。
2.2.1 libsora.stft()的分帧机制
(1) librosa.stft输出的帧数比正常计算的帧数多一帧，它分帧策略的准则是中心对齐的，即分帧点位于该帧的中心，首尾帧之外都有半个帧长度的padding。而padding部分使用0数据填充。
(2) 输出帧数的计算公式：signal_length/hop_length + 1
例：一帧是256个点，帧移取一半，也就是128，那么折叠的部分就是256-128=128。
正常的分帧方法可分为两个帧，第二帧的后半部分用0填充，或者一般来说，最后一帧可以直接舍弃。如下图：
而librosa的做法是会在数据的头部和尾部增加半个帧长度的“填充”，再进行分帧，如下图：
输入的数据：data=dat1+dat2
第一帧数据：128个0+dat1
第二帧数据：data=dat1+dat2
第三帧数据：dat2+128个0
每次的傅里叶变换都是256个点的输入。
2.2.2 测试代码
文件：libsora_test.py
import librosa
from librosa.core.spectrum import amplitude_to_db
import numpy as np
import soundfile as sf
import matplotlib.pyplot as pltdat_r = np.array([
0.000214,
0.000122,
-0.000061,
0.000275,
0.000092,
0.000336,
0.000153,
-0.000092,
0.000031,
0.000305,
0.000641,
0.000458,
0.000305,
0.000153,
0.000397,
0.000732,
0.000885,
0.000580,
0.000427,
0.000397,
0.000641,
0.001190,
0.001282,
0.000977,
0.000336,
0.000458,
0.001007,
0.001617,
0.001465,
0.000824,
0.000275,
0.000366,
0.001007,
0.001282,
0.000885,
0.000122,
-0.000336,
-0.000122,
0.000336,
0.000275,
-0.000153,
-0.000824,
-0.000916,
-0.000580,
-0.000519,
-0.000763,
-0.001099,
-0.001221,
-0.001007,
-0.000580,
-0.000793,
-0.000885,
-0.001038,
-0.000580,
-0.000275,
-0.000214,
-0.000549,
-0.000610,
-0.000061,
0.000702,
0.001190,
0.000763,
0.000305,
0.000275,
0.001068,
0.001923,
0.001953,
0.001190,
0.000488,
0.000519,
0.001251,
0.001740,
0.001282,
0.000397,
-0.000183,
0.000000,
0.000214,
0.000305,
-0.000092,
-0.000458,
-0.000732,
-0.000610,
-0.000671,
-0.000916,
-0.000946,
-0.000824,
-0.000702,
-0.001068,
-0.000977,
-0.001190,
-0.000671,
-0.000244,
-0.000122,
-0.000580,
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])res = librosa.stft(dat_r, n_fft=256, hop_length=128, win_length=256)
print("shape-res:", res.shape)
print("row-res:", res.shape[0]) #行数
print("col-res:", res.shape[1]) #列数
#np.savetxt('dat_r_fft.txt', res, fmt="%f") #保存结果矩阵(复数矩阵，a+bi的形式)
#np.savetxt('dat_r_fft.txt', np.real(res), fmt="%f") #保存每一帧的实部
#np.savetxt('dat_r_fft.txt', np.imag(res), fmt="%f") #保存每一帧的虚部lis0 = []
lis1 = []
lis2 = []
for i in range(res.shape[0]):lis0.append(np.real(res[i][0])) #取第一帧的实数部分(结果矩阵的第一列数据)lis1.append(np.real(res[i][1])) #取第二帧的实数部分(结果矩阵的第二列数据)lis2.append(np.real(res[i][2])) #取第三帧的实数部分(结果矩阵的第三列数据)np.savetxt('res_fft_0.txt', lis0, fmt="%f")
np.savetxt('res_fft_1.txt', lis1, fmt="%f")
np.savetxt('res_fft_2.txt', lis2, fmt="%f")2.2.3 运行结果分析
|
>>python librosa_test.py shape-res: (129, 3) #返回的矩阵大小：129行 x 3列，每一列就是一帧数据 #分成了三帧语音进行处理，每一帧是256个采样点，但是FFT的输 #出只保留了129个数据，没有输出冗余数据。--实数FFT的实现 row-res: 129 col-res: 3 |
为方便对比数据，将每一帧的运算结果的实数部分保存到txt文本中。
对比第二帧的结果，如下：可发现前129个数据近似相等[自己写的复数FFT算法有计算精度上的差异]。
同样可以对比验证，第一和第三帧的FFT结果，从而证实librosa的分帧机制。
2.3 librosa的短时傅里叶逆变换实现
对应的API是：librosa.istft
参数一致即可。逆变换会根据正变换的规则，处理重叠部分，还原数据帧。测试代码如下：
|
res = librosa.stft(dat_r, n_fft=256, hop_length=128, win_length=256) …… ires = librosa.istft(res, n_fft=256, hop_length=128, win_length=256) np.savetxt('ires.txt', ires, fmt="%f") |
对比结果，和输入序列数据是一致的。
3. 附录：librosa官网
Core IO and DSP — librosa 0.9.2 documentation
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