时域是什么_oct结果解读[通俗易懂]

时域OCT(TD-OCT) 我的理解

由于本人是先接触SS-OCT的 所以对TD-OCT有点不理解，百度查找了一些资料总结一下。

TD-OCT 光学系统跟SS-OCT差不多，如下图

引用知乎上的原理解释：https://zhuanlan.zhihu.com/p/46452072
时域OCT[1]SLD发出的低相干宽谱光在单模光纤中传递（如果使用多模光纤，模式色散会展宽宽光谱光，导致OCT的轴向分辨率下降），经过一个50:50的光纤耦合器（coupler）后，各有50%的光分别进入到测量臂和参考臂中照射到样品和参考镜上，来自样品的背向散射光和来自参考镜的反射光在耦合器上合并后被探测器（detector）收集。需要注意的是，只有当两臂的光程差在宽光谱光源相干长度的范围内（即光程差﹤光源的相干长度），两束光才可以发生干涉，这也使得OCT可以分辨出样品背向散射光的轴向长度分布。因此可以在参考臂利用步进电机在轴向平移参考镜（Reference），实现轴向扫描，得到样品的轴向深度分布图。同时在样品臂对光束进行横向扫描，可以得到样品的二维横截面图像，进而产生三维立体图像。
上述OCT技术是基于时间域的低相干干涉测量技术，通常称之为“时域OCT（Time Domain OCT, TD-OCT）”。时域OCT技术出现之后，由于其可以兼顾较高的成像深度和分辨率，迅速得到了研究者的广泛关注。但是在之后的研究中发现时域OCT在进行轴向扫描时，由于步进电机的限制，会致测量速度受到限制，机械装置的引入也会使系统的稳定性大打折扣。针对这些缺点，傅里叶域OCT（Fourier Domain OCT, FD-OCT)应运而生。基于测量臂与参考臂的干涉光谱与样品不同深度的背向散射光强度信息恰好是一对傅里叶变换对的关系这一原理，傅里叶域OCT避免了在测量时需要改变参考臂光程长度的问题，而是将代表着样品不同深度信息的光束统一收集后，通过傅里叶（逆）变换即可解算出样品深度的强度信息，实现了样品轴向信息的并行获取，从而省却了轴向扫描。

这里的”SLD发出的低相干宽谱光” 我认为是关键信息
找了一圈百度也没有找到SLD(超辐射发光二极管)，能发出什么波形的光(不正确的说法)，直到看到一段话:
如果光源在频域是宽带，那么在空间域就是窄带
这段话也是出于知乎的 https://zhuanlan.zhihu.com/p/159419026
然后又看到了大佬的文章 http://www.elecfans.com/d/845686.html里面的两个插图

我就在这文章里就武断地认为SLD光源发出的光的波的特性如下：

• 在时间上是分段的波形
• 形状上估计差不多是中心频率正弦波乘以正态分布的一个波形吧
• 如果对波形进行傳里叶变换，变换的结果相当于在中心频率上有一个差不多正态分布形状的频谱

这些特性是我个人猜测的，并没有查找到实际根据。

根据上面找到的信息，我自己用ipython做一下原理上的模拟如下

生成模拟SLD光源的光波形

import numpy as np
import math
import cmath

%pylab inline
x = [0] * 512

o = 10 #标准差越小 波形越窄 对应的频谱越宽
u = 255
centerfreq = 80
for i in range(512):
    t = 1 / math.sqrt(2 * math.pi ) * o
    t1 = math.exp(-(i - u)**2 / (2*o**2))
    x[i] = t*t1*math.sin(2 * math.pi * centerfreq * i / 511)

plt.plot(x)

#x = np.random.rand(64)
xfft = np.fft.fft(x)
xfft = np.abs(xfft)

plt.plot(xfft[:len(xfft)//2])
print("傳里叶变换后的中心频率 =",xfft.argmax())

傳里叶变换后的中心频率 = 80

也可以从正态分布的频域生成波形

print("从宽带谱域反过来生成波形")
specwave = [0]*512
specwidth = 10 #标准差越小 波形越窄 对应的频谱越宽
centerfreq = 80
for i in range(256):
    t = 1 / math.sqrt(2 * math.pi ) * specwidth
    t1 = math.exp(-(i - centerfreq)**2 / (2*specwidth**2))
    specwave[i] = t*t1
    if i>0 :  #频域理论上应该是对称的 (幅值相等 相位相反)
        specwave[512-i] = specwave[i]
specwave[256] = 0   
ifff = np.fft.ifft(specwave)
ifff = np.concatenate((ifff[-256:],ifff[:-256]))
plt.figure()
plt.text(0,0.2,"Spectral")
plt.plot(specwave)
plt.figure()
plt.text(0,0.2,"wave")
plt.plot(ifff.real)

从宽带谱域反过来生成波形

从一个SLD光源 获取的波长光谱 生成波形

specdata_wlen = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
                 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 
                 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
                 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 
                 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 
                 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 
                 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 
                 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 
                 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
                 0.0, 0.0, 0.0, 0.0, 0.0018315018315018315, 0.0018315018315018315, 
                 0.0018315018315018315, 0.0018315018315018315, 0.0018315018315018315,
                 0.0018315018315018315, 0.0018315018315018315, 0.0, 0.0, 0.0, 0.0, 
                 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.003663003663003663, 
                 0.003663003663003663, 0.003663003663003663, 0.003663003663003663,
                 0.003663003663003663, 0.003663003663003663, 0.003663003663003663, 
                 0.003663003663003663, 0.003663003663003663, 0.003663003663003663, 0.003663003663003663, 
                 0.007326007326007326, 0.007326007326007326,
                 0.007326007326007326, 0.007326007326007326, 0.007326007326007326, 
                 0.007326007326007326, 0.007326007326007326, 0.007326007326007326, 0.01098901098901099,
                 0.01098901098901099, 0.01098901098901099, 0.01098901098901099, 0.01098901098901099, 
                 0.01282051282051282, 0.01282051282051282, 0.01282051282051282, 0.016483516483516484,
                 0.016483516483516484, 0.016483516483516484, 0.016483516483516484, 0.020146520146520148, 
                 0.020146520146520148, 0.020146520146520148, 0.023809523809523808, 0.023809523809523808, 0.027472527472527472, 
                 0.027472527472527472, 0.027472527472527472, 0.027472527472527472, 0.029304029304029304, 0.031135531135531136, 
                 0.03296703296703297, 0.03663003663003663, 0.03663003663003663, 0.040293040293040296, 0.040293040293040296, 
                 0.04395604395604396, 0.04395604395604396, 0.045787545787545784, 0.04945054945054945, 0.04945054945054945, 
                 0.05311355311355311, 0.05311355311355311, 0.056776556776556776, 0.056776556776556776, 0.06227106227106227,
                 0.0641025641025641, 0.06593406593406594, 0.07326007326007326, 0.07326007326007326, 0.07509157509157509, 
                 0.07509157509157509, 0.08241758241758242, 0.08241758241758242, 0.08791208791208792, 0.09523809523809523, 
                 0.09523809523809523, 0.10256410256410256, 0.10256410256410256, 0.10805860805860806, 0.10805860805860806,
                 0.11538461538461539, 0.11538461538461539, 0.12087912087912088, 0.1282051282051282, 0.1282051282051282, 
                 0.13736263736263737, 0.13736263736263737, 0.1446886446886447, 0.1446886446886447, 0.15384615384615385, 
                 0.15384615384615385, 0.163003663003663, 0.17399267399267399, 0.17399267399267399, 0.18315018315018314, 
                 0.18315018315018314, 0.19230769230769232, 0.19413919413919414, 0.2032967032967033, 0.21245421245421245, 
                 0.21245421245421245, 0.22344322344322345, 0.22344322344322345, 0.2326007326007326, 0.2326007326007326, 
                 0.24175824175824176, 0.24358974358974358, 0.25457875457875456, 0.26556776556776557, 0.26556776556776557, 
                 0.2783882783882784, 0.2783882783882784, 0.2875457875457875, 0.2875457875457875, 0.29853479853479853, 
                 0.3076923076923077, 0.30952380952380953, 0.32051282051282054, 0.32051282051282054, 0.32967032967032966, 
                 0.32967032967032966, 0.34065934065934067, 0.34065934065934067, 0.3534798534798535, 0.3626373626373626, 
                 0.3626373626373626, 0.3717948717948718, 0.37362637362637363, 0.38278388278388276, 0.38278388278388276, 
                 0.39194139194139194, 0.39377289377289376, 0.40293040293040294, 0.41208791208791207, 0.41208791208791207, 
                 0.42124542124542125, 0.42124542124542125, 0.42857142857142855, 0.42857142857142855, 0.43772893772893773, 
                 0.44505494505494503, 0.44505494505494503, 0.45054945054945056, 0.45054945054945056, 0.45787545787545786, 
                 0.45787545787545786, 0.4633699633699634, 0.4652014652014652, 0.4706959706959707, 0.47619047619047616, 
                 0.47802197802197804, 0.47985347985347987, 0.47985347985347987, 0.48717948717948717, 0.48717948717948717,
                 0.4908424908424908, 0.4926739926739927, 0.4926739926739927, 0.49633699633699635, 0.49633699633699635, 0.5, 
                 0.5, 0.5, 0.5, 0.5036630036630036, 0.5036630036630036, 0.5036630036630036, 0.5036630036630036, 0.5036630036630036,
                 0.5036630036630036, 0.5036630036630036, 0.5036630036630036, 0.5036630036630036, 0.5036630036630036, 0.5036630036630036, 
                 0.5018315018315018, 0.5, 0.5, 0.5, 0.49633699633699635, 0.49633699633699635, 0.49633699633699635, 0.49633699633699635, 
                 0.4926739926739927, 0.4908424908424908, 0.4908424908424908, 0.48717948717948717, 0.48717948717948717, 0.48717948717948717,
                 0.48717948717948717, 0.4835164835164835, 0.4816849816849817, 0.47985347985347987, 0.47802197802197804, 0.47802197802197804, 
                 0.47435897435897434, 0.47435897435897434, 0.47435897435897434, 0.47435897435897434, 0.4706959706959707, 0.46703296703296704,
                 0.46703296703296704, 0.4633699633699634, 0.4633699633699634, 0.46153846153846156, 0.46153846153846156, 0.45787545787545786, 
                 0.45787545787545786, 0.4542124542124542, 0.4523809523809524, 0.45054945054945056, 0.44871794871794873, 0.4468864468864469,
                 0.44505494505494503, 0.44505494505494503, 0.4413919413919414, 0.4413919413919414, 0.4413919413919414, 0.43772893772893773,
                 0.43772893772893773, 0.4340659340659341, 0.4340659340659341, 0.43223443223443225, 0.43223443223443225, 0.43223443223443225,
                 0.42857142857142855, 0.42857142857142855, 0.4249084249084249, 0.4249084249084249, 0.4249084249084249, 0.4249084249084249,
                 0.42124542124542125, 0.4194139194139194, 0.4175824175824176, 0.4175824175824176, 0.4175824175824176, 0.4157509157509158, 
                 0.4157509157509158, 0.4157509157509158, 0.4157509157509158, 0.41208791208791207, 0.41208791208791207, 0.41208791208791207,
                 0.4084249084249084, 0.4084249084249084, 0.4084249084249084, 0.4084249084249084, 0.4084249084249084, 0.4084249084249084,
                 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477,
                 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477,
                 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477,
                 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477,
                 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.40476190476190477, 0.4065934065934066, 0.4084249084249084, 
                 0.4084249084249084, 0.4084249084249084, 0.4084249084249084, 0.4084249084249084, 0.4084249084249084, 0.4084249084249084, 
                 0.41208791208791207, 0.41208791208791207, 0.41208791208791207, 0.41208791208791207, 0.41208791208791207, 0.4157509157509158,
                 0.4157509157509158, 0.4157509157509158, 0.4157509157509158, 0.4175824175824176, 0.4175824175824176, 0.4175824175824176,
                 0.42124542124542125, 0.42124542124542125, 0.4249084249084249, 0.4249084249084249, 0.4249084249084249, 0.4267399267399267, 
                 0.42857142857142855, 0.42857142857142855, 0.42857142857142855, 0.43223443223443225, 0.43223443223443225, 0.4340659340659341,
                 0.4340659340659341, 0.4340659340659341, 0.43772893772893773, 0.43772893772893773, 0.4413919413919414, 0.4413919413919414, 
                 0.44505494505494503, 0.44505494505494503, 0.44505494505494503, 0.44505494505494503, 0.4468864468864469, 0.45054945054945056,
                 0.45054945054945056, 0.4542124542124542, 0.4542124542124542, 0.45787545787545786, 0.45787545787545786, 0.46153846153846156,
                 0.46153846153846156, 0.4633699633699634, 0.46703296703296704, 0.46703296703296704, 0.46703296703296704, 0.46703296703296704,
                 0.4706959706959707, 0.4706959706959707, 0.47435897435897434, 0.47435897435897434, 0.47619047619047616, 0.47985347985347987, 
                 0.47985347985347987, 0.4835164835164835, 0.4835164835164835, 0.4908424908424908, 0.4908424908424908, 0.4926739926739927, 
                 0.49633699633699635, 0.49633699633699635, 0.5, 0.5, 0.5036630036630036, 0.5036630036630036, 0.5073260073260073,
                 0.5091575091575091, 0.5091575091575091, 0.5128205128205128, 0.5128205128205128, 0.5164835164835165, 0.5164835164835165,
                 0.5201465201465202, 0.5201465201465202, 0.521978021978022, 0.5256410256410257, 0.5256410256410257, 0.532967032967033, 
                 0.532967032967033, 0.5366300366300366, 0.5366300366300366, 0.5384615384615384, 0.5421245421245421, 0.5421245421245421, 
                 0.5494505494505495, 0.5494505494505495, 0.5512820512820513, 0.5531135531135531, 0.554945054945055, 0.5567765567765568, 
                 0.5586080586080586, 0.5622710622710623, 0.5622710622710623, 0.5677655677655677, 0.5695970695970696, 0.5714285714285714, 
                 0.5714285714285714, 0.575091575091575, 0.5769230769230769, 0.5787545787545788, 0.5842490842490843, 0.5842490842490843,
                 0.5879120879120879, 0.5879120879120879, 0.5915750915750916, 0.5915750915750916, 0.5970695970695971, 0.6007326007326007,
                 0.6007326007326007, 0.6043956043956044, 0.6043956043956044, 0.6117216117216118, 0.6117216117216118, 0.6135531135531136, 
                 0.6153846153846154, 0.6172161172161172, 0.6172161172161172, 0.6245421245421245, 0.6263736263736264, 0.6282051282051282,
                 0.63003663003663, 0.6318681318681318, 0.6373626373626373, 0.6391941391941391, 0.6410256410256411, 0.6428571428571429, 
                 0.6428571428571429, 0.6501831501831502, 0.6501831501831502, 0.6538461538461539, 0.6538461538461539, 0.6593406593406593,
                 0.663003663003663, 0.663003663003663, 0.6703296703296703, 0.6703296703296703, 0.6721611721611722, 0.673992673992674,
                 0.6794871794871795, 0.6813186813186813, 0.6831501831501832, 0.6868131868131868, 0.6868131868131868, 0.6923076923076923, 
                 0.6923076923076923, 0.6959706959706959, 0.6959706959706959, 0.7014652014652014, 0.7051282051282052, 0.7051282051282052, 
                 0.7124542124542125, 0.7124542124542125, 0.7161172161172161, 0.7161172161172161, 0.7216117216117216, 0.7234432234432234, 
                 0.7252747252747253, 0.7252747252747253, 0.7307692307692307, 0.7344322344322345, 0.7344322344322345, 0.7417582417582418, 
                 0.7417582417582418, 0.7454212454212454, 0.7472527472527473, 0.7472527472527473, 0.7545787545787546, 0.7545787545787546, 
                 0.7619047619047619, 0.7619047619047619, 0.7637362637362637, 0.7655677655677655, 0.7710622710622711, 0.7710622710622711, 
                 0.7728937728937729, 0.7802197802197802, 0.7802197802197802, 0.7838827838827839, 0.7838827838827839, 0.7912087912087912, 
                 0.793040293040293, 0.793040293040293, 0.7967032967032966, 0.7967032967032966, 0.8040293040293041, 0.8040293040293041, 
                 0.8058608058608059, 0.8076923076923077, 0.8131868131868132, 0.8168498168498168, 0.8168498168498168, 0.8223443223443223,
                 0.8223443223443223, 0.826007326007326, 0.826007326007326, 0.8296703296703297, 0.8315018315018315, 0.836996336996337, 
                 0.8388278388278388, 0.8388278388278388, 0.8424908424908425, 0.8424908424908425, 0.8498168498168498, 0.8498168498168498, 
                 0.8553113553113553, 0.8589743589743589, 0.8589743589743589, 0.8626373626373627, 0.8626373626373627, 0.8663003663003663, 
                 0.8663003663003663, 0.8717948717948718, 0.8736263736263736, 0.8754578754578755, 0.8809523809523809, 0.8827838827838828,
                 0.8846153846153846, 0.8846153846153846, 0.8882783882783882, 0.8882783882783882, 0.891941391941392, 0.8937728937728938, 
                 0.8956043956043956, 0.8974358974358975, 0.8974358974358975, 0.9010989010989011, 0.9010989010989011, 0.9084249084249084, 
                 0.9084249084249084, 0.9120879120879121, 0.9175824175824175, 0.9175824175824175, 0.9212454212454212, 0.9212454212454212, 
                 0.924908424908425, 0.924908424908425, 0.9267399267399268, 0.9267399267399268, 0.9304029304029304, 0.9340659340659341, 
                 0.9340659340659341, 0.9377289377289377, 0.9377289377289377, 0.9413919413919414, 0.9413919413919414, 0.9432234432234432, 
                 0.9468864468864469, 0.9468864468864469, 0.9468864468864469, 0.9468864468864469, 0.9505494505494505, 0.9505494505494505, 
                 0.9542124542124543, 0.9542124542124543, 0.9560439560439561, 0.9597069597069597, 0.9597069597069597, 0.9633699633699634, 
                 0.9633699633699634, 0.967032967032967, 0.967032967032967, 0.967032967032967, 0.967032967032967, 0.967032967032967, 
                 0.9706959706959707, 0.9706959706959707, 0.9725274725274725, 0.9725274725274725, 0.9725274725274725, 0.9743589743589743,
                 0.9761904761904762, 0.9761904761904762, 0.9761904761904762, 0.9798534798534798, 0.9798534798534798, 0.9798534798534798, 
                 0.9798534798534798, 0.9835164835164835, 0.9853479853479854, 0.9853479853479854, 0.9853479853479854, 0.9871794871794872, 
                 0.989010989010989, 0.989010989010989, 0.989010989010989, 0.989010989010989, 0.989010989010989, 0.9926739926739927,
                 0.9926739926739927, 0.9926739926739927, 0.9926739926739927, 0.9926739926739927, 0.9926739926739927, 0.9926739926739927, 
                 0.9945054945054945, 0.9963369963369964, 0.9963369963369964, 0.9963369963369964, 0.9963369963369964, 0.9963369963369964, 
                 0.9963369963369964, 0.9963369963369964, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 0.9963369963369964, 0.9963369963369964, 
                 1.0, 1.0, 1.0, 0.9963369963369964, 0.9963369963369964, 0.9963369963369964, 0.9963369963369964, 0.9963369963369964, 
                 0.9963369963369964, 0.9926739926739927, 0.9926739926739927, 0.9926739926739927, 0.989010989010989, 0.989010989010989, 
                 0.989010989010989, 0.989010989010989, 0.9871794871794872, 0.9853479853479854, 0.9853479853479854, 0.9835164835164835, 
                 0.9835164835164835, 0.9798534798534798, 0.9798534798534798, 0.9798534798534798, 0.9798534798534798, 0.9798534798534798,
                 0.978021978021978, 0.9761904761904762, 0.9761904761904762, 0.9761904761904762, 0.9725274725274725, 0.9725274725274725,
                 0.9706959706959707, 0.9688644688644689, 0.967032967032967, 0.9633699633699634, 0.9633699633699634, 0.9597069597069597, 
                 0.9597069597069597, 0.9597069597069597, 0.9597069597069597, 0.9560439560439561, 0.9560439560439561, 0.9542124542124543, 
                 0.9505494505494505, 0.9505494505494505, 0.9468864468864469, 0.9468864468864469, 0.9432234432234432, 0.9432234432234432, 
                 0.9377289377289377, 0.9340659340659341, 0.9340659340659341, 0.9304029304029304, 0.9304029304029304, 0.924908424908425, 
                 0.924908424908425, 0.9212454212454212, 0.9212454212454212, 0.9175824175824175, 0.9120879120879121, 0.9120879120879121, 
                 0.9084249084249084, 0.9084249084249084, 0.9010989010989011, 0.8974358974358975, 0.8974358974358975, 0.891941391941392, 
                 0.891941391941392, 0.8882783882783882, 0.8882783882783882, 0.8827838827838828, 0.8809523809523809, 0.8791208791208791, 
                 0.8791208791208791, 0.8681318681318682, 0.8626373626373627, 0.8626373626373627, 0.8589743589743589, 0.8589743589743589,
                 0.8516483516483516, 0.8516483516483516, 0.8461538461538461, 0.836996336996337, 0.836996336996337, 0.8333333333333334, 
                 0.8333333333333334, 0.826007326007326, 0.826007326007326, 0.8205128205128205, 0.8205128205128205, 0.8131868131868132, 
                 0.8076923076923077, 0.8058608058608059, 0.8003663003663004, 0.8003663003663004, 0.793040293040293, 0.793040293040293,
                 0.7875457875457875, 0.7765567765567766, 0.7765567765567766, 0.7673992673992674, 0.7673992673992674, 0.7619047619047619,
                 0.7619047619047619, 0.7545787545787546, 0.7545787545787546, 0.7472527472527473, 0.7417582417582418, 0.7417582417582418,
                 0.7326007326007326, 0.7326007326007326, 0.7252747252747253, 0.7252747252747253, 0.717948717948718, 0.717948717948718, 
                 0.7087912087912088, 0.7032967032967034, 0.7014652014652014, 0.6959706959706959, 0.6959706959706959, 0.6868131868131868,
                 0.6868131868131868, 0.6758241758241759, 0.6703296703296703, 0.6703296703296703, 0.6611721611721612, 0.6593406593406593, 
                 0.6501831501831502, 0.6501831501831502, 0.6428571428571429, 0.6428571428571429, 0.6373626373626373, 0.6282051282051282, 
                 0.6263736263736264, 0.6208791208791209, 0.6208791208791209, 0.6117216117216118, 0.6117216117216118, 0.6007326007326007, 
                 0.6007326007326007, 0.5952380952380952, 0.5860805860805861, 0.5842490842490843, 0.5787545787545788, 0.5787545787545788, 
                 0.5677655677655677, 0.5677655677655677, 0.5622710622710623, 0.5531135531135531, 0.5512820512820513, 0.5457875457875457, 
                 0.5457875457875457, 0.5366300366300366, 0.5366300366300366, 0.5256410256410257, 0.5256410256410257, 0.5201465201465202, 
                 0.510989010989011, 0.5091575091575091, 0.5036630036630036, 0.5036630036630036, 0.4926739926739927, 0.4926739926739927, 
                 0.4835164835164835, 0.47802197802197804, 0.47802197802197804, 0.4706959706959707, 0.4706959706959707, 0.46153846153846156,
                 0.46153846153846156, 0.45054945054945056, 0.44505494505494503, 0.44505494505494503, 0.43772893772893773, 
                 0.43772893772893773, 0.42857142857142855, 0.42857142857142855, 0.42124542124542125, 0.42124542124542125, 
                 0.41208791208791207, 0.4065934065934066, 0.40476190476190477, 0.3956043956043956, 0.3956043956043956, 
                 0.3901098901098901, 0.3882783882783883, 0.38278388278388276, 0.38278388278388276, 0.37545787545787546,
                 0.3663003663003663, 0.3663003663003663, 0.3608058608058608, 0.358974358974359, 0.3534798534798535, 
                 0.3534798534798535, 0.3424908424908425, 0.3424908424908425, 0.336996336996337, 0.3315018315018315, 
                 0.32967032967032966, 0.3241758241758242, 0.3241758241758242, 0.31684981684981683, 0.31135531135531136, 
                 0.30952380952380953, 0.3058608058608059, 0.304029304029304, 0.29853479853479853, 0.29853479853479853, 
                 0.29120879120879123, 0.29120879120879123, 0.2838827838827839, 0.2838827838827839, 0.2783882783882784, 
                 0.27289377289377287, 0.27106227106227104, 0.26556776556776557, 0.26556776556776557, 0.25824175824175827,
                 0.25824175824175827, 0.25274725274725274, 0.24725274725274726, 0.2454212454212454, 0.24175824175824176, 
                 0.24175824175824176, 0.23626373626373626, 0.23626373626373626, 0.22893772893772893, 0.22893772893772893,
                 0.22344322344322345, 0.21978021978021978, 0.21978021978021978, 0.21245421245421245, 0.21245421245421245, 
                 0.20695970695970695, 0.20695970695970695, 0.2032967032967033, 0.2032967032967033, 0.19597069597069597,
                 0.19413919413919414, 0.19230769230769232, 0.18681318681318682, 0.18681318681318682, 0.18315018315018314,
                 0.18315018315018314, 0.17765567765567766, 0.17399267399267399, 0.17399267399267399, 0.16666666666666666, 
                 0.16666666666666666, 0.16483516483516483, 0.163003663003663, 0.1575091575091575, 0.15384615384615385, 
                 0.15384615384615385, 0.152014652014652, 0.15018315018315018, 0.14835164835164835, 0.14835164835164835, 
                 0.14102564102564102, 0.14102564102564102, 0.13736263736263737, 0.13553113553113552, 0.1336996336996337,
                 0.13186813186813187, 0.13186813186813187, 0.1282051282051282, 0.1282051282051282, 0.12454212454212454, 
                 0.12454212454212454, 0.11904761904761904, 0.11538461538461539, 0.11538461538461539, 0.11172161172161173,
                 0.11172161172161173, 0.10805860805860806, 0.10805860805860806, 0.1043956043956044, 0.10256410256410256, 
                 0.10073260073260074, 0.0989010989010989, 0.0989010989010989, 0.09523809523809523, 0.09523809523809523, 
                 0.09157509157509157, 0.09157509157509157, 0.09157509157509157, 0.08974358974358974, 0.08791208791208792,
                 0.08608058608058608, 0.08608058608058608, 0.08241758241758242, 0.08241758241758242, 0.07875457875457875,
                 0.07875457875457875, 0.07509157509157509, 0.07326007326007326, 0.07326007326007326, 0.07326007326007326, 
                 0.07326007326007326, 0.0695970695970696, 0.0695970695970696, 0.06593406593406594, 0.0641025641025641, 
                 0.06227106227106227, 0.06227106227106227, 0.06227106227106227, 0.06043956043956044, 0.05860805860805861,
                 0.056776556776556776, 0.056776556776556776, 0.056776556776556776, 0.05311355311355311, 0.05311355311355311, 
                 0.05311355311355311, 0.05311355311355311, 0.04945054945054945, 0.04945054945054945, 0.04945054945054945,
                 0.047619047619047616, 0.045787545787545784, 0.04395604395604396, 0.04395604395604396, 0.04395604395604396, 
                 0.04395604395604396, 0.040293040293040296, 0.040293040293040296, 0.040293040293040296, 0.040293040293040296,
                 0.040293040293040296, 0.03663003663003663, 0.03663003663003663, 0.03663003663003663, 0.03663003663003663, 
                 0.03296703296703297, 0.03296703296703297, 0.03296703296703297, 0.031135531135531136, 0.029304029304029304, 
                 0.029304029304029304, 0.029304029304029304, 0.029304029304029304, 0.029304029304029304, 0.027472527472527472,
                 0.027472527472527472, 0.027472527472527472, 0.027472527472527472, 0.027472527472527472, 0.023809523809523808,
                 0.023809523809523808, 0.023809523809523808, 0.023809523809523808, 0.023809523809523808, 0.020146520146520148, 
                 0.020146520146520148, 0.020146520146520148, 0.020146520146520148, 0.020146520146520148, 0.020146520146520148, 
                 0.016483516483516484, 0.016483516483516484, 0.016483516483516484, 0.016483516483516484, 0.016483516483516484, 
                 0.016483516483516484, 0.016483516483516484, 0.016483516483516484, 0.016483516483516484, 0.01282051282051282, 
                 0.01282051282051282, 0.01282051282051282, 0.01282051282051282, 0.01282051282051282, 0.01282051282051282,
                 0.01282051282051282, 0.01282051282051282, 0.01098901098901099, 0.01098901098901099, 0.01098901098901099, 
                 0.01098901098901099, 0.01098901098901099, 0.01098901098901099, 0.01098901098901099, 0.01098901098901099, 
                 0.01098901098901099, 0.01098901098901099, 0.01098901098901099, 0.007326007326007326, 0.007326007326007326,
                 0.007326007326007326, 0.007326007326007326, 0.007326007326007326, 0.007326007326007326, 0.007326007326007326,
                 0.007326007326007326, 0.007326007326007326, 0.007326007326007326, 0.007326007326007326]

print("THORLABS SLD850S-A20W 提取的 波长vs强度 谱域特性曲线 (800-900nm)")
plt.figure()
index = [800+ (900-800)*i/(len(specdata_wlen)-1) for i in range(len(specdata_wlen))]
plt.plot(index,specdata_wlen)
freqindex = [i*376/511 for i in range(512)] #从0Hz ~ 400e+6MHz
ifreqindex = np.array(freqindex,dtype=int)
lightspeed = 3e+5 #光速 3e+5 米/毫秒
specdata_freq = [0.0]*1024
for i in range(1,512):
    t = freqindex[i]
    t = lightspeed / t #求波长
    #按波长查找幅值
    if t < 800:
        continue
    if t > 900:
        continue
    t = (t - 800) / (900 - 800)
    t = t * (len(specdata_wlen) - 1)
    j = int(t)
    d = t - j
    if j + 1 >= len(specdata_wlen):
        j = len(specdata_wlen) - 1
        d = 0
    t = specdata_wlen[j]*(1-d) + specdata_wlen[j + 1]*(d)
    specdata_freq[i] = t

print("转换为频谱图 x轴单位:1e+6 MHz")
plt.figure()
plt.plot(ifreqindex,specdata_freq[:len(specdata_freq)//2])

#让频谱图对称
for i in range(1,512):
    specdata_freq[1024-i] = specdata_freq[i]
#plt.figure()
#plt.plot(specdata_freq)
sldlightwave = np.fft.ifft(specdata_freq)
sldlightwave = np.concatenate((sldlightwave[-512:],sldlightwave[:512]))

sindex = [""]*1024
eversteptime = 1/(1024e+6) #每个ifft结果的时间间隔 单位us
for i in range(1024):
    t = (512 - i) * eversteptime
    t = t * lightspeed #单位 = us * m/ms = 0.001m = mm
    t = t * 10000 #转换为 0.1um
    sindex[i] = t
    #print(sindex[i])

print("理论上的光的波动形状图 x轴单位:0.1微米 0.1um")
#plt.figure()
plt.figure(figsize=(10,5))
plt.xlim(-150,150)
plt.plot(sindex,sldlightwave.real)

THORLABS SLD850S-A20W 提取的 波长vs强度 谱域特性曲线 (800-900nm)
转换为频谱图 x轴单位:1e+6 MHz
理论上的光的波动形状图 x轴单位:0.1微米 0.1um

如果TD-OCT光学系统在参考臂电机移动情况下光电二极管采样，应该会接收到下面模拟的波形(假设只有一个位置全反射的样品，反射的强度为100%)

print("相移后合成的波形")
srcdata = x
#srcdata = sldlightwave.real[256:768] #如果屏幕就使用正态函数乘以正弦波 的波形
phaseshift = 0
i = 0
start = -22
end = 22
cnt = end - start + 1 
plt.figure(figsize=(20,20))
amp = [0.0] * cnt
for phaseshift in range(start,end + 1,1):
    xshift = np.concatenate((srcdata[-phaseshift:], srcdata[:-phaseshift]))
    add = xshift + srcdata
    amp[i] = np.mean(np.sqrt(add**2))
    i = i + 1
    plt.subplot(cnt,1,i)
    plt.plot(add)
    plt.text(0,0,str(phaseshift))

#print("合成波形的强度图")
#plt.figure()
#plt.plot(amp)

相移后合成的波形

估计光电二极管转换为电信号应该是波形的强度(跟均根方值成比例吧)，所以我估计可能光电二极管转换为电信号实际采样到波形会像下面模拟的这样(假设只有一个全反射的样品，反射的强度为20%)

#print("相移后合成的波形")
phaseshift = 0
i = 0
start = -128
end = 128
cnt = end - start + 1 
#plt.figure(figsize=(20,20))
amp = [0.0] * cnt
amp1 = [0.0] * cnt
for phaseshift in range(start,end + 1,1):
    xshift = np.concatenate((x[-phaseshift:], x[:-phaseshift]))
    add = 0.2 * xshift + srcdata
    amp[i] = np.mean(np.sqrt(add**2))
    amp1[i] = np.mean(np.sqrt(np.power(add[u+start//2 : u + end//2 +1],2)))
    i = i + 1
    #plt.subplot(cnt,1,i)
    #plt.plot(add)
    #plt.text(0,0,str(phaseshift))

print("合成波形的强度图(均根方值)")
print("左边相当于采样宽度很宽，右边相当于模拟实际光电二极管采样响应(一段时间内有效)")
plt.figure()
plt.subplot(1,2,1)
plt.plot(amp)
plt.subplot(1,2,2)
plt.plot(amp1)

合成波形的强度图(均根方值)
左边相当于采样宽度很宽，右边相当于模拟实际光电二极管采样响应(一段时间内有效)

从模拟结果来看，两个图在等光程(相移为0)时的波形振幅是最强的，这样的话跟上面找到的文章里提到的原理是吻合吧。这里我估计实际波形应该会像右图吧(光电二极管可能有转换周期之类，或者ADC有采样周期之类)。

问同事说用希尔伯特变换可以复原深度反射信息
先查资料找到一个个人认为比较有用的信息：
https://www.bilibili.com/read/cv2793412/

网页上，希尔伯特变换说是 $\frac{1}{t}$傅里叶变换是 -j和j，我对$\frac{e^{-jwt}}{t}$不会求积分，所以就不看公式了。希尔伯特变换说是对信号低频率部分相移$-\frac{\pi }{2}$，高频部分相移$\frac{\pi }{2}$。
希尔伯特变换用在信号解调方面的应用原理如下：
假设我们有一个低频信号a(t)，用高频载波信号cos(wt)对其调制结果为 $a(t)cos(wt)$，如果我们能从$a(t)cos(wt)$经过滤波可以得到$a(t)sin(wt)$的话，就可以合成一个复数信号：
$a(t)cos(wt)+ja(t)sin(wt)=a(t)\ast (cos(wt)+jsin(wt))=a(t)\ast e^{jwt}$
看公式就知到，要解调出原始信号a(t)，只需要对复数信号 $a(t)cos(wt)+ja(t)sin(wt)=a(t)\ast e^{jwt}$ 乘以$e^{-jwt}$ 就行了。至于怎么样从调制信号 $a(t)cos(wt)$ 获得 $a(t)sin(wt)$ 应该就是希尔伯特变换显身手的地方了。
先来复习一下欧拉公式:$e^{jx}=cosx+jsinx$
再平复习一下共轭复数: $(x+jy)\ast (x-jy)=x^2+y^2$
又复习一下离散傅里叶: $$X(k)=\sum _{n=0}^{N-1}{x}_n\ast e^{\frac{-j2\pi kn}{N}}$$
离散傅里叶结果相对于$k=\frac{N}{2}$共轭对称的，所以如果$k^{\prime }=1,..\frac{N}{2}-1$,有：
$X(k^{\prime })=\frac{Am{p}^2(X(N-k^{\prime }))}{X(N-k^{\prime })}$ ,Amp()代表幅值

记$e^{\frac{-j2\pi kn}{N}}$为$W_N^{kn}$
$$X(k)=\sum _{n=0}^{N-1}{x}_n\ast e^{\frac{-j2\pi kn}{N}}=\sum _{n=0}^{N-1}{x}_n{W}_N^{kn}$$
由于$W_N^{-kn}=W_N^{-kn}\ast W_N^{nN}=W_N^{(N-k)n}$可知，当$k>\frac{N}{2}$时的X(k)是$k<\frac{N}{2}$负频率共轭复数，所以：
$X(k^{\prime })=\frac{Am{p}^2(X(N-k^{\prime }))}{X(N-k^{\prime })}=\frac{Am{p}^2(X(k^{\prime }))}{X(-k^{\prime })}$

将高频信号$cos(\frac{2\pi cn}{N})$表示成复数的形式:
$$cos(\frac{2\pi cn}{N})=\frac{e^{\frac{-j2\pi cn}{N}}+e^{\frac{j2\pi cn}{N}}}{2}=\frac{W_N^{cn}+W_N^{-cn}}{2}$$

原始信号$x_n$ 经高频信号 $cos(\frac{2\pi cn}{N})$ 调制后的信号为
$x_n\ast \frac{W_N^{cn}+W_N^{-cn}}{2}$，它的傅里叶变换为：
$$X^{\prime }(k)=\sum _{n=0}^{N-1}{x}_n\ast \frac{W_N^{cn}+W_N^{-cn}}{2}\ast W_N^{kn}$$
对其进行化简
$$\begin{array}{rl}{X}^{\prime }(k)& =\sum _{n=0}^{N-1}{x}_n\ast \frac{W_N^{cn}+W_N^{-cn}}{2}\ast W_N^{kn}\\ & =\frac{1}{2}\sum _{n=0}^{N-1}{x}_n(W_N^{cn}+W_N^{-cn})W_N^{kn}\\ & =\frac{1}{2}\sum _{n=0}^{N-1}{x}_n(W_N^{(k+c)n}+W_N^{(k-c)n})\\ & =\frac{1}{2}\sum _{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}+\frac{1}{2}\sum _{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}\end{array}$$

可以看到 被调制后的信号分成了 $\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}$和$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}$两部分

假设原始信号$x_n$只是含有m个低频频率的信号(m小于$\frac{N}{2}$)，那么它的傅里叶变换只含有$X(-m+1),X(-m+2)...X(-1),X(0),X(1),...X(m-1)$，其余频率X(m++或 – -m) 为0，

负频率$X(-m+1),X(-m+2)...X(-1)$
相当于$X(N-m+1),X(N-m+2)...X(N-1)$
换句话说$x_n$只含有0,1,2…m 和 N-m+1,N-m+2,N-m+3…N-1两部分的频谱

假设高频信号$cos(\frac{2\pi cn}{N})$ 的c远大于m又小于$\frac{N}{2}$，

对于 $\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}$

可知$c\le k+c<(N+\frac{N}{2})$：

当$c\le k+c\le N-m$ 时($c>m$),

$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}=\frac{1}{2}X(k+c)=0$

当 $N-m+1\le k+c\le N-1$ 时,

$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}=\frac{1}{2}X(k+c)$

当 $N\le k+c\le N+m-1$ 时($0\le k+c-N\le m-1$),

$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}=\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}{W}_N^{-nN}=\frac{1}{2}X(k+c-N)$

当 $N+m\le k+c<N+\frac{N}{2}$ 时($m\le k+c-N<\frac{N}{2}$),

$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}=\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}{W}_N^{-nN}=\frac{1}{2}X(k+c-N)=0$

总结来说，对于$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}$，

当($N-m+1\le k+c\le N-1$)及($N\le k+c\le N+m-1$)时求和的值才不为零；而且，

当($N-c-m+1\le k\le N-c-1$),由于$m<<c<\frac{N}{2}$时，这时的k总是在$(\frac{N}{2},N-1]$之间
$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}$ 的值为原始信号$x_n$傅里叶变换的X(N-m+1),X(N-m+2),X(N-m+3)…X(N-1)的二分之一

当($N-c\le k\le N-c+m-1$),由于$m<<c<\frac{N}{2}$时，这时的k总是在$(\frac{N}{2},N-1]$之间
$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}$ 的值为原始信号$x_n$傅里叶变换的X(0),X(1),X(2)…X(m-1)的二分之一

对于$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}$，它代表着被调制后的信号的大于$\frac{N}{2}$倍频的频谱(或者说是负频率)，而且它的值是由原始信号的频谱合成的，按这样的排序
$\frac{1}{2}X(N-m-1),\frac{1}{2}X(N-m+2),\frac{1}{2}X(N-m+3)\dots \frac{1}{2}X(N-1),\frac{1}{2}X(0),\frac{1}{2}X(1),\frac{1}{2}X(2)\dots X(m-1)$
在调制后信号频谱的N-c频率处为中心进行放置

对于 $\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}$

可知$-c\le k-c<(N-\frac{N}{2})$：

当$-c\le k-c\le -m$ 时($N-c\le N+k-c\le N-m$)($c>m,c<\frac{N}{2}$),

$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}=\frac{1}{2}X(k-c)=0$

当$-m+1\le k-c<0$ 时($N-m+1\le N+k-c<N$)($c>m,c<\frac{N}{2}$),

$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}=\frac{1}{2}X(k-c)$

对应序列$\frac{1}{2}X(N-m+1),\frac{1}{2}X(N-m+2)...\frac{1}{2}X(N-1)$

当$0\le k-c<m$ 时($c>m,c<\frac{N}{2}$),

$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}=\frac{1}{2}X(k-c)$

对应序列$\frac{1}{2}X(0),\frac{1}{2}X(1)...\frac{1}{2}X(m-1)$

当$m\le k-c<N-\frac{N}{2}$ 时($c>m,c<\frac{N}{2}$),

$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}=\frac{1}{2}X(k-c)=0$

总结来说，对于$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}$，

当($-m+1\le k-c<0$)及($0\le k-c<m$)时求和的值才不为零；
也就是说($c-m+1\le k<c$)及($c\le k<c+m$)时，才有与原始信号频谱相关的值。

对于$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}$，它代表着被调制后的信号的小于$\frac{N}{2}$倍频的频谱，而且它的值是由原始信号的频谱合成的，按这样的排序
$\frac{1}{2}X(N-m+1),\frac{1}{2}X(N-m+2),\frac{1}{2}X(N-m+3)\dots \frac{1}{2}X(N-1),\frac{1}{2}X(0),\frac{1}{2}X(1),\frac{1}{2}X(2)\dots X(m-1)$
在调制后信号频谱的c频率处为中心进行放置

所以对于调制后的信号

其傅里叶变换可化为
$$\begin{array}{rl}{X}^{\prime }(k)& =\sum _{n=0}^{N-1}{x}_n\ast \frac{W_N^{cn}+W_N^{-cn}}{2}\ast W_N^{kn}\\ & =\frac{1}{2}\sum _{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}+\frac{1}{2}\sum _{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}\end{array}$$
$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}$代表着正半频率频谱，$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}$代表着负半频率频谱(或者说是大于$\frac{N}{2}$倍频的频谱)
而希尔伯特变换相当于正半频谱乘以-j，负半频谱乘以j,见下面推导：
$$\begin{array}{rl}H(X^{\prime }(k))& =j\frac{1}{2}\sum _{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}-j\frac{1}{2}\sum _{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}\\ & =j\frac{1}{2}\sum _{n=0}^{N-1}{x}_n{W}_N^{kn}{W}_N^{cn}-j\frac{1}{2}\sum _{n=0}^{N-1}{x}_n{W}_N^{kn}{W}_N^{-cn}\\ & =\sum _{n=0}^{N-1}{x}_n\frac{j{W}_N^{cn}-j{W}_N^{-cn}}{2}{W}_N^{kn}\end{array}$$
由上面的推度可知道
调制后的信号,傅里叶变换->希尔伯特变换->傅里叶逆变换 得到的是
$$x_n\frac{j{W}_N^{cn}-j{W}_N^{-cn}}{2}$$
而
$$\begin{array}{rl}\frac{j{W}_N^{cn}-j{W}_N^{-cn}}{2}& =\frac{j{e}^{\frac{-j2\pi cn}{N}}-j{e}^{\frac{j2\pi cn}{N}}}{2}\\ & =\frac{1}{2}(j(cos(\frac{2\pi cn}{N})-jsin(\frac{2\pi cn}{N})-cos(\frac{2\pi cn}{N})-jsin(\frac{2\pi cn}{N})))\\ & =\frac{1}{2}(j(-2\ast j\ast sin(\frac{2\pi cn}{N})))\\ & =sin(\frac{2\pi cn}{N})\end{array}$$
也就是
$$x_n\frac{j{W}_N^{cn}-j{W}_N^{-cn}}{2}=x_{n}sin(\frac{2\pi cn}{N})$$

对$x_n$通过高频载波信号$cos(\frac{2\pi cn}{N})$调制后的信号$x_{n}cos(\frac{2\pi cn}{N})$进行解调方法：

• 对$x_{n}cos(\frac{2\pi cn}{N})$进行傅里叶变换得到$X(k)$
• 对$k<\frac{N}{2}$, $X^{\prime }(k)=-jX(k)$
• 对$k>\frac{N}{2}$, $X^{\prime }(k)=jX(k)$
• $X^{\prime }(k)$进行傅里叶逆变换得到解析信号$x_{n}sin(\frac{2\pi cn}{N})$
• 将调制后的信号和解析信号合成复数信号$x_n^{\prime }=x_{n}cos(\frac{2\pi cn}{N})+j{x}_{n}sin(\frac{2\pi cn}{N})$
• 对合成的复数信号乘以$e^{\frac{-j2\pi cn}{N}}$，相乘后的结果实部原始信号$x_n$了
• 如果不知道高频调制信号的频率c,可以对复数信号每个点求复数的幅值就是原始信号的幅值了

下面用python进行这个过程试验

a = [0] * 512
c = [0] * 512
data = [0] * 512
for i in range(512):
    #a[i] = i / 511
    sigma = 10
    u = 256
    a[i] = 1/(math.sqrt(2 * math.pi)*sigma) * math.exp(-(i-u)**2 / (2 * sigma**2))
    #a[i] = 1.0 - math.cos(2 * math.pi * i /511)
    c[i] = math.cos(2 * math.pi * 100 * i /511)
    data[i] = a[i] * c[i]

datafft = np.fft.fft(data)
    
plt.figure(figsize=(20,10))
plt.subplot(311)
plt.plot(np.abs(np.fft.fft(a)))

plt.subplot(312)
plt.plot(np.abs(np.fft.fft(c)))

plt.subplot(313)
plt.plot(np.abs(np.fft.fft(data)))


for i in range(1,256): # 希尔伯特变换 Hilbert transform
    datafft[i] = datafft[i] * (0-1j)
    datafft[i+256] = datafft[i+256] * (0+1j)
datashift = np.fft.ifft(datafft) # 希尔伯特变换 Hilbert transform


#print(datashift)
plt.figure(figsize=(20,5))
plt.plot(data)
plt.plot(datashift.real)

datacomplex = [0+0j]*512
for i in range(512):  #生成解释信号波型
    datacomplex[i] = data[i] + (0+1j) * datashift[i]

dataz = [0+0j]*512
for i in range(0,512):  #乘以e^-jwt 获取原始被调制信号
    dataz[i] = datacomplex[i] * cmath.exp((0-1j) * 2 * math.pi * 100 * i/511) 

plt.figure(figsize=(20,10))
plt.subplot(211)
plt.plot(a)
plt.subplot(212)
plt.plot(a)
plt.plot(data)
plt.plot(np.real(dataz)) #这个是在知道调制信号的频率的情况下才有用的
#plt.plot(np.abs(datacomplex)) #不知道调制信号，且只要求得到原始信号的幅值

可以看到代码执行后的结果与公式推导是吻合的。

用希尔伯特变换的方法对TD-OCT模拟的信号解调

#orgsignal = data
orgsignal = amp
#print(np.mean(orgsignal))
submean = np.array(orgsignal,dtype=np.float) - np.mean(orgsignal)
plt.plot(submean)

#进行 hilbert变换
hilbertfft = np.fft.fft(submean) 
hcnt = len(hilbertfft)//2

for i in range(hcnt):
    hilbertfft[i] = 0-1j * hilbertfft[i]
    hilbertfft[i+hcnt] = 0+1j * hilbertfft[i+hcnt]

#hilbertfft[0] = 0 #不要零频
#hilbertfft[hcnt] = 0

hilbertsignal = np.fft.ifft(hilbertfft)
plt.plot(hilbertsignal.real)

complexsignal = submean +  (0+1j * hilbertsignal)
decodesignal = np.abs(complexsignal)

plt.figure()
plt.plot(decodesignal)

#如像解调出来的波形并不理想
#进行均值滤波 效果好一点
cnvcnt = 7
cnvkernel = [1/cnvcnt]*cnvcnt
cnvsignal = np.convolve(decodesignal,cnvkernel,'same') #valid same full

plt.figure()
plt.plot(cnvsignal)

#其实好像直接用原信号(没有负数的信号)进行平均滤波效果更好 但对于实际TD-OCT采集的信号是不是也不好说
cnvsignal1 = np.convolve(orgsignal,cnvkernel,'same') #valid same full
plt.figure()
plt.plot(0.5-cnvsignal1)

希尔伯特变换 再深入

对于调制后的信号 $x_n^{\prime }=x_n\ast cos(\frac{2\pi cn}{N})$，

假设原始信号$x_n$只是含有m个低频频率的信号(m小于$\frac{N}{2}$)，那么它的傅里叶变换只含有$X(-m+1),X(-m+2)...X(-1),X(0),X(1),...X(m-1)$，其余频率X(m++或 – -m) 为0，

负频率$X(-m+1),X(-m+2)...X(-1)$
相当于$X(N-m+1),X(N-m+2)...X(N-1)$
换句话说$x_n$只含有0,1,2…m 和 N-m+1,N-m+2,N-m+3…N-1两部分的频谱
假设高频载波信号$cos(\frac{2\pi cn}{N})$ 的c大于m又小于$\frac{N}{2}$

调制后的信号傅里叶变换可化为
$$\begin{array}{rl}{X}^{\prime }(k)& =\sum _{n=0}^{N-1}{x}_n\ast \frac{W_N^{cn}+W_N^{-cn}}{2}\ast W_N^{kn}\\ & =\frac{1}{2}\sum _{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}+\frac{1}{2}\sum _{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}\end{array}$$
$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k-c)n}$代表着正半频率频谱，$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_n{W}_N^{(k+c)n}$代表着负半频率频谱(或者说是大于$\frac{N}{2}$倍频的频谱)
而希尔伯特变换相当于正半频谱乘以-j，负半频谱乘以j,
$Hilbert(x_n^{\prime })=Hilbert(x_n\ast cos(\frac{2\pi cn}{N}))=x_{n}cos(\frac{2\pi cn}{N}-\frac{\pi }{2})=x_{n}sin(\frac{2\pi cn}{N})$

如果载波信号变为 $C\ast cos(\frac{2\pi cn}{N}+\psi )$ 其中C为幅值, c相当于频率，$\psi$为相位
代入上面的公式依然有：
$$\begin{array}{rl}{X}^{\prime }(k)& =\sum _{n=0}^{N-1}{x}_n\ast C\ast \frac{W_N^{(cn+N\psi /2\pi )}+W_N^{-(cn+N\psi /2\pi )}}{2}\ast W_N^{kn}\\ & =\frac{1}{2}\sum _{n=0}^{N-1}{x}_n\ast C\ast (W_N^{cn}{e}^{-j\psi }+W_N^{-cn}{e}^{j\psi })\ast W_N^{kn}\\ & =\frac{1}{2}\sum _{n=0}^{N-1}{x}_{n}C{W}_N^{(k+c)n}{e}^{-j\psi }+\frac{1}{2}\sum _{n=0}^{N-1}{x}_{n}C{W}_N^{(k-c)n}{e}^{j\psi }\\ & =\frac{1}{2}(cos\psi -jsin\psi )\sum _{n=0}^{N-1}{x}_{n}C{W}_N^{(k+c)n}+\frac{1}{2}(cos\psi +jsin\psi )\sum _{n=0}^{N-1}{x}_{n}C{W}_N^{(k-c)n}\\ & =\frac{1}{2}\left(\sum _{n=0}^{N-1}{x}_{n}cos\psi C{W}_N^{(k+c)n}-j\sum _{n=0}^{N-1}{x}_{n}sin\psi C{W}_N^{(k+c)n}\right)+\frac{1}{2}\left(\sum _{n=0}^{N-1}cos\psi x_{n}C{W}_N^{(k-c)n}+j\sum _{n=0}^{N-1}{x}_{n}sin\psi C{W}_N^{(k-c)n}\right)\end{array}$$
$x_n$乘以常数，频谱是不会变化的,k+c 和 k-c的特性还是没有改变，所以$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_{n}C{W}_N^{(k-c)n}{e}^{j\psi }$代表着正半频率频谱，$\frac{1}{2}{\sum }_{n=0}^{N-1}{x}_{n}C{W}_N^{(k+c)n}{e}^{-j\psi }$代表着负半频率频谱(或者说是大于$\frac{N}{2}$倍频的频谱)