转载（http://hi.baidu.com/zhaolinger_789/blog/category/%CA%FD%D7%D6%D0%C5%BA%C5%B4%A6%C0%ED）   
    假设一个信号，它含有2V的直流分量，频率为50Hz、相位为-30度、幅度为3V的交流信号，以及一个频率为75Hz、相位为90度、幅度为1.5V的交流信号。用数学表达式就是如下：S=2+3*cos(2*pi*50*t-pi*30/180)+1.5*cos(2*pi*75*t+pi*90/180)    式中cos参数为弧度，所以-30度和90度要分别换算成弧度。我们以256Hz的采样率对这个信号进行采样，总共采样256点。MATLAB程序如下：
clc;

clear;

Adc=2; %直流分量幅度

A1=3;   %频率F1信号的幅度

A2=1.5; %频率F2信号的幅度

F1=50; %信号1频率(Hz)

F2=75; %信号2频率(Hz)

Fs=256; %采样频率(Hz)

P1=-30; %信号1相位(度)

P2=90; %信号相位(度)

N=256; %采样点数

t=[0:1/Fs:N/Fs]; %采样时刻
%信号

S=Adc+A1*cos(2*pi*F1*t+pi*P1/180)+A2*cos(2*pi*F2*t+pi*P2/180);

%显示原始信号

subplot(411);plot(S);

title('原始信号');



Y = fft(S,N); %做FFT变换

Ayy = (abs(Y)); %取模

subplot(412);stem(Ayy(1:N)); %显示原始的FFT模值结果

title('FFT 模值');
Ayy=Ayy/(N/2);   %换算成实际的幅度

Ayy(1)=Ayy(1)/2;

F=([1:N]-1)*Fs/N; %换算成实际的频率值，Fn=(n-1)*Fs/N

subplot(413);stem(F(1:N/2),Ayy(1:N/2));   %显示换算后的FFT模值结果

title('幅度-频率曲线图');
Pyy=[1:N/2];

for i=1:N/2

Pyy(i)=angle(Y(i)); %计算相位

Pyy(i)=Pyy(i)*180/pi; %换算为角度

end;

subplot(414);stem(F(1:N/2),Pyy(1:N/2));   %显示相位图

title('相位-频率曲线图');

 

 

FFT结果：
   Y(1)=512

   Y(51)=3.3255e+002 -1.9200e+002i

   Y(76)=3.4350e-012 +1.9200e+002i


   

FFT幅值结果：

   abs(Y(1))=512

   abs(Y(51))=384

abs(Y(76))=192

FFT相位结果：

   angle(Y(1))=0

   angle(Y(51))=-0.5236 rad=-30.000°

    angle(Y(76))=1.5708 rad= 90.000°

另外：atan2(b,a)是求坐标为(a,b)点的角度值，范围从-pi到pi。例如：

    atan2(0,512)=0
    atan2(-1.9200e+002,3.3255e+002 )=-0.5236 rad=-30.000°
   

求具体信号的幅值：

   Adc= abs(Y(1))/N=512/256=2             %直流分量是仅除以N
   A1= abs(Y(51))/N*2=384/256*2=3      %交流分量是除以N后还要乘以2
   A2= abs(Y(76))/N*2=192/256*2=1.5
				
转载于:https://blog.51cto.com/kurtlen/386312

	

导读
天津大学精仪学院姚建铨院士团队在《激光与光电子学进展》发表了题为“多晶材料随机准相位匹配研究进展”的综述文章，被选为当期的封面文章。
文中概括了随机准相位匹配技术的发展过程，对近年来超快激光作用多晶陶瓷通过随机准相位匹配技术混频的一些典型实验进行了详细介绍，并探讨了随机准相位匹配技术常用的材料——多晶陶瓷的处理、建模问题，最后展望了该技术的发展前景。
封面文章| 刘科飞,钟凯,姚建铨. 多晶材料随机准相位匹配研究进展[J]. 激光与光电子学进展, 2020, 57(21): 210001
撰稿| 刘科飞 钟凯(天津大学)
01
光学频率变换技术的背景
激光入射到非线性晶体中时，强大的光电场会使电子的发生振荡，使之极化，会发出与激光频率不同频率的谐波，除了与激光频率相同的基次谐波(线性极化)，另一种是二次极化谐波(非线性极化)，它的频率是激光频率的两倍，由它在晶体中相干增长产生的光束便称为倍频光，激发这束倍频光的激光称为基频光。从量子的观点来看，这是通过晶体的虚能级使两个低能量光子合并为一个高能量光子的过程。若使基频光的能量源源不断地转换到倍频光，需要保持基频光激发的二次极化谐波和倍频光在晶体中位置时时刻刻相同，然而由于晶体的色散导致基频光和倍频光的折射率不同，进而导致两束光在晶体中群速度不同，无法实现倍频光的持续增长，一般情况下，仅在几十微米的相干长度内持续增长，此时相位失配最大，之后倍频光在下一个相干长度内会不断减弱，以此形成倍频光的周期振荡。为解决这一问题而发展的非线性光学相位匹配技术，可使直接发出的激光波长高效地转换到各个波段。利用各向异性晶体的双折射特性，使一定偏振的基频光沿晶体的特定方向入射，或者改变晶体的温度，实现角度或者温度相位匹配(PM)——即使基频光和倍频光在晶体中特定方向传播时的折射率相同；另一种技术，倍频光在晶体中传播一个相干长度开始减弱时，反转晶体方向，重新使相位失配为零，可以使基频光能量持续转换到倍频光，这便是依赖周期反转晶体的准相位匹配(QPM)技术。上述倍频过程的相位匹配技术，在差频、和频等二阶非线性过程中是相通的。尽管传统的PM，QPM技术可以实现高效的转换，但是PM过程往往对晶体的要求苛刻，并不是所有激光波长和任意晶体都可以满足角度或者温度PM条件，QPM器件制作工艺复杂，此外，由于PM及QPM往往具有较小的允许带宽，难以在任意波长处实现超短脉冲激光泵浦的超宽带频率变换，这些不足限制了其在中红外光谱检测，光频率梳等需要宽带光源的应用。各向同性的材料没有双折射效应，能不能实现相位匹配呢？答案是肯定的，需要通过随机准相位匹配技术实现。
02
随机准相位匹配简介
随机准相位匹配(RQPM)的原型可以追溯到六十年代测量非线性材料的粉末倍频实验，1968年，Kurtz和Perry以晶质石英为基准，通过粉末倍频分析技术研究了大量材料的非线性系数的大小，当时实验所得到的一些结论，与RQPM的结论相符合。2004年，M. Baudrier-Raybaut等人研究了多晶陶瓷材料的差频，提出了随机准相位匹配(Random QPM)的概念。如图1(d)所示，多晶陶瓷随机的晶畴结构分布，导致了相位的随机波动，相对相位的随机波动过程带来的RQPM，可以使光学各向同性的材料一定程度上实现相位匹配，实现信号光的增长[1]。RQPM过程不需要入射光特殊的偏振态，无需制作特定周期畴结构的极化材料，最重要的是，允许三波互作用发生在非常大的带宽中，因此RQPM过程具有宽带响应特性，可以在很宽的波段内实现激光的频率变换。PM和QPM需要依赖特殊切向的晶体，或者周期反转结构的器件，RQPM利用的是材料的无序畴结构，近些年，多晶材料以其独特的性质在RQPM光学频率变换领域越来越多的关注。图1  (a) 相位失配下，光场周期振荡；(b) 完全无序的气体液体，无法产生相干辐射；(c) 准相位匹配过程中，光矢量线性增长；(d) 多晶材料中，光矢量缓慢增长
03
多晶材料RQPM的研究现状
3.1 多晶材料的优势以ZnSe为代表的一类红外半导体材料，具有良好的物理、光学及机械特性，是最理想的产生中红外激光的非线性材料，相对于PM或者QPM的晶体，其价格低廉，易于获得。不过多晶材料RQPM的平均非线性增益较低，纳秒激光泵浦时转换效率远低于常规PM和QPM，因此一直没有实用化的器件产生。对于飞秒激光泵浦的宽带OPO，一方面由于需要严格控制色散，通常采用厚度很小的非线性材料，大大缩小了RQPM与PM及QPM的非线性增益差距；另一方面，材料对飞秒激光的抗损伤性能比纳秒激光高三个数量级，可以通过紧聚焦极大地提高泵浦光功率密度，增强非线性增益。3.2 多晶陶瓷的RQPM OPOOPO通过谐振腔使参量信号多次反馈振荡来增强非线性效应，可以实现很高的转换效率。多晶材料OPO的实现，真正地证明了RQPM的实用性。2017年，美国中佛罗里达大学Ru等对多晶ZnSe进行了倍频特性测试，发现倍频效率统计均值与材料厚度成正比，且与QPM的定向外延生长的砷化镓(OP-GaAs)晶体相比，多晶ZnSe样品的某些区域倍频效率能够接近OP-GaAs。以倍频测试结果作为指导，使用Cr:ZnS飞秒激光同步泵浦基于多晶ZnSe的OPO，实现了宽带中红外激光输出[2]。该OPO的实验装置如图2(a)所示的蝶形腔结构，泵浦阈值小于100mW，输出功率可达20mW，输出光谱覆盖3-7.5μm。各项指标均接近OP-GaAs OPO的水平。图2 (a)ZnSe多晶陶瓷随机准相位匹配OPO的装置示意图，右下角为布儒斯特角放置ZnSe的照片；(b)输出的3-7.5μm的宽带光谱；(c)腔长失谐图包括ZnSe在内的大量红外半导体材料如ZnSe、ZnS、ZnTe、ZnO、GaP、GaN等，这些物化性能优异的材料却无法制成实用的QPM器件，多晶材料的RQPM OPO为解决上述问题提供了一种可行的方法。3.3 多晶基质RQPM混频产生宽带光谱IPG Photonics公司在超短脉冲Cr:ZnS/ZnSe中红外激光器领域开展了大量的研究工作。2017年，Vasilyev等人制作了EDLF泵浦的中红外Cr:ZnS/Cr:ZnSe的超短脉冲激光器，不同晶粒尺寸的Cr:ZnS飞秒激光振荡器中最高观测到四次谐波信号，并在放大器中通过RQPM混频实现了覆盖1.8-4.5μm的超连续谱，如图3所示[3]。图3 多晶Cr:ZnS飞秒激光器的RQPM谐波及超连续谱。(a) 基频光的一至四次谐波；(b) 1.8-4.5μm超连续谱2019年，使用飞秒Cr:ZnS激光器搭建了中红外光频梳系统，多晶基质的RQPM光谱宽度覆盖两个倍频程，使测量载波包络偏移频率更为便利[4]。并在GaSe和ZGP晶体内通过脉冲内不同光谱分量的混频将超宽带光谱拓展到近18μm的远红外[5-6]。同时，德国马普所的Zhang等人利用Ho:YAG飞秒激光泵浦多晶ZnSe/ZnS 材料，利用RQPM混频实现2.7-20μm的超宽带光谱，平均功率达到了16mW，达到与双折射晶体转换效率相当的水平[7]。如上所述，近年来超快激光技术的快速发展开启了研究多晶材料RQPM非线性光学频率变换理论与技术的新篇章。3.4 RQPM理论与多晶材料处理多晶的RQPM过程受到晶粒不规则形态的影响，且其有效非线性系数与晶粒的晶向随机分布密切相关。晶向导致有效非线性系数的随机分布，图4所示为光束与多晶作用的示意图，不同的颜色表示不同的晶粒。图4 一定截面积的光束通过多晶材料示意图。(a)主视图；(b)侧视图近年来，RQPM理论都在试图逼真地还原上述的模型。Kawamori等使用材料领域成熟的软件生成多晶模型，经过网格化获得任意空间维度的晶粒随机分布情况，从二阶极化率张量变换的角度描述非线性系数的随机变化，较准确地还原了上述物理模型[8]。图5为我们使用Neper生成的多晶模型及其网格化的形态。图5 利用Neper生成的多晶材料及其网格化的形态通过蒙特卡洛计算，Kawamori证实了信号光强随互作用长度线性增长的结论，生成的功率分布直方图，与实验数据相一致。多晶陶瓷由晶粒生长，烧结，压制而成，Chen等通过大量的实验，系统研究了ZnSe陶瓷的处理过程对晶粒尺寸的影响[9]，证明在目前商用多晶陶瓷材料的基础上，经过高温处理及退火能够满足绝大部分三波互作用过程的RQPM条件。
04
展望
得力于超快激光技术的快速发展，以多晶ZnSe为代表的红外半导体材料，结合RQPM技术，可为超宽带中红外连续谱及光频梳产生等热门研究领域提供一种相对低成本的新途径。今后的研究中，进一步完善RQPM基本理论，期望可以推出实用化RQPM商用的器件，随着各种问题的解决，多晶材料RQPM有望广泛应用于紫外，到太赫兹波段的光源。
参考文献
[1] Baudrier-Raybaut M, Haidar R, Kupecek P, et al. Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials[J]. Nature, 2004, 432: 374-376.
[2] Ru Q, Lee N, Chen X, et al. Optical parametric oscillation in a random polycrystalline medium[J]. Optica, 2017, 4(6): 617-618.
[3] Vasilyev S, Moskalev I, Mirov M, et al. Ultrafast middle-IR lasers and amplifiers based on polycrystalline Cr:ZnS and Cr:ZnSe[J]. Optical Materials Express, 2017, 7(7): 2636-2650.
[4] Vasilyev S, Smolski V, Peppers J, et al. Middle-IR frequency comb based on Cr: ZnS laser[J]. Optics Express, 2019, 27(24): 35079-35087.
[5]  Vasilyev S, Moskalev I S, Smolski V O, et al. Super-octave longwave mid-infrared coherent transients produced by optical rectification of few-cycle 2.5-μm pulses[J]. Optica, 2019, 6(1): 111-114.
[6] Vasilyev S, Moskalev I, Smolski V, et al. Multi-octave visible to long-wave IR femtosecond continuum generated in Cr: ZnS-GaSe tandem[J]. Optics Express, 2019, 27(11): 16405-16413.
[7] Zhang J, Fritsch K, Wang Q, et al. Intra-pulse difference-frequency generation of mid-infrared (2.7–20μm) by random quasi-phase-matching[J]. Optics Letters, 2019, 44(12): 2986-2989.
[8] Kawamori T, Ru Q, Vodopyanov K L. Comprehensive Model for Randomly Phase-Matched Frequency Conversion in Zinc-Blende Polycrystals and Experimental Results for ZnSe[J]. Physical Review Applied, 2019, 11(5): 054015.
[9] Chen X, Gaume R. Non-stoichiometric grain-growth in ZnSe ceramics for χ(2) interaction[J]. Optical Materials Express, 2019, 9(2): 400-409.
课题组介绍
天津大学激光与光电子研究所(姚建铨院士团队)，是“光学工程”国家重点学科、光电信息技术教育部重点实验室以及微光机电系统技术教育部重点实验室(B类)的重要组成部分。
主要从事激光与光电子技术研究，具体包括全固态激光及非线性光学频率变换技术、高性能光纤激光技术、太赫兹光子学、激光及太赫兹技术应用、微纳光电子技术及器件、光纤传感技术等。
近十年来承担973、863、国家重点研发计划、国家自然科学基金等项目及课题30余项，取得了丰硕的科研成果，曾获国家发明二等奖、天津科技进步二等奖、军队科技进步一等奖和中科院特等奖等，在国内外重要刊物发表论文1200 余篇，授权发明专利 40 余项。
推荐阅读：陶绪堂教授：领跑世界的“中国牌”晶体光学青年| 刘兆军教授与晶体光纤激光技术研究，做核心材料和器件亮点 | 基于ADP晶体的钕玻璃激光高效五次谐波转换技术
END
由于微信公众号试行乱序推送，您可能没办法准时收到“爱光学”的文章。为了让您第一时间看到“爱光学”的新鲜推送， 请您：
1. 将“爱光学”点亮星标(具体操作见文末)
2. 多给我们点“在看”
欢迎爆料
新闻线索、各类投稿、观点探讨、故事趣事
留言/邮件，我来让你/事红
爆料请联系：lvxuan@siom.ac.cn
点在看联系更紧密
写留言
	

 
01简介

使用普通的万用表测量交流信号的时候，通常会遇到 万用表的频率响应 的问题。使用可以联网的示波器可以获得它采集到的数据，进而可以计算出所测量的交流信号的有效值和相位。
这里通过实验来确定使用示波器测量交流信号的参数方法中的以下问题：

测量的精度有多高？ 测量的频率范围有多宽？
获得高精度的测量参数，对于示波器参数如何设置？

▲ 实验所使用的示波器：ds6104

实验所使用的DS6104是一款可以通过联网读取各通道采集数据的示波器，为自动化测量提供了很好的支持。对它的 DS6104局域网络编成接口 可以从网络中下载相关的文档。
DS6104对于每个通道采样数据点个数$N = 1400$，采样的有效位数为8bit。下面显示了DS6104采集一个有效值为$E = 2.39V$，频率为$f = 1000Hz$的正弦波波形。该信号是由 DS345 信号发生器产生。该信号发生器同时还可以产生同相的同步方波信号，可以便于示波器用于同步采集信号的波形。

▲ 通过DS6104读取的信号发生器所产生的1000Hz信号及其同步方波信号

 
02测量方法

假设被测量信号的频率为$f$，有效值为$E$，那么信号的表达式为：$f\left( t \right) = \sqrt 2  \cdot E \cdot \sin \left( {2\pi f \cdot t + \theta } \right)$示波器采样时间间隔为$T_s$，采样点个数$N$。所得到的采样数据为：$f\left[ n \right] = \sqrt 2  \cdot E \cdot \sin \left( {2\pi f \cdot nT_s  + \theta } \right),\,\,\,\,n = 0,1,...,N - 1$
从数据中计算信号的$E,\theta$，采用以下方法。首先计算序列与$\cos \left[ {nT_s } \right],\sin \left[ {nT_s } \right]$的投影（内积）：

$Y_s  = {1 \over N}\sum\limits_{n = 0}^{N - 1} {f\left[ n \right] \cdot \sin \left( {2\pi f \cdot nT_s } \right)}$$Y_c  = {1 \over N}\sum\limits_{n = 0}^{N - 1} {f\left[ n \right] \cdot \cos \left( {2\pi f \cdot nT_s } \right)}$
将上述两个公众是的$f\left[ n \right]$代入，并进行化简。对于$Y_s$可以化简如下：

$Y_s  = {1 \over N}\sum\limits_{n = 0}^{N - 1} {\sqrt 2  \cdot E \cdot \sin \left( {2\pi f \cdot nT_s  + \theta } \right) \cdot \sin \left( {2\pi f \cdot nT_s } \right)}$$= {{\sqrt 2  \cdot E} \over N}\sum\limits_{n = 0}^{N - 1} {{1 \over 2}\left[ {\cos \left( \theta  \right) - \cos \left( {4\pi f \cdot nT_s  + \theta } \right)} \right]}$$= {E \over {\sqrt 2 }}\cos \left( \theta  \right) + {E \over {\sqrt 2  \cdot N}}\sum\limits_{n = 0}^{N - 1} {\cos \left( {4\pi f \cdot nT_s  + \theta } \right)}$
根据 等差角度cos，sin序列之和公式 ：

$\sum\limits_{n = 0}^{N - 1} {\cos \left( {a + nd} \right)}  = R \cdot \cos \left[ {a + \left( {N - 1} \right){d \over 2}} \right]$$\sum\limits_{n = 0}^{N - 1} {\sin \left( {a + nd} \right)}  = R \cdot \sin \left[ {\alpha  + \left( {N - 1} \right){d \over 2}} \right]$
其中：$R = {{\sin \left( {{{N \cdot d} \over 2}} \right)} \over {\sin \left( {{d \over 2}} \right)}}$
当$\sin \left( {{d \over 2}} \right) = 0$时，$R = N$。

对于前序列采样公式，$d = 4\pi f \cdot T_s$，N=1400。下面假设f=1，对ts取不同值绘制出R对应的曲线：

▲ ts的不同取值下的R的取值，其中N=1400

可以看出，当ts取(0.5)整数倍数的时候，对应的R值很大。当ts偏离(0.5)整数倍数时，R迅速减少。下面给出Ts取0~0.之间的值对应的R的曲线：

▲ ts取值0~0.1之间的数值时对应的R取值

▲ ts取值0.45~0.55之间时，对应的R的取值

从上面分析可以看出，当ts取值范围在$0.1{1 \over f}$到$0.45{1 \over f}$之间时，对应正弦波每个周期采样点在2.5~10之间，R的取值都会远远小于N。因此，$Y_s$就可以近似为：

$Y_s  \approx {E \over {\sqrt 2 }}\cos \left( \theta  \right)$
同样的分析可以知道：$Y_c  \approx {E \over {\sqrt 2 }}\sin \left( \theta  \right)$
然后在计算信号的幅值和相位：

$E = \sqrt 2 \cdot\sqrt {Y_s^2  + Y_c^2 }$

$\theta  = \arctan \left( {{{Y_c } \over {Y_s }}} \right)$
def data2ap(tt, y, freq):
    sint = [sin(t * 2 * pi * freq) for t in tt]
    cost = [cos(t * 2 * pi * freq) for t in tt]
    ys = sum([a * b for a,b in zip(y, sint)]) / len(tt)
    yc = sum([a * b for a,b in zip(y, cost)]) / len(tt)
    amp = sqrt(ys**2 + yc**2)*sqrt(2)
    phase = atan2(ys,yc)
    return amp,phase


通过对一个1000Hz测试正弦波的测量，所获得的测量结果为：E=2.4382V, $\theta  = 1.3108$。

示波器的设置：时基时间T$Tb = 500\mu s$，幅值尺度$V_b  = 2V$。
使用FLUKE45数字万用表的交流档测量该交流信号的有效值为：$E_f  = 2.394V$。可以看出与示波器所测得得到的结果之间有1.85%的误差。
 
03测量性能

1.测量数值精度
测量100次信号的幅值和相位，统计相应的平均值和方差：

幅值：$E_{mean}  = 2.4383V,\,\,\,\,Var\left( E \right) = 3.51 \times 10^{ - 7}$
相位：$\theta _{mean}  = 1.3112,\,\,Var\left( \theta  \right) = 6.2134 \times 10^{ - 8}$

▲ 测量100个数据的分布

2.示波器时基长度对测量结果的影响
（1）直接采集信号进行测量
设置示波器的时基参数，可以改变数据采集时间间隔，同时也影响了每帧数据（对于DS6104，数据长度为1400）对应信号的周期个数。

▲ 不同时基下采集到的信号波形

下面显示了对于时基信号从30us变化到5000us过程中，200个信号的幅值和相位的测量数据。

▲ 不同的时下测量幅度和相位的变化

在上面分析由于不同时基的变化，对应采样时间的变化对于测量结果的影响，在ts小于0.002秒之前似乎还能够通过理论分析得到这个结果。但是对于ts大于0.002秒之后的这种变化，感觉就无法从理论上来分析了。
在博文 对示波器测量正弦波幅值和相位仿真实验 中通过理论仿真来验证不同时基ts下的测量误差，可以看到在前面ts大于0.002秒之后的部分的确不在理论可以分析的范围之内。所以这部分误差需要在实际测量中加以避免。
a=[2.13,2.84,2.27,2.40,2.60,2.32,2.45,2.52,2.35,2.47,2.48,2.36,2.48,2.45,2.37,2.48,2.44,2.39,2.48,2.42,2.41,2.48,2.41,2.43,2.47,2.40,2.41,2.45,2.40,2.44,2.43,2.38,2.44,2.44,2.41,2.42,2.41,2.40,2.45,2.42,2.42,2.43,2.40,2.42,2.45,2.42,2.40,2.43,2.40,2.43,2.45,2.38,2.42,2.43,2.41,2.44,2.40,2.39,2.43,2.42,2.42,2.45,2.39,2.40,2.44,2.42,2.43,2.41,2.40,2.42,2.43,2.42,2.40,2.41,2.40,2.43,2.43,2.43,2.41,2.40,2.24,2.40,2.24,2.37,2.24,2.36,2.20,2.37,2.18,2.34,2.42,2.32,2.42,2.32,2.40,2.31,2.40,2.28,2.41,2.27,2.38,2.25,2.38,2.22,2.38,2.21,2.36,2.20,2.34,2.17,2.34,2.41,2.32,2.41,2.30,2.41,2.30,2.39,2.28,2.39,2.25,2.39,2.24,2.37,2.23,2.36,2.20,2.36,2.18,2.34,2.17,2.32,2.41,2.32,2.40,2.30,2.40,2.27,2.40,2.27,2.38,2.25,2.37,2.22,2.38,2.21,2.35,2.19,2.34,2.16,2.34,2.41,2.32,2.41,2.29,2.41,2.29,2.39,2.27,2.39,2.25,2.39,2.23,2.37,2.22,2.36,2.19,2.35,2.17,2.33,2.15,2.32,2.41,2.31,2.40,2.29,2.40,2.27,2.40,2.26,2.38,2.24,2.37,2.21,2.37,2.20,2.35,2.18,2.34,2.15,2.33,2.13,2.31,2.41,2.29,2.41,2.29,2.38,2.26,2.40]
p=[1.06,1.32,1.15,1.20,1.28,1.17,1.25,1.23,1.20,1.26,1.29,1.28,1.26,1.23,1.31,1.28,1.22,1.31,1.28,1.36,1.31,1.28,1.36,1.31,1.28,1.36,1.36,1.32,1.25,1.22,1.46,1.40,1.36,1.31,1.25,1.50,1.45,1.40,1.36,1.31,1.26,1.49,1.45,1.40,1.36,1.30,1.55,1.50,1.45,1.40,1.36,1.59,1.38,1.33,1.28,1.23,1.46,1.41,1.37,1.31,1.25,1.21,1.45,1.39,1.34,1.29,1.24,1.48,1.42,1.37,1.32,1.27,1.51,1.46,1.40,1.36,1.30,1.25,1.49,1.37,1.76,1.40,1.81,1.47,1.85,1.52,1.90,1.56,1.95,1.61,1.28,1.64,1.32,1.71,1.37,1.75,1.42,1.81,1.47,1.85,1.51,1.90,1.56,1.95,1.59,2.00,1.66,2.03,1.71,2.09,1.75,1.41,1.80,1.46,1.83,1.51,1.89,1.54,1.94,1.61,1.99,1.65,2.04,1.70,2.07,1.75,2.14,1.79,2.18,1.84,2.22,1.48,1.14,1.53,1.18,1.56,1.22,1.61,1.27,1.65,1.31,1.69,1.35,1.73,1.39,1.77,1.43,1.84,1.47,1.86,1.52,1.17,1.55,1.22,1.60,1.26,1.64,1.30,1.68,1.35,1.73,1.38,1.77,1.47,1.80,1.46,1.85,1.51,1.89,1.55,1.95,1.59,1.25,1.63,1.29,1.67,1.34,1.71,1.37,1.75,1.42,1.79,1.45,1.86,1.49,1.88,1.54,1.92,1.58,1.96,1.62,2.01,1.66,1.33,1.70,1.36,1.85,1.41,1.79,1.45]
tbase=[0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.01]

（2）通过加窗方法进行测量
对采集数据通过加窗处理，可以避免由于对数据的截断所引起的频率特性的波动，进而提高对信号测量的精度。
常用到的窗口包括有三角波窗口，升余弦（Hanning）窗口等。

▲ 三角窗口与升余弦（Hanning）窗口

下面是统一正弦波在三角窗口和Hanning窗口加权下的波形。可以看出，经过加窗处理之后，数据在在时域内不再会因为截取而不连续了。

▲ 加窗后的数据波形（左：三角窗；右：Haning窗口）

由于加窗对数据幅度有了改变，所以会直接影响测量的结果。下面显示不同加窗方法处理后的幅度和相位。幅度会因为增加三角窗口和Hanning窗口而减少1倍左右，相位测量不受窗口方法的影响。




窗口
幅度
相位




矩形窗
2.4400
1.4182


三角窗
1.2209
1.4177


汉宁窗
1.2226
1.4175


下面对数据经过加窗处理之后，看示波器的时基大小对测量结果的影响。
A.使用三角窗测量信号的幅度和相位
测量过程中信号经过三角波加窗。

▲ 使用三角窗测量信号的幅值和相位

测量的幅值和相位的结果如下：

▲ 使用三角窗测量信号的幅度和相位

a=[1.71,2.31,2.42,2.36,2.42,2.43,2.41,2.44,2.43,2.42,2.44,2.43,2.43,2.44,2.43,2.43,2.44,2.43,2.43,2.43,2.43,2.44,2.43,2.43,2.44,2.44,2.42,2.43,2.43,2.43,2.42,2.42,2.43,2.43,2.43,2.41,2.42,2.43,2.42,2.43,2.43,2.42,2.42,2.43,2.43,2.43,2.42,2.42,2.43,2.43,2.45,2.41,2.42,2.43,2.43,2.43,2.41,2.42,2.42,2.43,2.43,2.43,2.42,2.42,2.43,2.43,2.43,2.41,2.42,2.42,2.43,2.43,2.41,2.42,2.42,2.43,2.43,2.43,2.42,2.41,2.33,2.40,2.33,2.40,2.32,2.39,2.31,2.39,2.30,2.38,2.41,2.37,2.41,2.37,2.41,2.36,2.41,2.35,2.41,2.34,2.40,2.34,2.40,2.32,2.39,2.31,2.39,2.30,2.38,2.29,2.37,2.41,2.37,2.41,2.36,2.41,2.35,2.41,2.34,2.40,2.33,2.40,2.33,2.40,2.32,2.39,2.31,2.38,2.30,2.38,2.29,2.37,2.41,2.36,2.41,2.36,2.41,2.35,2.41,2.34,2.40,2.33,2.40,2.32,2.39,2.31,2.39,2.30,2.38,2.29,2.37,2.41,2.37,2.41,2.36,2.41,2.35,2.41,2.34,2.41,2.33,2.40,2.32,2.39,2.31,2.39,2.30,2.38,2.29,2.37,2.28,2.37,2.41,2.36,2.41,2.35,2.41,2.34,2.40,2.34,2.40,2.33,2.39,2.32,2.39,2.31,2.38,2.29,2.38,2.28,2.37,2.27,2.36,2.41,2.35,2.41,2.35,2.40,2.34,2.42]
p=[0.66,1.15,1.20,1.17,1.23,1.22,1.24,1.20,1.22,1.25,1.27,1.30,1.25,1.22,1.32,1.27,1.22,1.32,1.27,1.37,1.32,1.27,1.37,1.32,1.27,1.37,1.36,1.31,1.26,1.21,1.45,1.41,1.36,1.31,1.26,1.50,1.45,1.40,1.35,1.31,1.26,1.49,1.45,1.40,1.35,1.31,1.55,1.50,1.45,1.40,1.36,1.59,1.38,1.33,1.28,1.23,1.46,1.41,1.37,1.31,1.26,1.21,1.45,1.39,1.34,1.29,1.24,1.47,1.42,1.37,1.32,1.28,1.51,1.45,1.40,1.36,1.30,1.25,1.49,1.37,1.76,1.40,1.81,1.47,1.85,1.52,1.90,1.56,1.95,1.61,1.28,1.64,1.32,1.71,1.37,1.76,1.42,1.80,1.47,1.85,1.51,1.89,1.56,1.95,1.59,1.99,1.66,2.04,1.70,2.09,1.75,1.42,1.80,1.46,1.82,1.51,1.89,1.54,1.94,1.61,1.99,1.65,2.04,1.70,2.08,1.75,2.13,1.79,2.18,1.84,2.23,1.48,1.14,1.53,1.18,1.56,1.22,1.62,1.27,1.65,1.31,1.69,1.35,1.73,1.39,1.77,1.43,1.84,1.47,1.86,1.52,1.18,1.56,1.22,1.60,1.26,1.64,1.30,1.69,1.35,1.73,1.38,1.77,1.47,1.80,1.46,1.85,1.51,1.89,1.55,1.95,1.59,1.25,1.63,1.29,1.67,1.34,1.71,1.37,1.75,1.42,1.80,1.45,1.86,1.50,1.88,1.54,1.92,1.58,1.96,1.62,2.01,1.66,1.32,1.70,1.36,1.85,1.41,1.79,1.45]
tbase=[0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.01]

B.使用Hanning窗口加权：
▲ 使用Hnning窗测量信号的幅值和相位

▲ 使用Hanning窗测量信号的幅值和相位

a=[1.67,2.19,2.49,2.45,2.43,2.45,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.44,2.43,2.43,2.43,2.44,2.42,2.43,2.43,2.43,2.44,2.42,2.43,2.43,2.43,2.43,2.44,2.42,2.43,2.43,2.43,2.44,2.42,2.43,2.43,2.43,2.45,2.42,2.43,2.43,2.43,2.43,2.42,2.42,2.43,2.43,2.43,2.44,2.42,2.43,2.43,2.43,2.43,2.42,2.43,2.43,2.43,2.43,2.42,2.42,2.43,2.43,2.43,2.43,2.42,2.41,2.34,2.40,2.35,2.40,2.34,2.40,2.33,2.39,2.33,2.39,2.42,2.38,2.42,2.38,2.41,2.37,2.41,2.36,2.42,2.36,2.41,2.36,2.40,2.35,2.40,2.34,2.40,2.33,2.39,2.32,2.38,2.42,2.38,2.42,2.38,2.41,2.37,2.41,2.36,2.41,2.35,2.40,2.35,2.40,2.34,2.40,2.33,2.39,2.32,2.39,2.32,2.38,2.42,2.38,2.41,2.37,2.41,2.37,2.41,2.36,2.41,2.35,2.40,2.34,2.40,2.34,2.40,2.33,2.39,2.32,2.39,2.42,2.38,2.42,2.37,2.41,2.37,2.41,2.36,2.41,2.35,2.40,2.35,2.40,2.34,2.40,2.33,2.39,2.32,2.39,2.31,2.38,2.42,2.38,2.41,2.37,2.41,2.36,2.41,2.36,2.40,2.35,2.40,2.34,2.40,2.33,2.39,2.32,2.39,2.31,2.38,2.30,2.38,2.41,2.37,2.41,2.37,2.41,2.36,2.42]
p=[0.54,1.09,1.22,1.21,1.23,1.22,1.25,1.20,1.23,1.25,1.27,1.30,1.25,1.22,1.32,1.27,1.22,1.32,1.27,1.37,1.32,1.27,1.37,1.32,1.27,1.37,1.36,1.31,1.26,1.21,1.45,1.41,1.36,1.31,1.26,1.50,1.45,1.40,1.35,1.31,1.26,1.49,1.45,1.40,1.35,1.31,1.55,1.50,1.45,1.40,1.36,1.59,1.38,1.33,1.28,1.23,1.46,1.41,1.37,1.31,1.26,1.21,1.45,1.39,1.34,1.29,1.24,1.47,1.42,1.37,1.32,1.28,1.51,1.45,1.40,1.36,1.30,1.25,1.49,1.37,1.76,1.40,1.81,1.47,1.85,1.52,1.90,1.56,1.95,1.61,1.28,1.64,1.32,1.71,1.37,1.75,1.42,1.80,1.47,1.85,1.51,1.89,1.56,1.95,1.59,1.99,1.66,2.04,1.70,2.09,1.75,1.42,1.80,1.46,1.82,1.51,1.89,1.54,1.94,1.61,1.99,1.65,2.04,1.70,2.08,1.75,2.13,1.79,2.18,1.84,2.23,1.48,1.14,1.53,1.18,1.56,1.22,1.62,1.27,1.65,1.31,1.69,1.35,1.73,1.39,1.77,1.43,1.84,1.47,1.86,1.52,1.18,1.56,1.22,1.60,1.26,1.64,1.30,1.69,1.35,1.73,1.38,1.77,1.47,1.80,1.46,1.85,1.51,1.89,1.55,1.95,1.59,1.25,1.63,1.29,1.67,1.34,1.71,1.37,1.75,1.42,1.79,1.45,1.86,1.50,1.88,1.54,1.92,1.58,1.96,1.62,2.01,1.66,1.32,1.70,1.36,1.85,1.40,1.79,1.45]
tbase=[0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.01]

C.对比矩形窗、三角窗、Hanning窗的效果
通过对比可以看出，经过加窗处理之后，时基长度小于2ms之内，测量的结果收到时基长度的影响就大大减弱了。

▲ 对比矩形窗、三角窗、Hanning窗对于幅度测量的影响

特别是在时基设置为0.25~0.7毫秒左右（对应1000Hz的信号，示波器显示大约3.5至10个周期）使用Hanning窗口所测量的幅度谱基本上不受时基长短的 影响。

▲ 对比矩形窗、三角窗、Hanning窗对于幅度测量的影响(局部)

#------------------------------------------------------------
def winddata(len, mode = 0):
    if mode == 0:
        return [1] * len
    elif mode == 1:         # Triangle Window
        win = [1] * len
        for i in range(len):
            if i < len / 2:
                win[i] = i / ((len + 1) / 2)
            else: win[i] = (len-i) / ((len + 1) / 2)
        return win
    elif mode == 2:
        win = [1] * len
        for i in range(len):
            theta = (i - (len - 1) / 2) * pi * 2 / len
            win[i] = (1 + cos(theta)) / 2
        return win
    else: return [1] * len
#------------------------------------------------------------
def data2ap(tt, y, freq):
    sint = [sin(t * 2 * pi * freq) for t in tt]
    cost = [cos(t * 2 * pi * freq) for t in tt]
    ys = sum([a * b for a,b in zip(y, sint)]) / len(tt)
    yc = sum([a * b for a,b in zip(y, cost)]) / len(tt)
    amp = sqrt(ys**2 + yc**2)*sqrt(2)
    phase = atan2(ys,yc)
    return amp,phase
#------------------------------------------------------------
def ds6104data(freq, win=0):
    x,y = ds6104readcal(1)
    if win == 1:
        wind = winddata(len(x), 1)
        y = [d * w * 2 for d,w in zip(y, wind)]
    elif win == 2:
        wind = winddata(len(x), 2)
        y = [d*w*2 for d,w in zip(y, wind)]
    a,p = data2ap(x,y,freq)
    return a, p, x, y


 
04示波器采样模式对测量精度的影响

示波器采样模式包括有四种：正常采样模式、平均采样模式、峰值采样模式以及高分辨率采样模式。下面通过对比实验，查看这些采样模式对于测量精度的影响。、
下面实验中，采样信号为1000Hz，有效值为2.3V左右的正弦波信号。示波器的时基设为0.7ms，处理数据使用Hanning窗加窗。
1.正常采样模式
测量结果：

幅值：2.44283V；相位：1.29056

▲ 正常采样模式下的信号的波形

2.平均采样模式
测量结果：

幅值：2.44128；相位：1.2282

▲ 平均采样模式下信号的波形

3.峰值采样模式
测量结果：

幅值：2.44415V；相位：1.2903

▲ 峰值采样模式下的信号波形

4.高分辨率采样模式
测量结果：

幅值：2.4243V；相位：1.4068

▲ 高分辨率采样模式下的信号波形

对比以上四种示波器采样模式，对于信号的幅度没有太多的影响，对于信号的相位，高分辨率采样则出现了与其它三类较大的差异。根据示波器实际波形显示，采用高分辨率采样模式所得到的结果与实际相位是最接近的。




采样模式

幅值
相位




1
正常采样
2.44283
1.29056


2
平均采样
2.44128
1.2282


3
峰值采样
2.44415
1.2903


4
高分辨率采样
2.4243
1.4068


 
05不同频率对测量结果的影响

对于信号频率从10Hz变化到10MHz，使用相同的方法进行测量它的幅值和相位，结果如下：




频率
幅值
相位




10.0
2.3879
1.3380


100.0
2.4073
1.2545


1000.0
2.4385
1.3111


10000.0
2.5401
1.2231


100000.0
2.5516
1.2740


1000000.0
2.5952
1.0833


10000000.0
2.3971
-0.6503


#------------------------------------------------------------
def ds6104settimebase(freq, period):
    timebase = period / freq / 14
    ds6104timebase(timebase)   


 
06结论

通过对DS6104联网示波器采集到的数据进行加窗计算，可以获得被测量信号的幅值和相位。这种方法所得到的数值精度在1000Hz时精度很高。同样，这种方法所能适应的频率范围可以是示波器的有效显示频率范围。
为了提高测量精度，示波器在设置参数的需要尽可能满足：

设置示波器通道增益，尽可能使得信号波形达到屏幕满幅值，这样可以最大利用示波器采样8bit的精度；
设置示波器时基，使得能够采样到7个周期左右的数据；
示波器的采样模式使用“高分辨率”采样模式；

在处理数据时，建议使用增加Hanning窗口，来减少对数据截断所带来的幅值波动。

此文将下面链接中的PPT以文章的形式写在这里， 作为笔记使用，希望和各位小伙伴多多交流https://wenku.baidu.com/view/0f5ef8f74693daef5ef73dfd.html
一.  驻定相位原理（POSP）
（注：1.使用POSP的场景：经常应用于积分式，尤其是傅里叶变换和逆傅里叶变换。
2. 使用POSP来估计傅里叶变换的条件：信号相位变化非常快，以至于在做积分时，只有在驻相点附近的积分才不为零。）
首先，观察积分式：
其中， 
 是包络， 
 是相位，由于 
 的正部分和负部分的面积相互抵消，上式的积分接近于零。只有在 
 点（驻相点）附近，由于相位变化率很小，相位有很长时间的滞留，才能使上式的积分显著不为零。
二. 窄带信号频谱
以窄带信号(高频扰动信号) 
  的傅里叶变换为例，应用驻定相位原理。
 的傅里叶变换为：
类比式（1），式（2） 中
 是包络信息， 
 是相位信息。积分时正部分和负部分面积相抵消，上式积分接近零。只有在 
 点附近，由于相位变化率很小，相位值有很长时间的滞留，才能使上式积分显著不为零。
用 
 表示
驻定相位点。则 
对于相位项 
 在驻定相位点附近进行泰勒展开：
忽略高次项，且 
 ，上式可以写为：
代入（2）式，
做变量代换 
 ,则 
代入信号的频域表达式（6）中，
式中积分为菲尔涅积分。若积分上限较大[1]，则非尔涅积分趋于 
 .
则信号的频域表达式为：
三 线性调频信号（LFM）的频谱（利用驻定相位原理）
设LFM信号为：
 其傅里叶变换为：
 作变量代换 
则 
且其中， 
注：菲尔涅积分
则矩形包络LMF信号的频谱精确解为：
其幅度谱和相位谱分别为： 
当时宽带宽积足够大时，有 
由于精确解难以计算，所以利用公式8估值。上述第二小节中，窄带信号的频谱为公式（8），那么对于矩形包络的LFM信号，
结合式8和式14， 利用驻定相位原理得到LFM信号的幅度和相位谱分别是
其中， 
[2]
参考
^这里有个问题：因为菲涅尔积分来说，积分上限较大时，积分式趋于常数。对于此积分，积分区间是在驻定相位附近，是否满足上限较大这一条件呢
^第二个问题：如果t_k 不唯一，那么频谱是什么样子呢。此时X（w）是不同tk时 式（8）的叠加吗？