相关系数与信噪比

Question

我感兴趣的是一个参考信号和一系列其他信号之间的相关性。为此，我使用了斯皮尔曼的柯夫。此外，我想解释一下，为什么某些数据的相关性更强，而其他数据的更弱的。显而易见的答案是：“嗯，信号是相似的/不是相似的”。我的想法是，我希望它们都是相似的，这就是为什么我对它们感兴趣的原因。

假设我有这三个信号：

ref = [0.153698000000000    0.159227000000000   0.153578000000000   0.135617000000000   0.106725000000000   0.0720750000000000  0.0383580000000000  0.00419000000000000 -0.0406150000000000 -0.104511000000000  -0.178757000000000  -0.237172000000000  -0.248983000000000  -0.201136000000000  -0.112051000000000  -0.0171030000000000 0.0556220000000000  0.100198000000000   0.127225000000000   0.146429000000000   0.156469000000000   0.148586000000000   0.120207000000000   0.0822790000000000  0.0465630000000000  0.0127520000000000  -0.0302530000000000 -0.0892330000000000 -0.151656000000000  -0.187429000000000  -0.171909000000000  -0.106635000000000  -0.0207050000000000 0.0504730000000000  0.0890630000000000  0.101382000000000   0.102716000000000   0.103612000000000   0.101693000000000   0.0877250000000000  0.0557510000000000  0.00836400000000000 -0.0466150000000000 -0.101411000000000  -0.149962000000000  -0.184511000000000  -0.192687000000000  -0.163118000000000  -0.0969680000000000 -0.0117410000000000 0.0659480000000000  0.116307000000000   0.136674000000000   0.136510000000000   0.127171000000000   0.112167000000000   0.0907370000000000  0.0624430000000000  0.0266490000000000  -0.0188730000000000 -0.0752460000000000 -0.139144000000000  -0.194234000000000  -0.216676000000000  -0.188924000000000  -0.114953000000000  -0.0216350000000000 0.0608270000000000  0.116310000000000   0.147004000000000   0.166052000000000   0.179028000000000   0.180012000000000   0.161279000000000   0.121606000000000   0.0727670000000000  0.0258380000000000  -0.0186780000000000 -0.0675800000000000 -0.127120000000000  -0.188126000000000  -0.225266000000000  -0.212062000000000  -0.145823000000000  -0.0534320000000000 0.0279870000000000  0.0772470000000000  0.0977090000000000  0.108232000000000   0.121362000000000   0.131944000000000   0.126989000000000   0.100732000000000   0.0629320000000000  0.0262270000000000  -0.00548300000000000    -0.0427550000000000 -0.0963450000000000 -0.158377000000000  -0.203218000000000  -0.203878000000000  -0.149980000000000  -0.0592800000000000 0.0337540000000000  0.0997230000000000  0.128674000000000   0.133482000000000   0.128745000000000   0.122315000000000   0.112027000000000   0.0905840000000000  0.0587200000000000  0.0247430000000000  -0.00688800000000000    -0.0431370000000000 -0.0940400000000000 -0.156244000000000  -0.207234000000000  -0.217930000000000  -0.171546000000000  -0.0821420000000000 0.0148240000000000  0.0863420000000000  0.119430000000000   0.125936000000000   0.125051000000000   0.127984000000000   0.128781000000000   0.115835000000000   0.0852330000000000  0.0443400000000000  0.00490900000000000 -0.0255570000000000 -0.0544690000000000 -0.0948830000000000 -0.148332000000000  -0.197016000000000  -0.212808000000000  -0.179499000000000  -0.103302000000000  -0.0123390000000000 0.0617030000000000  0.104228000000000   0.119877000000000   0.122972000000000   0.125183000000000   0.125265000000000   0.114951000000000   0.0870480000000000  0.0447450000000000  -0.000855000000000000   -0.0396970000000000 -0.0696810000000000]

x= [0.271434000000000   0.288274000000000   0.209192000000000   0.0655440000000000  -0.0820200000000000 -0.148026000000000  -0.108114000000000  0.00370200000000000 0.0903880000000000  0.0670870000000000  -0.0543520000000000 -0.198260000000000  -0.266923000000000  -0.210663000000000  -0.0721470000000000 0.0476940000000000  0.0976820000000000  0.0815330000000000  0.0550950000000000  0.0509430000000000  0.0654310000000000  0.0761700000000000  0.0716700000000000  0.0496400000000000  0.0296730000000000  0.00932700000000000 -0.0220210000000000 -0.0918530000000000 -0.191917000000000  -0.273027000000000  -0.278303000000000  -0.193660000000000  -0.0558200000000000 0.0496270000000000  0.0822850000000000  0.0492800000000000  0.00158300000000000 -0.0250730000000000 -0.000159000000000000   0.0643930000000000  0.133856000000000   0.165200000000000   0.124040000000000   0.0217580000000000  -0.0914600000000000 -0.162790000000000  -0.156950000000000  -0.0734400000000000 0.0277270000000000  0.0829160000000000  0.0694490000000000  0.0286590000000000  0.0242190000000000  0.0770240000000000  0.144502000000000   0.164170000000000   0.111881000000000   0.0246120000000000  -0.0525930000000000 -0.0932270000000000 -0.117220000000000  -0.162008000000000  -0.231990000000000  -0.283092000000000  -0.257563000000000  -0.140892000000000  -0.0109250000000000 0.0575570000000000  0.0398860000000000  -0.0119950000000000 -0.0311610000000000 0.00895200000000000 0.0767310000000000  0.111818000000000   0.0770950000000000  -0.00480400000000000    -0.0648260000000000 -0.0590600000000000 -0.0239010000000000 -0.0113260000000000 -0.0503820000000000 -0.119773000000000  -0.166668000000000  -0.153385000000000  -0.0951590000000000 -0.0134120000000000 0.0758570000000000  0.169250000000000   0.265192000000000   0.320269000000000   0.312399000000000   0.248779000000000   0.170381000000000   0.116544000000000   0.0825490000000000  0.0229900000000000  -0.112957000000000  -0.311565000000000  -0.477718000000000  -0.528972000000000  -0.441705000000000  -0.277521000000000  -0.132652000000000  -0.0641130000000000 -0.0469650000000000 0.00307500000000000 0.145157000000000   0.348731000000000   0.493012000000000   0.479475000000000   0.325143000000000   0.114432000000000   -0.0427650000000000 -0.100921000000000  -0.0959820000000000 -0.103768000000000  -0.157386000000000  -0.232400000000000  -0.260288000000000  -0.205974000000000  -0.107415000000000  0.00374900000000000 0.0959960000000000  0.175779000000000   0.259562000000000   0.343460000000000   0.392091000000000   0.364989000000000   0.262129000000000   0.115872000000000   -0.0238940000000000 -0.112717000000000  -0.167219000000000  -0.228616000000000  -0.308254000000000  -0.357301000000000  -0.323026000000000  -0.200829000000000  -0.0399150000000000 0.0773370000000000  0.106666000000000   0.0934510000000000  0.134411000000000   0.283691000000000   0.484582000000000   0.615659000000000   0.578599000000000   0.376662000000000   0.121242000000000   -0.0499920000000000 -0.0908270000000000 -0.0512570000000000 -0.0251360000000000]

y=[0.0805620000000000   0.0193840000000000  0.0317950000000000  0.0923790000000000  0.157971000000000   0.196456000000000   0.213476000000000   0.217857000000000   0.199800000000000   0.132178000000000   0.00511700000000000 -0.129106000000000  -0.200164000000000  -0.161512000000000  -0.0533460000000000 0.0447720000000000  0.0777510000000000  0.0457860000000000  -0.000266000000000000   -0.00166500000000000    0.0469650000000000  0.100767000000000   0.105057000000000   0.0487410000000000  -0.0139600000000000 -0.0339380000000000 -0.0268990000000000 -0.0467280000000000 -0.117983000000000  -0.191721000000000  -0.206983000000000  -0.126591000000000  0.0368150000000000  0.187842000000000   0.245185000000000   0.207442000000000   0.113449000000000   0.0277730000000000  -0.0201070000000000 -0.0340330000000000 -0.0418780000000000 -0.0577930000000000 -0.0687330000000000 -0.0774580000000000 -0.112039000000000  -0.203065000000000  -0.325518000000000  -0.408340000000000  -0.384561000000000  -0.233895000000000  -0.000200000000000000   0.206361000000000   0.301673000000000   0.258327000000000   0.158405000000000   0.0850420000000000  0.0638960000000000  0.0601920000000000  0.0267320000000000  -0.0495640000000000 -0.140141000000000  -0.216581000000000  -0.246500000000000  -0.220618000000000  -0.159008000000000  -0.0645680000000000 0.0638910000000000  0.195499000000000   0.292969000000000   0.314935000000000   0.253050000000000   0.141489000000000   0.0485870000000000  0.0141340000000000  0.00904500000000000 -0.0209880000000000 -0.102024000000000  -0.191704000000000  -0.229835000000000  -0.193077000000000  -0.117835000000000  -0.0788180000000000 -0.132543000000000  -0.227262000000000  -0.261066000000000  -0.146183000000000  0.0732720000000000  0.288526000000000   0.393861000000000   0.375488000000000   0.312956000000000   0.280953000000000   0.293536000000000   0.294787000000000   0.228394000000000   0.0812080000000000  -0.0979610000000000 -0.219818000000000  -0.247000000000000  -0.203883000000000  -0.152293000000000  -0.128943000000000  -0.114403000000000  -0.0782120000000000 -0.0208240000000000 0.0301690000000000  0.0414160000000000  0.00930500000000000 -0.0224010000000000 -0.0127020000000000 0.0374780000000000  0.0839880000000000  0.0805340000000000  0.0179880000000000  -0.0578820000000000 -0.103678000000000  -0.0949510000000000 -0.0417700000000000 0.0216980000000000  0.0582560000000000  0.0632220000000000  0.0520980000000000  0.0350020000000000  0.0327130000000000  0.0430200000000000  0.0648000000000000  0.0662060000000000  0.0380750000000000  0.00131200000000000 -0.0341550000000000 -0.0591350000000000 -0.0705680000000000 -0.0688930000000000 -0.0640920000000000 -0.0680670000000000 -0.0878340000000000 -0.112712000000000  -0.137633000000000  -0.136924000000000  -0.0989540000000000 -0.0408820000000000 0.0148740000000000  0.0569500000000000  0.0903410000000000  0.119637000000000   0.161266000000000   0.213775000000000   0.254719000000000   0.255163000000000   0.198274000000000   0.106108000000000   0.0180750000000000  -0.0376770000000000]

corrcoef(ref,x) = 0.7448 (我不知道为什么会得到4个值作为correcoef的结果)。corrcoef(ref,y) = 0.6860。

现在，我不能用时间来判断原因或稍微弱一点的相关性，所以我试着观察频率：

​

现在，我猜是因为每个信号的信噪比。我想计算每个信号的信噪比，但这是我被困的地方。我找到的所有与信噪比有关的公式都事先知道哪一个是信号，哪个是噪声，但我不知道。我想计算信噪比的方法是取最高峰值的能量，然后除以其余峰值的能量，但我不知道如何在matlab中做到这一点。我想，对于最高的峰值，我可以有一个光标指向，但我如何计算其余的峰值的总和能量？

PS:我试过Matlab的a = snr(x) ;，我得到了

Undefined function 'snr' for input arguments of type 'double'.

即使我列出工具箱>>ver，信号处理工具箱版本6.19，它是一个列表。谢谢!

稍后编辑:如果我尝试这样做会怎样：

Energy = abs(fft(ref, NbOfPoints))  //get the magnitudes for FFT in NbOfPoint
Energy = Energy(1:NbOfPoints/2)     // keep the first half, the others are mirrored
[HP index] = max(Energy)            // get the magnitude of the highest peak and it's position
Energy(index)=0                     // set the Highest Peak to 0
SNR = HP / sum(Energy)              // compute magnitude of highest peak / sum of others

这有关系吗？结果明显偏低( 0.02 ~ 0.03 )，不知道这是否是一种比较信号和噪声的公认方法。也许我应该在最高峰周围选择一些豆子，而不是最高峰呢？比如(HP-2:HP+2)？

Answer 1

当你试图解释为什么信号在时间域上或多或少相关时，你必须仔细地阅读信号的光谱域。虽然我理解你试图量化每个信号的信噪比的方法，但你也说你不知道什么是信号和什么是噪音。参考信号的图显示主谱峰存在谐波，这也是信号的一部分(而不是噪声)。

如果将两个信号的x-y图与参考信号相对应，这就是正在执行的相关操作。

plot (ref,x,'.')
hold on
plot (ref,y,'.r')

因此，在逐点(或逐个样本)的基础上，与x或y线性增长相关的参考信号增加，信号线性变化越大，它们的相关性就越强。

有时，看看互相关，看看信号中是否有任何滞后，是很有用的。如果是这样的话，您可能会发现峰值相关并不发生在滞后0(这相当于运行一个简单的相关)，而是在信号之间存在一定程度的滞后。

[LAGS,C] = xcorr ([ref',x'])
plot(LAGS,C)

您可能还需要进行一致性分析，以查看两个信号之间的频率内容是相似的。这将帮助您决定信号是否位于光谱峰值是全部噪声，或者是否有其他的光谱内容(包括谐波)是实际的“信号”。