After all previous posts on Auto Correlation function attached is my simulation of Auto Correlation function using Excel. Emphasis is given to signal-noise ratio and its effect on AC function.

Conclusion (FWIW) - at given magnitude of a periodic signal, the AutoCorr(t) magnitude will be dependant of noise. In other words the noise in AutoCorr function is filtered out just partially.

Dave

Autocorr_D.xls (85 KB, 44 downloads)

I believe that the random function which you used simulates a uniform distribution. Perhaps, more appropriate would be a Gaussian distribution.

This should be approximately from a Gaussian distribution with mean 0 and std of .333. Excel is not always the best statistical tool. Scale as needed.


-0.7357 -0.49787 0.164046 -0.53777 -0.28427
-0.74447 -0.03241 -0.76915 0.165239 0.251366
0.053145 0.203663 0.152636 -0.18245 0.255499
0.285489 -0.67513 0.315545 0.249825 -0.30673
0.331537 -0.36681 -0.02367 -1.0085 0.020564
0.444408 -0.12276 0.090939 0.423781 0.152024
-0.23567 -0.60751 0.207186 0.140083 -0.20324
0.14046 0.628238 0.03705 0.262609 -0.06015
0.078748 -0.05129 0.020943 0.432905 0.509843
-0.31285 -0.33772 0.196407 -0.32388 -0.22866
-0.20076 0.150542 0.122461 0.736315 0.064682
-0.2656 0.115692 -0.20737 0.024525 0.3718
0.17958 -0.00925 0.770378 0.165606 0.423789
0.408133 0.502336 0.225743 0.056288 0.267241
-0.17316 0.28534 0.201074 -0.07753 0.20465
0.490651 0.562806 -0.06728 -0.48969 -0.11184
0.560489 0.042102 0.016187 -0.35122 -0.31649
0.456553 -0.0753 -0.01251 -0.08175 -0.63601
0.047625 0.164937 -0.04032 -0.14914 0.103189
0.571231 -0.19513 -0.12292 0.317008 0.596891
0.760041 0.239337 -0.42684 0.06023 -0.22996
0.635089 0.059582 0.476761 -0.31945 0.669579
0.135832 -0.01828 0.530334 0.533889 -0.38539
-0.9955 0.003708 -0.648 -0.54609 0.183567
0.006151 0.14375 -0.10274 0.097873 0.285721
-0.42554 -0.1177 -0.07109 0.109049 -0.19832
-0.52332 0.210993 0.001673 -0.19427 -0.40654
-0.32927 -0.17624 -0.34168 0.157786 0.097345
-0.4548 -0.63434 0.284747 0.111599 0.547589
0.361646 -0.03823 -0.11346 -0.26204 0.633669
-0.06935 -0.08608 -0.43896 0.195696 -0.2187
-0.522 -0.35951 -0.20545 -0.28015 0.645958
-0.21867 0.138819 -0.05623 0.002206 -0.50254
0.138028 0.068564 -0.15806 0.033081 -0.50766
-0.73211 -0.75023 -0.1252 0.081374 0.070859
-0.11984 -0.09739 -0.55019 -0.37998 -0.30638
0.178226 0.09041 -0.07962 -0.25029 -0.21681
-0.18572 -0.06753 -0.8157 0.035386 0.024587
0.023589 0.320038 0.440828 0.200761 -0.47224
-0.04106 -0.22417 0.492886 -0.11148 -0.49207
0.140132 -0.13891 -0.69586 -0.29815 0.15111
0.324135 -0.19995 0.226932 -0.35131 0.326102
-0.86225 -0.08942 0.510461 0.246385 -0.17124
0.213619 0.359314 -0.27481 0.148509 0.282454
-0.28066 -0.08313 -0.09806 -0.11976 0.357855
0.145177 0.178251 0.297753 -0.43255 -0.08862
0.04602 -0.02161 0.182765 -0.15636 -0.77789
-0.04054 -0.75736 0.202527 -0.15376 -0.39653
-0.07643 0.634207 0.528742 -0.34751 0.029597
-0.11367 0.740591 -0.21853 0.091649 -0.61496
0.501041 -0.0806 0.360937 0.178789 0.42002
-0.00645 -0.25323 0.523308 -0.15861 -0.60349
-0.01123 0.070747 -0.2283 -0.26813 0.119422
0.082881 -0.26781 -0.03608 -0.23334 -0.32598
-0.29584 -0.21915 -0.23141 0.160123 -0.04067
-0.00948 -0.14924 -0.41414 0.05111 0.33069
-0.04532 0.346328 -0.279 0.425655 0.066697
0.232531 0.003856 -0.11016 -0.03806 0.284596
-0.38012 0.253718 -0.20018 -0.30711 -0.06352
-0.34529 0.407398 -0.02907 -0.19591 -0.51581
0.392524 0.162298 0.249235 -0.30008 -0.23795
0.131131 -0.36948 -0.13663 -0.16881 -0.05044
0.176984 0.631264 -0.42326 0.857951 0.19517
-0.57463 0.046471 -0.21421 0.228469 -0.39646

This message has been edited. Last edited by: William_C._Foiles,

Used Gaussian distribution for the Random function and slightly enhenced the Excel file.
Will double the data block size in the next update in order to see its effects.

Dave

Autocorr_D.xls (258 KB, 15 downloads)

Yes, longer comparison interval will improve signal to noise ratio of the autocorrelation output waveform.

With the Gaussian noise it has mean zero.

The interpretation of the standard deviation, std, for vibration is that it is the RMS of the distribution (approximately for the sample as well). So, instead of putting in a pk noise one dials in an rms level of noise.

With this type of random noise it is more likely to get small noise than large noise loosely speaking, but it is possible to get very large noise. Whereas, the uniform distribution produces noise in an interval. One generally expects Gaussian type stuff or often related distributions.

One might argue that 1X noise would have a Rayleigh distribution for amplitude and a uniform distribution (0 - 360 degrees) for phase, but this just comes from looking at a binormal distribution in amplitude and phase.

Attached is AutoCorrelation function with a larger data block size.

Autocorr_D_large.xls (390 KB, 14 downloads)

You have the correlation coefficient plotted.

Take your sum ($k$146) and divide by N/2 to get the variance of 14.11. Take the square root and you get 3.7566608, which is the rms value of the signal. The noise was from a distribution of 10*.333 = 3.333. The signal was a sine wave with rms = .70711, but the sine wave does not have an integral number of cycles in the plotted data, which could cause a deviation from its true rms.