Physical meaning of autocorrelation

[jollage]

Hi All,

I was in a process of processing my vibration-test data. I now generated a plot of the autocorrelation function of the object acceleration. Please see the attachements (the second attachment is the close-up for small tau's).

The x-axis in the plot is the time delay tau. You can see from the second attachment that, at tau=0,the autocorrelation reaches its maximum.

My question is, as for the signal like this, in the long time delay (tau=60s, 70s), the autocorrelation still does not decay too much. What could this behavior imply? In terms of the noise frequency, what can we say?

Any comments will be welcome. Thanks in advance.

Mz

 

[Jano L.]

The autocorrelation function gives the amount of correlation of values of the original function at times delayed by ##\tau##. If the autocorrelation function vanishes quickly, it means that the value of the function at some time cannot be predicted with success from neighboring values.

If it does not vanish quickly, it means that the behavior of the function is less random; from the value at time 0 one can estimate possible values at future times with better success. For example, if the function is periodic, after one wave one can make reasonable estimate what the next values will be, and the autocorrelation function comes out also periodic. But if the function is position of Brownian particle, the next values are hard to predict successfully, and also the autocorrelation function decays very fast (exponentially).

 

[sophiecentaur]

If the autocorrelation does not die away then there must be some significant periodic elements in the signalbecause the delayed signal correlates well with an earlier version of itself.

 

[Andy Resnick]

Hi All,

I was in a process of processing my vibration-test data. I now generated a plot of the autocorrelation function of the object acceleration. Please see the attachements (the second attachment is the close-up for small tau's).

Any comments will be welcome. Thanks in advance.

Mz

A few comments:

I think of the autocorrelation as a measure of how deterministic the system is- a perfectly deterministic system will always have a normalized autocrrelation of '1', while a perfectly random system will have have an autocorrelation that looks like a delta function. For real systems with dissipation present, the autocorrelation function will generally follow a decaying exponential, and the width of the exponential is a measure of the (de-)correlation time (or length, if we extend the analysis to cover spatially random systems such as reflection from a rough surface). Multiple dissipation processes result in multi-exponential autocorrelation functions- the dynamic light scattering field is a good place to get familiar with interpreting autocorrelation curves.

Without knowing any details about what generated the data (what is vibrating? How was it excited?) it's tough to parse your data. However, when we see a low-frequency component like your first graph has, it's due to drift in our detector.

 

[pmacias]

Hello Mz,

The autocorrelation function is a measure of the similarity between a signal and a delayed version of itself. In other words, it tells us how much a signal is correlated with itself at different time intervals. The maximum value of the autocorrelation function at tau=0 indicates that the signal has a high degree of self-similarity.

In terms of the noise frequency, the behavior of the autocorrelation function at long time delays (tau=60s, 70s) can provide information about the underlying frequency components of the signal. If the autocorrelation function does not decay too much at these longer time delays, it suggests that there may be low frequency components present in the signal. This could indicate that the vibration-test data contains low frequency noise or that the object being tested has low frequency vibrations.

It is important to note that the autocorrelation function is just one tool for analyzing signals and should be used in conjunction with other techniques to fully understand the characteristics of the signal. I hope this helps to clarify the physical meaning of autocorrelation in your data analysis. Best of luck with your research.

Sincerely,