PHYS 381/504: Random Variables, Correlation Functions and Power Spectral Density

Random variables

In experimental science, often one encounters a signal that seems to fluctuate versus time. An example such a signal is shown below. Several features seem to characterize this signal:

1. The signal is a random variable.



2. It has zero mean.
3. Its average magnitude seems relatively well-defined with values far greater than this magnitude unlikely.
4. The statistical properties of the signal do not change in time. (It is an ergodic process.)
5. The signals separated by large time intervals are statistically independent.
6. However, for small time intervals, the signal does not change appreciably.

Correlation functions

In dealing with random variables, we are lead to the probability densities for those variables. Therefore, let's start by trying to describe mathematically the properties listed above by writing down a joint probability density [] which specifies the probability density that the signal at time is and that the signal at time is . (Be sure that you understand the difference between a probability density and a probability.)

A joint probability density that seems to yield the properties listed above is a so-called jointly Gaussian (or bivariate Gaussian) probability density:

                                              (1)

and are parameters. is a constant, independent of . It will turn out that is a function of that is always less than 1.

It is straightforward to show that --- the probability density for , irrespective of --- is Gaussian with zero mean, and variance , independent of , or equivalently time, as suggested in our list. (Do it --- use Mathematica to carry out the integral.)

We noted above that and are statistically independent for large . Statistical independence implies that . This is only the case, if . Therefore, Eq. 1 can only be applicable to our case, if as . (Another way to say that and are statistically independent is to say that they are uncorrelated.)

On the other hand, it is straightforward to show that as , the variance of goes to zero, i.e. for . Thus, we recover the observation that the signal does not change appreciably for small time intervals only if for .

Thus, our candidate joint probability density seems consistent with the properties of our signal provided behaves as described. It is plausible then that our random process is characterized by two quantities and . Our goal now is to figure out how to measure these quantities.

Using Eq. 1, we may show that

                                                                                             (2)

and that

                                                                               (3)

where the angle brackets denote an average. (Do this --- again use Mathematica.) These results inform us how, in principle, to determine and from an experimental determination of . How to determine is straightforward. To determine , Eq. 3 tells us to take the signal at time and the signal at time and multiply them together. We do this for every value of , while keeping fixed, and then average all these products together. This quantity is the correlation function or autocorrelation function of the random process in question. The correlation function concept is very important in both experimental and theoretical condensed matter physics. Correlation functions are important experimentally, because usually, if something can be measured, it is a correlation function. Correlation functions are important theoretically, because usually, if something can be calculated, it is a correlation function. And that's what physics is about: making and testing quantitative predictions.

Power spectral density

So far, our discussion has occurred in the time domain. However, time and frequency domains are related by Fourier transformation. Given a random variable as a function of time acquired at a sampling frequency for a total time , we (or LabVIEW, at least) can calculate the discrete Fourier transform

                                                                   (4)

with , , and , ..., is an integer, and . Note that the maximum frequency for which is available is . This maximum frequency, equal to one half of the sampling frequency, is called the Nyquist frequency ().

A little thought will convince you that a finite sampling frequency means that fluctuations with an actual frequency of will appear in this measurement to have a frequency , where is the integer that puts in the frequency range sampled. This is the phenomenon of aliasing. Aliasing will NOT be important if your PSD does NOT have much weight at frequencies above . Whether this is the case must be determined on a case-by-case basis.

Rather than outputing itself, in fact, LabVIEW outputs the so-called power spectral density (PSD): . Note that the units of PSD as measured are , if is in volts. Sometimes people take the square-root and the units then would be .

One reason for outputing the PSD rather than , is that the Fourier transform of the PSD is the correlation function, , discussed above. Because of their Fourier transform relationship, an experimental determination of either or as a function of its argument is sufficient for the determination of the other.

Acknowledgement: Much of the discussion of this section is based on unpublished lecture notes by Tom Greytak (MIT) originally given in MIT's sophomore stat. mech. class, 8.044.