Autocorrelation of white noise.

In summary, white noise cannot be rigorously defined as a stochastic process, in the same way that the Dirac delta function does not exist as a function. White noise exists only as a distributional derivative of a Brownian motion.
  • #1

vanesch

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I'm stuck with an elementary thing, it must be something obvious but I can't see what's wrong.

Here it goes. I was writing up some elementary course material for an instrumentation course, and wanted to quickly introduce "white noise".

Now, the usual definition of white noise is something like a stationary random process such that E( X(t) ) = 0 for all t and a flat power spectral density.

On the other hand, the autocorrelation function is defined as R(tau) = E (X(t) X(t+tau) ) (independent of t).

But here's the problem. The Wiener-Khinchine theorem states that the power spectral density equals the Fourier transform of the autocorrelation function, so a flat power spectral density comes down to a Dirac for the autocorrelation. And for example on Wiki, you find that as a defining property of white noise.

But the autocorrelation of white noise E (X(t) X(t) ) is nothing else but sigma-squared.

So it would seem that the autocorrelation function is everywhere 0, except in 0, where it is a finite number.

What am I missing here ?
 
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  • #2
White noise cannot be defined rigorously in any of these ways. White noise does not exist as a stochastic process, in the same way that the Dirac delta function does not exist as a function.

There is no (measurable) continuous time stochastic process X that satisfies E[X(t)] = 0, var(X(t)) = σ2, with X(s) and X(t) independent whenever s ≠ t. If we allow var(X(t)) to be infinite, then we can construct such a process, but of course it cannot be continuous. Such a definition, however, is completely useless, because we need the integral of white noise to be Brownian motion. (And it would not be for such a process.)

To rigorously define white noise, we could start with a Brownian motion, B(t). Each sample path, t → B(ω,t), has a derivative in the space of generalized functions on the positive half-line. Call this derivative W(ω). Then W(ω) is our white noise process. Strictly speaking, it does not have pointwise values. We can only integrate it against test functions. Formally, we would have

φ(t)W(t) dt = -∫ φ'(t)B(t) dt
= ∫ φ(t) dB(t),

where this last integral is the Ito integral.

If we want to consider the "process" σW, then this is the distributional derivative of σB, and var(σB(t + h) - σB(t)) = σ2h. If we want to look at difference quotients of σB (which diverge, of course), then we have

var((σB(t + h) - σB(t))/h) = σ2/h.

So even heuristically, the variance of white noise at a single point in time should be infinite. For a more accurate heuristic, we might say that

var((1/h)∫tt+h σW(s) ds) = σ2/h.
 
  • #3
Thanks !
 

1. What is autocorrelation of white noise?

Autocorrelation of white noise is a statistical measure that quantifies the relationship between a series of random data points. It measures the degree to which a data point is related to its neighboring data points. In other words, it measures how similar a data point is to the points that come before and after it.

2. How is autocorrelation of white noise calculated?

The autocorrelation of white noise is calculated by taking a series of data points and comparing them to their neighboring data points. This is done by computing the correlation coefficient, which is a measure of the strength and direction of the relationship between two variables. The correlation coefficient ranges from -1 to 1, with 0 indicating no correlation and a value of 1 or -1 indicating a perfect positive or negative correlation, respectively.

3. What is the significance of autocorrelation of white noise?

The autocorrelation of white noise is significant because it can help determine the presence of patterns or trends in a series of random data. If the autocorrelation is high, it indicates that there is a strong relationship between data points, which may suggest the presence of a pattern or trend. On the other hand, a low autocorrelation may indicate that the data is truly random and does not follow any specific pattern.

4. How does autocorrelation of white noise impact statistical analysis?

The presence of autocorrelation in a dataset can affect the accuracy and reliability of statistical analyses. When autocorrelation is present, it means that the data points are not independent of each other, which violates one of the assumptions of many statistical tests. This can lead to incorrect conclusions and biased results.

5. Can autocorrelation of white noise be prevented?

No, autocorrelation cannot be prevented as it is a natural result of random data. However, it can be reduced by using techniques such as randomization or shuffling of data points. Additionally, it is important to be aware of the presence of autocorrelation in data and adjust statistical analyses accordingly to mitigate its impact.

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