### What is Noise?

Consider the spectrum analysis of the following sequence:

x = [-1.55, -1.35, -0.33, -0.93, 0.39, 0.45, -0.45, -1.98]In the absence of any other information, this is just a list of numbers. It could be temperature fluctuations in some location from one day to the next, or it could be some normalization of successive samples from a music CD. There is no way to know if the numbers are ``random'' or just ``complicated''.

^{7.2}More than a century ago, before the dawn of quantum mechanics in physics, it was thought that there was no such thing as true randomness--given the positions and momenta of all particles, the future could be predicted exactly; now, ``probability'' is a fundamental component of all elementary particle interactions in the Standard Model of physics [59].

It so happens that, in the example above, the numbers were generated
by the `randn` function in matlab, thereby simulating normally
distributed random variables with unit variance. However, this cannot
be definitively inferred from a finite list of numbers. The best we
can do is estimate the *likelihood* that these numbers were
generated according to some normal distribution. The point here is
that any such analysis of noise *imposes the assumption* that the
noise data were generated by some ``random'' process. This turns out
to be a very effective model for many kinds of physical processes such
as thermal motions or sounds from turbulent flow. However, we should
always keep in mind that any analysis we perform is carried out in
terms of some underlying *signal model* which represents
*assumptions* we are making regarding the nature of the data.
Ultimately, we are fitting models to data.

We will consider only one type of noise: the *stationary
stochastic process* (defined in Appendix C). All such noises can
be created by passing white noise through a linear
time-invariant (stable) filter [263]. Thus, for purposes of this book,
the term *noise* always means ``filtered white noise''.

#### Testing for White Noise

To test whether a set of samples can be well modeled as white
noise, we may compute its *sample autocorrelation* and verify
that it approaches an *impulse* in the limit as the number of
samples becomes large; this is another way of saying that successive
noise samples are *uncorrelated*. Equivalently, we may break the
set of samples into successive blocks across time, take an FFT of
each block, and average their squared magnitudes; if the resulting
average magnitude spectrum is *flat*, then the set of samples
looks like white noise. In the following sections, we will describe
these steps in further detail, culminating in *Welch's method*
for noise spectrum analysis, summarized in §6.9.

**Next Section:**

Matlab for the Periodogram

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Why Analyze Noise?