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''.

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 [#!JOSFP!#]. Thus, for purposes of this book,
the term *noise* always means ``filtered white noise''.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University