The direct-power-spectrum (DPS) algorithm outlined above is insufficient
for estimating the PSD of a stationary noise
signal because the variance of the estimated PSD is proportional to the
value of the actual PSD. For the third part of this lab you will try to
reduce the variance of the PSD estimate by windowing the
autocorrelation of the noise signal and computing the fft.

The autocorrelation of a sequence is the correlation of the
sequence with itself:

Rm=∑
i
=0N−1−|m|xixi+|m|
R
m
i
0
N
1
m
x
i
x
i
m

(2) where

m∈−(N−1)−(N−2)…N−1
m
N
1
N
2
…
N
1
For non-deterministic signals, the autocorrelation here is an estimate
of the actual autocorrelation.
As |m|m is
increased, the number of samples used in the autocorrelation decreases.
The windowed-DPS algorithm is equivalent to taking the FFT of the
autocorrelation of the windowed data. Windowing of the data
adds even more noise to the autocorrelation estimate, since the
autocorrelation is performed on a distorted version of the original
signal. An improvement can be made by constructing an accurate estimate
of the autocorrelation (using a rectangular window),
applying a window and computing the FFT. The motivation for applying
the window at the latter stage is that emphasis should be given to
accurate autocorrelation values while less accurate values should be
de-emphasized or discarded.

A good empirical characterization of a random process requires
sufficient data, and both of the PSD-estimation algorithms defined
above can be extended to accomodate more data. There is one caveat,
however: many
real-world processes are modeled as *short-time*
stationary processes (non-stationary models are hard to deal with),
so there is a practical limit to how much data is available for a PSD
estimate.
Additional data is added to the direct-PSD estimation algorithm by
adding multiple spectra together, thereby smoothing the PSD estimate.
Additional data is added to the windowed-autocorrelation method
by computing the autocorrelation of the total data set before
windowing. You will explore the windowed-autocorrelation method on
the DSP.