The goal of linear-prediction is to model a signal as a linear combination of
its past/future.
Yule-Walker equations
For a discrete-time signal x, we want to approximate the sample x[n] as a
linear combination xp[n] of the k preceding samples:
xp[n] = -c[1] * x[n-2] - ... - c[k-1] * x[n-k-1]
The best approximation in the mean-square sense is a tuple(c[1], ..., c[k])
such as the squared error:
e = xp - x
Is minimal. Noting p(x) = xp, and x^{-k} the signal x^{-k}[n] = x[n-k], since p
is a linear combination of the (x^-1, ..., x^-k), we know that the error p(x) -
x is minimal for p the orthogonal project of x on the vector space V spanned by
the x^-1, ..., x^-k. In particular, the error e is then orthogonal to any
vector in V:
TODO: decent blob for above
System Message: WARNING/2 (\begin{pmatrix}
-R[1] \\
-R[2] \\
\vdots \\
-R[p]
\end{pmatrix}
=
\begin{pmatrix}
R[0] & \overline{R}[1] & \dots & \overline{R}[p-1] \\
R[1] & R[0] & \ddots & \vdots \\
\vdots & \ddots & \ddots & \overline{R}[1]\\
R[p-1] & \dots & R[1] & R[0]
\end{pmatrix}
\begin{pmatrix}
a[1] \\
\vdots \\
\vdots \\
a[p]
\end{pmatrix})
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