![\( s[n]=\sum\limits _{i=1}^{P}s[n-i]a[i]+e[n] \)](img9.png)
Now that we've identified the audio sample damaged by the click (whose true values are unknown), we need to estimate the unknown samples. First we start by finding P coefficients a[i] for the autoregressive model, which is:
(1)
where s[n] is the signal value for the sample number n, and e[n] is the error term of the autoregressive model for sample n. For a simple example we'll let P=2 and n=6. The coefficients are determined using ordinary least squares techniques using good samples before and behind the unkown section. (1) can be rearranged:
(2)
![\( e[n]=s[n]-\sum\limits _{i=1}^{P}s[n-1]a[i] \)](img10.png)
This can be further written in matrix notation as:
e = A * s
s (the original row vector of signal data) is modified, so that all the unknown sample values are set to zero, i.e. s = [s1 s2 0 0 s5 s6 ].
A is constructed so the product A * s will produce (2):
A =
![\( \left[ \begin{array}{cccccc}
-a_{2} & -a_{1} & 1 & 0 & 0 & 0\\
0 & -a_{2} & ...
...-a_{2} & -a_{1} & 1 & 0\\
0 & 0 & 0 & -a_{2} & -a_{1} & 1
\end{array}\right] \)](img11.png)
Au is now constructed from the columns in A corresponding to the unknown signal columns in s:
Au =
![\( \left[ \begin{array}{cc}
1 & 0\\
-a_{1} & 1\\
-a_{2} & -a_{1}\\
0 & -a_{2}
\end{array}\right] \)](img12.png)
(And then a matrix algebra miracle occurs), ``the minimum variance unbiased missing estimator for the missing data 1'' is
