Given N samples of speech, we would like to compute estimates to that result in the best fit. One reasonable way to define ``best fit'' is in terms of mean squared error. These can also be regarded as ``most probable'' parameters if it is assumed the distribution of errors is Gaussian and a priori there were no restrictions on the values of .
The error at any time, , is:
Hence the summed squared error, E, over a finite window of length N is:
The minimum of E occurs when the derivative is zero with respect to each of the parameters, . As can be seen from equation 67 the value of E is quadratic in each of the therefore there is a single solution. Very large positive or negative values of must lead to poor prediction and hence the solution to must be a minimum.
Figure 38: Schematic showing single minimum of a quadratic
Hence differentiating equation 67 with respect to
and setting equal to zero gives the set of p equations:
rearranging equation 69 gives:
Define the covariance matrix, with elements :
Now we can write equation 70 as:
or in matrix form:
Hence the Covariance method solution is obtained by matrix inverse:
Note that is symmetric, i.e. , and that this symmetry can be expoited in inverting (see ).
These equations reference the samples .