|Posted by:||Jack (NOSP…@THANK.YOU)|
|Date:||Mon, 15 May 2006|
I have read some litterature about linear prediction.
However, I haven't been able to find any litterature
that explains in mathematical terms exactly why
minimization of the variance of the residual leads
to an estimate of the all-pole filter coefficients.
The known output of a 10th order unknown all-pole filter is:
where the unknown u[k] is defined to be random noise with
variance 1 and the unknown, constants a[q] are the coefficients
of the all-pole filter.
Now...if I send x[k] through a FIR filter I get:
where the constants b[q] are the coefficients
of the FIR filter.
I can see that choosing b[q]=a[q] leads to
but I don't understand why minimizing the
variance of e[k] guarantees that
the resulting estimates of b[q] are
close to a[q]
How do I prove that mathematically?
I haven't been able to find such a proof
Maybe some of you guys could help me?