double SPcorFilt (double Ed, const float rxx, const float r, float h,
Find filter coefficients to minimize the mean-square error
This procedure finds the filter coefficients for a linear filter which
minimizes the mean-square error. Consider a filter with N coefficients,
with coefficient h(i) corresponding to lag Nd+i. The filter output is
y(k) = SUM h(i) x(k-i-Nd) ,
where x(i) is the input signal. The filter error is
e(k) = d(k) - y(k) ,
where d(k) is the desired signal. To minimize the mean-square filtering
R h = r,
where R is a symmetric positive definite correlation matrix, h is a vector
of filter coefficients and r is a vector of correlation values. The matrix
R and and vector r are defined as follows
R(i,j) = E[x(k-i-Nd) x(k-j-Nd)], for 0 <= i,j < N,
r(i) = E[d(k) x(k-i-Nd], for 0 <= i < N.
For this routine, the matrix R must be symmetric and Toeplitz. Then
R(i,j) = rxx(|i-j|).
The solution is determined using Levinson's method. The resulting
mean-square filtering error can be expressed as
ferr = Ed - 2 h'r + h'R h
= Ed - h'r ,
where Ed is the mean-square value of the desired signal,
Ed = E[d(k)^2] .
<- double SPcorFmse
Resultant filter mean-square error
-> double Ed
Signal energy for the desired signal. This value is used only for the
computation of the mean-square error.
-> const float rxx
N element vector of autocorrelation values. Element rxx[i] is the
autocorrelation at lag i.
-> const float r
N element vector of cross-correlation values. Element r[i] is the
cross-correlation at lag Nd+i.
<- float h
N element vector of filter coefficients. Coefficient h[i] is the filter
coefficient corresponding to lag Nd+i.
-> int N
Number of elements in each of the vectors rxx, h and r.
Author / revision
/ Revision 1.6 2003/05/09
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