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<HTML>
<HEAD>
<TITLE>TSP (libtsp/SP) - SPcorFilt</TITLE>
</HEAD>
<BODY BGCOLOR="#FFFACD">
<H2>SPcorFilt</H2>
<HR>
<H4>Routine</H4>
<DL>
<DT>
double SPcorFilt (double Ed, const float rxx[], const float r[], float h[],
int N)
</DL>
<H4>Purpose</H4>
<DL>
<DT>
Find filter coefficients to minimize the mean-square error
</DL>
<H4>Description</H4>
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
<PRE>
N-1
y(k) = SUM h(i) x(k-i-Nd) ,
i=0
</PRE>
where x(i) is the input signal. The filter error is
<P>
<PRE>
e(k) = d(k) - y(k) ,
</PRE>
<P>
where d(k) is the desired signal. To minimize the mean-square filtering
error, solve
<P>
<PRE>
R h = r,
</PRE>
<P>
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
<P>
<PRE>
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.
</PRE>
<P>
For this routine, the matrix R must be symmetric and Toeplitz. Then
<P>
<PRE>
R(i,j) = rxx(|i-j|).
</PRE>
<P>
The solution is determined using Levinson's method. The resulting
mean-square filtering error can be expressed as
<P>
<PRE>
ferr = Ed - 2 h'r + h'R h
= Ed - h'r ,
</PRE>
<P>
where Ed is the mean-square value of the desired signal,
<P>
<PRE>
Ed = E[d(k)^2] .
</PRE>
<H4>Parameters</H4>
<DL>
<DT>
<- double SPcorFmse
<DD>
Resultant filter mean-square error
<DT>
-> double Ed
<DD>
Signal energy for the desired signal. This value is used only for the
computation of the mean-square error.
<DT>
-> const float rxx[]
<DD>
N element vector of autocorrelation values. Element rxx[i] is the
autocorrelation at lag i.
<DT>
-> const float r[]
<DD>
N element vector of cross-correlation values. Element r[i] is the
cross-correlation at lag Nd+i.
<DT>
<- float h[]
<DD>
N element vector of filter coefficients. Coefficient h[i] is the filter
coefficient corresponding to lag Nd+i.
<DT>
-> int N
<DD>
Number of elements in each of the vectors rxx, h and r.
</DL>
<H4>Author / revision</H4>
P. Kabal Copyright (C) 1997
/ Revision 1.4 1997/10/10
<H4>See Also</H4>
<A HREF="SPcorFmse.html">SPcorFmse</A>,
<A HREF="SPcorXpc.html">SPcorXpc</A>
<P>
<HR>
Main Index <A HREF="../libtsp.html">libtsp</A>
</BODY>
</HTML>
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