## SPcorFmse

#### Routine

double SPcorFmse (const float h[], double Ed, const float rxx[], const float r[], int N)

#### Purpose

Calculate the mean-square filtering error

#### Description

This function calculates the mean-square error for a linear filter. Consider a filter with N coefficients, with coefficient h(i) corresponding to lag Nd+i. The filter output is

             N-1
      y(k) = SUM h(i) x(k-i-Nd) ,
             i=0

where x(i) is the input signal. The filter error is

      e(k) = d(k) - y(k) ,

where d(k) is the desired signal. The mean-square filtering error is E[e(k)^2] or in vector-matrix notation

      ferr = Ed - 2 h'r + h' R h ,

The mean-square value E0, matrix R and vector p are defined as follows

          Ed = E[d(k)^2]
      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, viz.

      R(i,j) = rxx(|i-j|).

Linear prediction can be cast into the above form, if we let Nd=1. Also for linear prediction, usually d(k)=x(k), giving r(i)=rxx(i).

#### Parameters

<- double SPcorFmse
  Resultant mean-square error
-> const float h[]
  N element vector of filter coefficients. Coefficient h[i] is the filter coefficient corresponding to lag Nd+i.
-> 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.
-> int N
  Number of elements in each of the vectors rxx, h and r.

#### Author / revision

P. Kabal / Revision 1.12 2003/05/09

#### See Also

SPcorFilt

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