## SPcorXpc

#### Routine

double SPcorXpc (const float rxx[], float pc[], int Np)

#### Purpose

Find predictor coefficients from autocorrelation values

#### Description

This routine calculates the coefficients for a linear predictor which minimizes the resulting mean square error given the autocorrelation of the input signal. Consider a linear predictor with Np coefficients,

              Np
      y(k) = SUM p(i) x(k-i) ,
             i=1

where x(k) is the input signal. The prediction error is

      e(k) = x(k) - y(k) .

To minimize the mean-square prediction error, solve

      R p = r,

where R is a symmetric positive definite covariance matrix, p is a vector of predictor 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) x(k-j)],  for 1 <= i,j <= Np,
        r(i) = E[x(k) x(k-i)],    for 1 <= i <= Np.

The resulting mean-square prediction error can be expressed as

      perr = Ex - 2 p'r + p'R p
           = Ex - p'r ,

where Ex is the mean-square value of the input signal,

      Ex = E[x(k)^2].

For this routine, the matrix R must be symmetric and Toeplitz. Then

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

If the correlation matrix is numerically not positive definite, or if the prediction error energy becomes negative at some stage in the calculation, the remaining predictor coefficients are set to zero. This is equivalent to truncating the autocorrelation coefficient vector at the point at which it is positive definite.

This subroutine solves for the predictor coefficients using Durbin's recursion. This algorithm requires

      Np    divides,
      Np*Np multiplies, and
      Np*Np adds.

Predictor coefficients are usually expressed algebraically as vectors with 1-offset indexing. The correspondence to the 0-offset C-arrays is as follows.

      p(1) <==> pc[0]       predictor coefficient corresponding to lag 1
      p(i) <==> pc[i-1]     1 <= i < Np

#### Parameters

<- double SPcorXpc
  Resultant mean-square prediction error
-> const float rxx[]
  Np+1 element vector of autocorrelation values. Element rxx[i] is the autocorrelation at lag i.
<- const float pc[]
  Np element vector of predictor coefficients. Coefficient pc[i] is the predictor coefficient corresponding to lag i+1.
-> const float rxx[]
  Np+1 element vector of autocorrelation values. Element rxx[i] is the autocorrelation at lag i.
-> int Np
  Number of predictor coefficients

#### Author / revision

P. Kabal / Revision 1.19 2003/05/09

#### See Also

SPcovXpc, SPpcXec

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