SPcorPmse

Routine

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

Purpose

Calculate the mean-square prediction error

Description

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

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

The mean-square prediction error is E[e(k)^2] or in vector-matrix notation

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

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

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

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

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

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 SPcorPmse

Resultant mean-square prediction error

-> 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

See Also