Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 108

of the linear filter with impulse response g j . The conditional equations for the
optimum Wiener prediction filter are found to be

k = 1 g k φ ff ( m − k )

N

= φ df ( m ), m = 1,..., N . (2.71)

The summation starts at k = 1, so there is one less equation, since the member

of the time sequence to be predicted cannot be included in the calculation of the
prediction. The perfect prediction would be f j itself. Thus, d j = f j ,and

E d k f k − j E f k f k − j φ df ( j ) = = = φ ff ( j ). (2.72)

The unit prediction equations (2.71) then become

k = 1 g k φ ff ( m − k )

N

= φ ff ( m ), m = 1,..., N (2.73)

with matrix form

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

φ ff (0) ··· φ ff ( − N + 1) g 1

. . .

φ ff (1)
. . .

. . . . . .

. . .

= . (2.74)

φ ff ( N − 1) ··· φ ff (0) g N φ ff ( N )

The quality of the unit prediction is measured by the error sequence j .Itis
given by

k = 1 g k f j − k

N

k = 0 γ k f j − k ,

N

j = d j − h j = f j − = (2.75)

with γ 0 =− g N . Thus, the filter 1,γ 1 ,...,γ N convolved with

the sequence f j gives the prediction error sequence directly. It is called the predic-
tion error filter .

= 1,γ 1 =− g 1 ,...,γ N