Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 206

is replaced by its conjugate, φ ∗ ff ( k ). Then, the coe cient matrix of the predic-

tion error equations for backward prediction becomes the conjugate of that for
forward prediction. Taking conjugates of the equations for backward prediction,
we find that

⎝ ⎠ ⎝ ⎠

1
γ ∗ 1
. . .
γ ∗ N

P N + 1
0
. . .
0

⎝ ⎠

φ ff (0) ··· φ ff ( − N )

. . . . . .

. . .

=

. (2.474)

φ ff ( N ) ··· φ ff (0)

By comparison with the equations (2.80) for forward prediction, the prediction
error filter for backward prediction is just the complex conjugate of that for forward
prediction.
Returning to the equations (2.473) for N

=

1, forward prediction gives the error

sequence

j

= f j + 1 + γ 1,1 f j , j = 1,..., M − 1, (2.475)

+ 1,1

while backward prediction gives the error sequence

B
j ,1

f j + γ ∗ 1,1 f j + 1 = j = 1,..., M − 1. (2.476)

For the assumed stationarity of the given sequence, the best estimate of P 2 is the
average of the forward and backward prediction values,

f j + 1

2

M − 1

+ γ 1,1 f j

f j + γ ∗ 1,1 f j + 1

1
2 ( M

2

P 2 = + . (2.477)

− 1)

j = 1

The Burg algorithm is data adaptive in the sense thatγ 1,1 can be chosen to minimise
the value of P 2 calculated from the data. For a minimum, the partial derivatives of
P 2 with respect to the real and imaginary parts of γ 1,1 vanish. Then,

f j f ∗ j + 1 + γ ∗ 1,1 f ∗ j

M − 1

f ∗ j f j + 1 + γ 1,1 f j

∂ P 2
∂Reγ 1,1 =

1
2 ( M

+

− 1)

j = 1

+ γ ∗ 1,1 f j + 1 f j + 1 f ∗ j + γ 1,1 f ∗ j + 1 f ∗ j + 1 f j + + = 0, (2.478)

and

f j f ∗ j + 1 + γ ∗ 1,1 f ∗ j

M − 1

f ∗ j f j + 1 + γ 1,1 f j

∂ P 2
∂Imγ 1,1 =

i
2 ( M

−

1 ) −

j = 1

+ f ∗ j + 1 f j + γ ∗ 1,1 f j + 1 − f j + 1 f ∗ j + γ 1,1 f ∗ j + 1 = 0. (2.479)