Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 207

Adding i × (2.479) to (2.478) gives

f ∗ j f j + 1 + γ 1,1 f j

f j + 1 f ∗ j + γ 1,1 f ∗ j + 1

M − 1

+ = 0. (2.480)

j = 1

Solving for the value of γ 1,1 that minimises P 2 ,weget

M − 1 M − 1

f j + 1 f ∗ j f j + 1 f ∗ j + 1 +

f j f ∗ j γ 1,1 =− 2

j = 1 j = 1

2

M − 1 M − 1

j ,0 j + 1,0 j ,0

2

F B ∗ F B ∗

=− 2

j + 1,0

+ , (2.481)

j = 1 j = 1

if we interpret the forward prediction error sequence as

F
j + 1,0

= f j + 1 , (2.482)

and the backward prediction error sequence as

B
j ,0

= f j (2.483)

for N 0 is just the unit impulse and

the two prediction error sequences are both just the original data sequence itself.
For this value of γ 1,1 , equations (2.473) give the recurrence relations

= 0. In fact, the prediction error filter for N =

1

− γ 1,1 γ ∗ 1,1 P 1 . φ ff (1) =− γ 1,1 φ ff (0) and P 2 = (2.484)

The next step is reminiscent of the Levinson algorithm (Section 2.2.3). We begin
with the system of equations
⎝

⎠ ⎝ ⎠ ⎝ ⎠

φ ff (0) φ ff ( − 1) φ ff ( − 2) 1

γ 1,1
0

P 2
0
Δ 2

φ ff ( 1 ) φ ff ( 0 ) φ ff ( 1 ) − = , (2.485)

φ ff ( 2 ) φ ff ( 1 ) φ ff ( 0 )

where 2 is defined by the third equation, and the first two equations are those

(2.473) for N

Δ

=

1. Reversing the order of equations and unknowns yields the

system

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

φ ff (0) φ ff (1) φ ff (2) 0

γ 1,1
1

Δ 2

φ ff ( 1 ) φ ff ( 0 ) φ ff ( 1 )

− = 0

P 2

. (2.486)

φ ff (

− 2 ) φ ff ( − 1 ) φ ff ( 0 )