Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 204

with

+···+ γ N z N Γ ( z ) = 1 + γ 1 z (2.459)

as the z-transform of the prediction error filter and

h − N z − N h 1 z − 1 H ( z ) = +···+ (2.460)

as the z-transform of the sequence h j .AsexplainedinSection2.1.5, Ψ ( z ) can be

factored as

F ( z ) F ∗ (1/ z ∗ ) F ( z ) R ( z )/ z N Ψ ( z ) = = , (2.461)

where F ( z ) is the z-transform of a wavelet,

f N z N F ( z ) = f 0 + f 1 z +···+ , (2.462)

and

f N + f N − 1 z f 0 z N R ( z ) = +···+ (2.463)

is the z-transform of the reverse wavelet. Multiplying the z-transform of the predic-
tion error equations (2.458) by z N / R ( z ) and then performing polynomial division,

we find that

z N P N + 1
R ( z ) +

z N H ( z )
R ( z )

Γ ( z ) F ( z ) =

f 1
f ∗ 2

h 1 z − 1
f 0 +··· .

P N + 1
f 0 −

P N + 1 z − 1 = +···+ (2.464)

0

The left-hand side of this expression is a polynomial of degree 2 N , while the right-
hand side, except for the first constant term, consists of terms in negative powers
of z . Thus,

P N + 1
f 0 =

Γ ( z ) F ( z ) = f 0 . (2.465)

The reciprocal of Ψ ( z ) can then be written

2
P N + 1 Γ

1

F ( z ) F ∗ (1/ z ∗ ) = | f 0 |

1

Γ ∗ (1/ z ∗ )

( z ) =

( z )

Ψ

Γ ∗ (1/ z ∗ )
P N + 1 .

= Γ

( z )

(2.466)

The condition (2.454) for the entropy rate to be an extremum (a maximum) with
respect to the unknown autocorrelations provides the identification

( z )
( P N + 1 ) 1/2

2 f N Γ

G ( z ) = (2.467)