Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 203

c ∗ − k .Then The realness of 1/ S ( f ) requires that c k =

N

1
S ( f ) =

1
2 f N

c − k e − i 2π fk Δ t . (2.451)

k =− N

If C ( z ) is the z-transform of c N , c N − 1 ,..., c − N + 1 , c − N ,then C ( z ) can be factored, as
in(2.58),togive

G ( z ) G ∗ (1/ z ∗ ) C ( z ) = (2.452)

with G ( z ) a polynomial of degree N . As described in Section 2.1.5, G ( z ) can be put
in minimum delay form so that its roots lie outside the unit circle in the complex
z -plane, while those of G ∗ (1/ z ∗ ) lie inside the unit circle. If

Φ

( z ) is the z-transform
of the infinite autocorrelation series in expression (2.336) for the spectral density,
as outlined in Section 2.3.2,

e − i 2π f Δ t 2 f N S ( f ) =Φ ( z )f r z = . (2.453)

Then, (2.451) can be replaced with

1
Φ

1
4 f N

1
4 f N

G ( z ) G ∗ (1/ z ∗ ), z e − i 2π f Δ t

( z ) =

C ( z ) = = . (2.454)

As well as satisfying the extremum conditions (2.449), the spectral density must
also be consistent with the known autocorrelations

φ( − N ),φ( − N + 1),...,φ(0),...,φ( N − 1),φ( N ). (2.455)

If these have the z-transform Ψ ( z ), then like terms in the z-transform Φ ( z )must

be identical where the two overlap. ( z ) itself is indefinitely long. To apply this

restriction we use the prediction error equations (2.80).
The prediction error equations can be expressed in the form

Φ

k = 0 γ k φ( j

N

0

− = =

k ) P N + 1 δ

j ,

j 0,1,..., N . (2.456)

The sum can be converted to a convolution, giving

− N γ k φ( j

N

0
j

− k ) = P N + 1 δ + h j , j =− N ,...,0,..., N , (2.457)

sinceγ k vanishes for k negative and the sequence h j is taken to vanish for 0 ≤ j ≤ N .

Then, on taking z-transforms of both sides of this equation, we get

Γ ( z ) Ψ ( z ) = P N + 1 + H ( z ), (2.458)