Time-Series Models - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 103

in the polar form:

p p

( ) {}

∑ ∏

− n − 1

{} ()

a ⇒ A z = a z = 1 − p z ⇒ p

n n n n

n = 0 n = 1

and the stability condition:

p

<∀ 1 n

[4.15]

4.2.2.2. Moments and spectra
If we limit ourselves to zero mean signals, the general theorems given in Chapter
1 on the multispectra make it possible to obtain the general relation [NIK 93]:

( ) ( ) ( )

S v , … , v S v , … , v . Hv , … , v

[4.16] =

xx x 1 n − 1 nn n 1 n − 1 1 n − 1

with:

( )

( ) ( ) ( )

( ) ( ) ( )

[ ]

Hv ,, … v h p2 i v π … h p2 i v π h * p2 i π v … v

= + +

1 n − 1 1 n − 1 1 n − 1

If we consider the whiteness of the excitation n ( k ) at the considered order:

( ) ( )

S v , … , v = C , … , 0 ∀ v , … , v

nn … n 1 n − 1 nn … n 1 n − 1

we obtain an expression, which generalizes equation [4.12] to all orders. This helps
obtain useful spectral relations for this topic. However, the equation for generating
the signal x ( k ) of equation [4.7] helps write a direct fundamental relation on the
moments.

To start with, let us consider the autocorrelation function of x ( k ), γ xx ( m ) . It is easy
to directly show (for example, see [KAY 88]) from equation [4.7] that:

( ) () () ( )

γ mExkxkm = * −

xx

p q

∑ ∑

( )

2

( )

= a γ mn − + σ b h * mn − 0 ≤ ≤ mq

[4.17]

n x n

n = 1 n = m

p

∑

( )

= a γ mn − mq ≥ + 1

n x

n = 1