Spectral Analysis by Stationary Time Series Modeling - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 142

p

() () ( )

⎦ ∑

( )

2

()

γ mExkxkm = ⎡ − ⎤ = − a γ mn − + σδ m

[6.13]

*

⎣

xx x xx n

n = 1

which we call Yule-Walker equations, which were already mentioned in Chapter 4.
This result shows that the autocorrelation of x ( k ) satisfies the same recursion as the
signal. A large number of estimation methods solve the Yule-Walker equations by
replacing the theoretic autocorrelation () by an estimation ()

ˆ xx

γ m γ m

:

xx

− 1

⎡

()

*

()

*

( )

⎤ − ⎡

()
()

γ ˆ 0 γ ˆ 1 γ ˆ p 1 γ

γ

ˆ 1 ⎤

xx xx xx xx

xx

⎢ ⎥ ⎢

⎥

*

ˆ 2

() () ( )

⎢ ˆ ˆ ˆ ⎥ ⎢

γ 1 γ 0 γ p

⎥

ˆ

xx xx xx

a =− ⎢

[6.14]

⎥ ⎢

⎥

⎢ ⎥ ⎢

⎥

()

γ ˆ p

⎢

( ) ( )

⎥ ⎢

ˆ ˆ ⎥

γ p − 1 γ 0

⎣ ⎦

xx

⎣ ⎦

xx xx

ˆ

r

ˆ

-1

x

R

x

ˆ x

It is the autocorrelation method and is the (estimated) autocorrelation R

()

matrix. When

ˆ xx γ is the biased estimator of the correlation, the poles are always
inside the unit circle, which is not the case with the non-biased estimator which
gives, however, a better estimation. The Levinson-Durbin algorithm [LEV 47]
provides an order recursive solution of the system [6.14] in O ( p 2 ) computational
burden. In order to reduce the noise influence and to obtain a better estimation, the
system [6.14] can use a larger number of equations (>> p ) and be solved in the least-
squares sense (LS) or in the total least-squares sense (TLS) [VAN 91]. These
methods are called LSYW and TLSYW (see [DUC 98]):

( )

− 1

ˆˆ ˆ

H H

ˆ ˆ

LSYW a =− RRRr

xx xx

( )

− 1

ˆˆ ˆ

H 2

min

H

ˆ ˆ

TLSYW a =− RR − σ I Rr

xx xx

[6.15]

⎡

()

*

( )

⎤

()

ˆ ˆ ˆ

γ 0 γ p − 1 ⎡ γ 1 ⎤

xx xx xx

⎢ ⎥

⎢ ⎥

ˆ

ˆ

R r =

⎢ ⎥

⎢ ⎥

x x

⎢ ⎥

( ) ( )

⎢

( )

⎥

ˆ ˆ ˆ

γ N − 1 γ N − p γ N

⎣ ⎦

⎣

xx 0 xx 0

⎦

xx 0

ˆ x ⎡ ⎣ ⎦

Rr , I the identity
matrix and N 0 the number of equations. The value of N 0 should be at most of the
order of

where

σ

is the smaller singular value of the matrix

min

N when the non-biased estimator of the correlation is used so as not to
make correlations with a strong variance intervene.

2

The TLS solution is more efficient than the LS method because it minimizes the
errors in ˆ r and in

ˆ x

R

at the same time, but it presents a disadvantage which we