Spectral Analysis by Stationary Time Series Modeling - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 145

and an order p = 10. Figure 6.1 represents the plot of the poles estimated in the
complex plane for a signal realization.

We notice that the poles linked to the noise of the LSMYW method are clearly
damped in relation to those obtained with TLSMYW which sometimes fall outside
the unit circle. When the order is overestimated, the TLSMYW method should be
avoided. However, for a correct order, the estimation by TLSMYW is better than
that of LSMYW.

.

.

Figure 6.1. Influence of the order overestimation for the LSMYW and TLSMYW methods

Up to now, the LSMYW method is one of the most efficient of those that we
have described [CHA 82, STO 89]. Stoica and Söderström [S Ö D 93] have given the
asymptotic performances of this method in the case of a noisy exponential (white
noise) of circular frequency ω = 2 π f and amplitude A :

4

( )

22 p + 1

1 σ

⎡

2

⎤

() ( )

u

Var ω ˆ = lim E ω ω ˆ − = [6.21]

⎢ ⎥

⎣ ⎦

4

( )

2
0

N A

3 pp + 1 N

N →∞

where p represents the order and N 0 ≥ p the number of equations. This expression
can be compared to the asymptotic Cramer-Rao bound of the circular frequency ω
(see section 3.4.1):

2

6 σ

()

u

CRB ω = [6.22]

23

A N

Or we could think that the variance of HOYW [6.21] can become smaller than
the Cramer-Rao bound. Actually, this is not the case at all if it is only asymptotically
that it can become very close to Cramer-Rao. [S Ö D 91] compares this method to the
Root-MUSIC and ESPRIT methods (see Chapter 8) in terms of precision and