Spectral Analysis by Stationary Time Series Modeling - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 147

+∞

1

() ()

= ∑

− n

bz = with cz cz

n

()

cz

n = 0

q

∑

()

c =− b c + δ k

k n k − n

n = 1

We obtain a relation between the MA coefficients and the coefficients of the
equivalent AR model:

q

∑

()

c =− b c + δ k

k n k − n

n = 1

δ ( k ) standing for the Kronecker symbol (δ (0) = 1 and δ ( k ) = 0, for any k ≠ 0).

In practice, we choose to estimate an AR model of high order M such as
M >> q. By using the AR parameters estimated this way, we obtain a set of
equations in the form:

q

=+ ∑

()

ˆ ˆ

ε kc bc [6.23]

k n k − n

n = 1

Ideally, ()

ε k should be null everywhere except for k = 0 where it should be
equal to 1. As the order of the estimated AR is not infinite, it's nothing of the sort
and the estimation of the MA parameters should be done by minimizing a quadratic
criterion of the form:

() 2

ε ∑

k

k

The index k varies on a domain, which differs according to the used estimation
methods. Indeed, equation [6.23] is not unlike the linear prediction error of an AR
model in which the parameters would be the b n and the signal ˆ c . From this fact, the
AR estimation techniques can be envisaged with, as particular cases: k = 0, ..., M + q
corresponding to the method of autocorrelations and k = q , ..., M corresponding to
the covariance method.

6.2.3. Estimation of Prony parameters

The classic estimation method of the Prony model parameters is based on the
recursion expression of a sum of exponentials: