Spectral Analysis by Stationary Time Series Modeling - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 148

p p

()

∑∑

k
nn

( )

= = − − 1for ≥ [6.24]

x k B z a x k k p

n

n = 1 n = 1

which leads to:

p p

( )

∑

( )

∑

( ) ( )

xk = axkn − + aukn pk N − ≤ ≤ − 1 a = 1 [6.25]

n n 0

n = 1 n = 0

We find the well-known equivalence between a sum of p noisy exponentials and
the ARMA(p, p) model for which the AR and MA coefficients are identical. The
poles z n are deduced from the polynomial roots:

p p

( )

()

∑

− n

∏

− 1

az = az = 1 − z z [6.26]

n n

n = 0

n = 1

The classic estimation method of the AR parameters in the estimation procedure
of the Prony model parameters consists of minimizing the quadratic error:

2

p

N − 1

∑∑

() ( )

e = min

n

x k + a x k − n [6.27]

n

a

kp = n = 1

which, in a matrix form, is written as:

2

( ) ( ) ()

⎡⎤

⎡ xp − 1 x 0 ⎤ ⎡ ⎤

⎢⎥

a xp

1

⎢ ⎥ ⎢ ⎥

min +

⎢⎥

⎢ ⎥ ⎢ ⎥

⎢⎥

a

[6.28]

( ) ( ) ()

⎢ ⎥ ⎢ ⎥

xN − 1 xN p − a xN

⎣ ⎦ ⎣ ⎦

⎣⎦

p

2

min Xa + x

a

and leads to the least-squares solution a :

LS

( ) 1

−

H H

a =− X X X x . [6.29]

LS

This method is sometimes called LS-Prony. When the system is solved in the
total least-squares sense, we speak of TLS-Prony. We will note that the matrix X H X
is the estimation of the covariance matrix, with one multiplying factor (

1

N − ) . This
method is thus identical to the covariance method for the estimation of the AR