Spectral Analysis by Stationary Time Series Modeling - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 149

coefficients and of the signal poles. This method can be slightly changed in the case
of

p non-damped real sinusoids to force the estimated poles to be of unit module:

2

2

a

⎡ ⎤
⎢ ⎥
⎢ ⎥

1

( ) ( ) ( )

⎡ xNp * −+ 1 xN * ⎤ ⎡ xNp * − ⎤

⎢ ⎥ a

⎢ ⎥

p

⎢ ⎥

− 1

⎢ ⎥

2

⎢ ⎥ ⎢ ⎥

⎢ ⎥

a

() ( ) ()

()

⎢ ⎥ ⎢ ⎥

x *1 x * p x * 0

p

⎢ ⎥ +

min ⎢ ⎥ ⎢ ⎥ [6.30]

2

⎢ ⎥

( ) ( )

xp − 1 x 0 xp

a

⎢ ⎥ ⎢ ⎥

a

n

⎢ ⎥

p

⎢ ⎥

− 1

⎢ ⎥

⎢ ⎥

2

⎢ ⎥ ⎢ ⎥

⎢ ⎥

( ) ( ) ()

xN − 1 xN p − xN

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

⎢ ⎥
⎢ ⎥
⎢ ⎣ ⎦

a

1

1

We speak then of the harmonic Prony method . Other estimation methods of the
AR parameters , implementing the correlation, can be used: LSYW and LSMYW , for
example.

Once the poles are estimated ()

ˆ n

z , the complex amplitudes are obtained by

solving the Vandermonde system:

1 1 1

⎡ ⎤

()
()
()

⎡ x 0 ⎤

⎢ ⎥

ˆ z z ˆ z ˆ ⎢ ⎥

⎢

1 2 p

⎥ ⎡⎤

B x 1

1

⎢ ⎥

⎢ ⎥ ⎢⎥

2 2 2

ˆ ˆ ˆ ⎢ ⎥

z z z

≈ x 2

⎢

2 p

⎥ ⎢⎥ 1

[6.31]

⎢ ⎥

⎢⎥

⎢ ⎥

B

⎢ ⎥

⎣⎦

p

⎢ ⎥

⎢ ⎥

( )

xM − 1

MM − 1 − 1 M

p

− 1

⎢ ⎥

z ˆ ˆ z ˆ z ⎣ ⎦

⎣

2

⎦

1

ˆ

VB ≈ x

Solutions in the least-squares sense and in the total least-squares sense can be
envisaged. It was shown in [DUC 97] that the total least-squares solution is less
efficient in terms of bias than that of the classic least-squares. On the other hand, the
number of equations M of the Vandermonde system can be chosen in an optimal
manner, as detailed in [DUC 95].