Subspace-based Estimators - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 214

When the noise subspace is determined using the propagator, the
pseudospectrum:

1

()

S f = [8.43]

propa

H H

OO () ()

a f QQa f

yields an estimator of the frequencies. The solutions are the arguments of the
maxima of [8.43].

8.5.2.1. Propagator estimation using least squares technique

The advantage of the propagator is the simple way in which we can obtain it
starting from the covariance matrix.

In the no noise case , the covariance matrix of the data Γ xx [8.4] becomes:

H

Γ = AA Γ [8.44]

xx ss

By using the partition [8.37] of the matrix of the complex sinusoid vectors and
by decomposing the covariance matrix:

H H

[ ]

⎡ ⎤

Γ = AAAA Γ , Γ = GH , [8.45]

xx ss 1 ss 2

⎣ ⎦

where G and H are the matrices of dimension M × P and M × ( M - P ) respectively,
the definition [8.38] implies:

HGP [8.46]

This important relation means that the last M - P columns of the covariance
matrix without the noise belong to the subspace spanned by the first P columns. The
propagator P is thus the solution of the overdetermined linear system [8.46].

Using the hypothesis that the sine waves are not totally correlated, a hypothesis
which is made in the model and which is also necessary for MUSIC, the matrix G is
of full rank and the propagator is obtained by the least-squares solution:

−
PGGGH

( ) 1

H H

[8.47]

In the noisy case , the decomposition [8.45] of the matrix [8.4] is always possible,
but the relation [8.46] is no longer valid. An estimation P of the propagator can be
obtained by minimizing the following cost function: