Subspace-based Estimators - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 217

When the noise variance is estimated, it can then be substracted from the
covariance matrix Γ xx so that the propagator is then extracted from [8.51]:

( ) ( )

†

ˆ

2 2

PG σ ˆ H σ ˆ [8.57]

Let us note that this estimation of the noise variance was taken again in [STO 92]
and the authors have especially shown that this method was less complex than the
search of the smallest eigenvalue of Γ xx . Actually, we evaluate here at
( M - P ) P 2 + P 3 /3 the number of multiplications necessary to the calculation of Π,
then at ( M - P ) 2 the number of multiplications necessary to the calculation of

ˆ σ

Along the same line, a joint estimation of ( P , σ 2 , P ) was obtained in [MAR 90a].
It should be noted that the joint determination of the number of complex sine waves,
of the noise variance and of the propagator (and therefore of the noise and signal
subspaces) remains less complex in calculation than the search of the eigenvalues of
a matrix of dimension M × M and this being all the more so as the number of
observations is bigger compared to the number of frequencies to estimate. The
asymptotic performance of the propagator method was analyzed and compared to
those of MUSIC and of the other “linear” methods in [MAR 98, Chapter 10]. There
it is shown that the propagator method has performance which is superior to the
other “linear” methods and performance equivalent to MUSIC.

8.6. The ESPRIT method

The ESPRIT method (Estimation of Signal Parameters via Rotational Invariance
Techniques) [ROY 86] initially designed for the array processing can be applied to
spectral analysis. Its main advantage is that it considerably reduces the cost in
calculations of the estimate of the frequencies. Actually, this technique, rather than
requiring minima or maxima in a pseudospectrum, uses the search of the
eigenvalues of a matrix of dimension P × P from where we can directly extract the
estimates of the pure frequencies contained in the signal. In array processing, this
complexity reduction is obtained with the constraint that the antenna possesses an
invariant form by translation ( translational displacement invariance ). In other
words, the antenna is made up of doublets of sensors obtained by the same
translation. It is the case particularly when the antenna is uniform linear (ULA
model). We will directly give the application of the ESPRIT method to the spectral
analysis and for the more general case of the antenna processing we recommend the
reader to refer to [MAR 98, Chapter 3].