Subspace-based Estimators - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 208

The uniqueness of the P roots on the unit circle for determining the P
frequencies result from Theorem 8.2. It is important to note that P ( z ) possesses
2 M - 2 symmetric roots in relation to the unit circle, i.e. if z i is the solution, then 1/ z i
is also the solution. Thus, there are M - 1 roots of module less than or equal to 1
(inside the unit circle) and the searched frequencies thus correspond to the P roots
which are on the unit circle. In the practical case, we have only

ˆ xx

Γ

and we thus

choose as estimate the P roots which are closest to the unit circle.

The relation between the polynomial version of MUSIC and the pseudo-spectral
version is established by noting that:

M − 1

( )

2

j 2 π f

∏

j 2 π f

[8.26] Pe = a e − z

i

i = 1

j 2π f

where a is a constant. P (

e

) is thus the sum of the squares of the distances of

j e is aligned with a root close
to the unit circle. We show that the performance of the pseudo-spectral and
polynomial versions of MUSIC is the same [MAR 98, Chapter 9]. The interest of
this version resides in the complexity. Searching the zeros of a polynomial is much
less costly in calculation than searching the peaks in a pseudo-spectrum.

j 2π f 2π

e with the roots of P ( z ); it is minimal when

The MUSIC method is known for its superior performance, especially in
resolution, compared to the other methods. In particular, contrary to the methods
presented before, the estimates of the pure frequencies obtained by the MUSIC
method converge to their true values while the number of samples N used in the
estimation of the covariance [8.5] and [8.6] tends to infinity. The MUSIC method is
thus a high-resolution method.

However, the limitations of the MUSIC method are important. First, the MUSIC
method is based on a precise modeling of the noise by supposing that the covariance
matrix of the noise is known with a multiplying factor. In the case of array
processing, the MUSIC method is also less robust with a poor knowledge of the
propagation model (i.e. the form a of the source vectors) than the beamforming
method (equivalent to the periodogram method) or the minimum variance method.
The MUSIC method also requires a determination of the number of complex sine
waves before building the estimator of the frequencies. The computational
complexity of the MUSIC method, particularly for the search of the eigenelements
of the covariance matrix, of the order of M 3 is an important constraint, which pushes
the searchers to conceive less complex algorithms (see the section on linear
methods). Finally, we will refer to [MAR 98, Chapters 9 and 11] for the study of the
performance and of the robustness of the MUSIC method.