ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 117

Clearly, if noise subspace E n is unreliably estimated (e.g. via eigenvectors of the SCM
when the noise is non-Gaussian or impulsive), then the obtained MUSIC DOA esti-
mators are unreliable. For robust estimation of noise subspace one may use for
example, eigenvectors of M -estimators of scatter, or, eigenvectors of the sample
plug-in estimate

n X

n
i¼ 1 kzk 2 z i z i

1

C sgn W C sgn ( F n ) ¼

(2 : 33)

of the sign covariance matrix C sgn ( z ) as in [59].
A weighted signal subspace fitting (SSF) approach, for example, finds DOAs via
criterion function

Tr[ P A E s YE H

u¼ arg mi u

]

s

where P A ¼ IA ( A H A ) 1 A H is a projection matrix onto the noise subspace and Y
is some weighting matrix. The estimated optimal weighting matrix Y opt is a diagonal
matrix, whose diagonal elements are certain functions of the estimated eigenvalues of
the covariance matrix C ( z ). Hence, reliable and accurate estimation of DOAs via
weighted SSF approach requires robust estimation of the signal subspace E s and eigen-
values of the covariance matrix. These can be obtained, for example, using eigenvec-
tors and eigenvalues of robust M -estimators instead of the SCM.

B EXAMPLE 2.8
Four independent random signals, QPSK, 16-PSK, 32-QAM and BPSK signal of
equal power s s , are impinging on a k ¼ 8 element ULA with l= 2 spacing from
DOAs 2 10 8 ,15 8 ,10 8 and 35 8 . The simulation setting is as in Example 2.3,
except that now we consider two different noise environments. In the first setting,
noise n has circular Gaussian distribution CN k ( s n I ), and in the second setting noise
has circular Cauchy distribution CT k ,1 ( s n I ). Note that the Cauchy distribution does
not have finite variance and s n is the scale parameter of the distribution. In both
simulation settings, the signal to noise ratio (SNR) is 10 log 10 ( s s =s n ) ¼ 20 dB
and the number of snapshots is n ¼ 300. The number of signals ( d ¼ 4) is assumed
to be known a priori . We then estimated the noise subspace E n from eigenvectors
of the SCM C , sample sign covariance matrix (2.33) andMLT(1) estimator. Typical
MUSIC spectrums associated with different estimators are shown in Figure 2.6 for
both the Gaussian and Cauchy noise settings. All the estimators are able to resolve
the four sources correctly in the Gaussian noise case: in fact, the differences in the
spectrums are very minor, that is, they provide essentially the same DOA estimates.
In the Cauchy noise case, however, MUSIC based on the classical sample estimator
C is not able to resolve the sources. The robust estimators, the sign covariance
matrix and the MLT(1) estimator, however, yield reliable estimates of the
DOAs. Based on the sharpness of the peaks, the MLT(1) estimator is performing
better than the sample sign covariance matrix C sgn .