ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 102

0 : 05 dB and the number of snapshots is n ¼ 300. Since the noise and the sources are
circular, also the marginals z i of the array output z are circular as well, so P ( z ) ¼ 0.
Then, based on 500Monte-Carlo trials, the null hypothesis of (second-order) circu-
larity was falsely rejected (type I error) by GLRT test at a¼ 0 : 05 level in 5.6 per-
cent of all trials. Hence we observe that the GLRT test performs very well even
though the Gaussian data assumption under which the GLRT test statistic l n was
derived do not hold exactly. (Since the source RV s is non-Gaussian, the observed
array output z ¼ Asþn is also non-Gaussian.)
We further investigated the power of the GLRT in detecting noncircularity. For
this purpose, we included a fourth source, a BPSK signal, that impinges on the
array from DOA 35 8 . Apart from this additional source signal, the simulation set-
ting is exactly as earlier. Note that the BPSK signal (or any other purely real-valued
signal) is noncircular with circularity coefficient l¼ 1. Consequently, the array
output z is no longer second-order circular. The calculated GLRT-test statistic
n ln l n correctly rejected at the a¼ 0 : 05 level the null hypothesis of second-
order circularity for all 500 simulated Monte-Carlo trials. Hence, GLRT test was
able to detect noncircularity of the snapshot data (in conventional thermal circular
Gaussian sensor noise) despite the fact that source signals were non-Gaussian.
2.4 TOOLS TO COMPARE ESTIMATORS
2.4.1 Robustness and Influence Function
In general, robustness in signal processing means insensitivity to departures from
underlying assumptions. Robust methods are needed when precise characterization
of signal and noise conditions is unrealistic. Typically the deviations from the assump-
tions occur in the form of outliers, that is, observed data that do not follow the pattern
of the majority of the data. Other causes of departure include noise model class selec-
tion errors and incorrect assumptions on noise environment. The errors in sensor array
and signal models and possible uncertainty in physical signal environment (e.g. propa-
gation) and noise model emphasize the importance of validating all of the assumptions
by physical measurements. Commonly many assumptions in multichannel signal pro-
cessing are made just to make the algorithm derivation easy. For example, by assuming
circular complex Gaussian pdfs, the derivation of the algorithms often leads to linear
structures because linear transformations of Gaussians are Gaussians.
Robustness can be characterized both quantitatively and qualitatively. Intuitively,
quantitative robustness describes how large a proportion of the observed data can
be contaminated without causing significant errors (large bias) in the estimates. It is
commonly described using the concept of breakdown point. Qualitative robustness
on the other hand characterizes whether the influence of highly deviated observations
is bounded. Moreover, it describes the smoothness of the estimator in a sense that
small changes in the data should cause only small changes in the resulting estimates.
We will focus on the qualitative robustness of the estimators using a very powerful tool
called the influence function (IF).