ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 137

Table 2.3 Average computation times (in milliseconds) over 100 runs for array
snapshot data sets without outliers and with four outliers generated as z out
gut1 gut2 gut3 gut4 gut5 Fica Jade Cfpa Kmg d1 Fobi

Without
outliers

1.15 2.04 2.02 8.70 8.76 25.73 5.87 9.08 9.98 14.77 1.71

With
outliers

1.19 2.16 2.12 10.13 10.49 59.52 6.36 17.30 17.61 17.82 1.77

fast algorithm of [49]. Also observe that the occurrence of outliers severely increases
the computation times of the iterative fixed-point algorithms fica , kmg and cfpa ,
whereas computation times for the other methods are only slightly affected by outliers.

2.9 CONCLUSION

In this chapter we focused on multichannel signal processing of complex-valued sig-
nals in cases where the underlying ideal assumptions on signal and noise models do
not necessarily hold. We considered departures from two key assumptions, that is cir-
cularity of the signal and / or noise as well as Gaussianity of the noise distribution. A
comprehensive description of the full second-order statistics of complex random vec-
tors was provided since the conventional covariance matrix alone suffices only in the
case of circular signals. A detection scheme for noncircularity was developed. This
facilitates using the full second-order statistics of the signals and appropriate algor-
ithms in the presence of noncircularity. Moreover, estimators and multichannel
signal processing algorithms that also take into account the noncircularity of the sig-
nals and are robust in the face of heavy-tailed noise were introduced. Their robustness
and efficiency were analyzed. Example applications in beamforming, subspace-based
direction finding and blind signal separation were presented.

2.10 PROBLEMS

2.1 Show that circularity coefficient l ( z ) satisfies (2.3) and (2.4).

2.2 Based on the definition q n of the GLRT decision statistic, derive the equation
(2.13) for l n and verify using arguments based on the properties of eigenvalues
that the test statistic l n is invariant under invertible linear transformations of
the data.

2.3 MATLAB assignment. Let us have a uniform linear array of eight elements. Two
QPSK modulated signals are impinging on the array from directions of arrival
72 8 and 66 8 . Our N ¼ 200 observations are contaminated by complex white
second order circular Gaussian noise such that the signal to noise ratio (SNR)
is 10 dB. Plot the MUSIC pseudo-spectrum. Study the traces of array covariance
and pseudo-covariance matrices. What can you say about the circularity of the
observations?