ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 89

Figure 2.1 A uniform linear array (ULA) of k sensors with sensor displacement d
receiving plane waves from d far-field point sources.
where v¼ 2 p ( d=l ) sin ( u ) depends on the signal wavelength l , the DOA u of the
signal with respect to broadside, and the sensor spacing d . The source signal vector
s is modeled as either deterministic or random, depending on the application.
In blind signal separation (BSS) based on independent component analysis (ICA),
both the mixing system A and the sources s are unknown. The goal in ICA is to solve
the mixing matrix and consequently to separate the sources from their mixtures
exploiting only the assumption that sources are mutually statistically independent.
In this chapter, we consider the noiseless ICA model.
Common assumptions imposed on the signal model (2.1) are as follows:
A SSUMPTION (A1) noise n and / or source s possess circularly symmetric
distributions.
In addition, in the process of deriving optimal array processors, the distribution of the
noise n is assumed to be known also, the conventional assumption being that
A SSUMPTION (A2) noise n possesses circular complex Gaussian distribution.
Furthermore, if s is modelled as stochastic, then s and n are both assumed to be inde-
pendent with circular complex Gaussian distribution, and consequently, sensor output
z also has k -variate circular complex Gaussian distribution.
In this chapter, we consider the cases where assumptions (A1) and (A2) do not
hold. Hence we introduce methods for array processing and ICA that work well at cir-
cular and noncircular distributions and when the conventional assumption of normal-
ity is not valid. Signal processing examples on beamforming, subspace-based DOA
estimation and source-signal separation are provided. Moreover, tests for detecting
noncircularity of the data are introduced and the distributions of the test statistics
are established as well. Such a test statistic can be used as a guide in choosing the
appropriate array processor. For example, if the test rejects the null hypothesis of cir-
cularity, it is often wiser to choose a method that explicitly exploits the noncircularity