ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 119

of sources d is often not known and needs to be estimated from the data. The com-
monly used minimum description length (MDL)-based information theoretical
criterion, obtains the estimate d for the number of signals d as an integer
p [ (0, 1, ... , k 1) which minimizes the criterion [60]

! ( kp ) n

( Q i ¼ p þ 1 l i ) 1 = ( kp )
1

1
2 p (2 k p ) log n

MDL( p ) W log þ

kp P i¼pþ 1 l i

where l 1 , l 2 , ... , l k denote the (ordered) eigenvalues of the SCM C arranged in des-
cending order. Instead of using the eigenvalues of SCM, it is desirable for purposes
of reliable estimation in non-Gaussian noise to employ eigenvalues of some robust
estimator of covariance, for example, M -estimator of scatter, instead of the SCM.
We demonstrate this via a simulation study.
The ULA contains k ¼ 8 sensors with half a wavelength interelement spacing.
Two uncorrelated Gaussian signals with equal power 20 dB from DOAs u 1 ¼ 5 8
and u 2 ¼ 5 8 are impinging on the array. The components of the additive noise n are
modeled as i.i.d. with complex symmetric a -stable (S a S) distribution [56] with dis-
persion g¼ 1 and values a ranging from a¼ 1 (complex Cauchy noise) to a¼ 2
(complex Gaussian noise). Simulation results are based on 500 Monte Carlo runs
with n ¼ 300 as the sample size. Figure 2.7 depicts the relative proportion of correct
estimation results using MDL criterion, when the eigenvalues are obtained from SCM
C and robust MLT(1), HUB(0.9) and HUB(0.5) estimators. The performance of the
classic MDL employing the SCM is poor: it is able to estimate the number of signals

Figure 2.7 Simulation results for estimation of number of sources using the MDL criterion
based on the SCM, HUB(0.9), HUB(0.5) and MLT(1)-estimators. There are d ¼ 2 Gaussian
source signals in SaS distributed noise for 1 a 2. The number of sensors is k ¼ 8 and
number of snapshot is n ¼ 300.