ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 88

Many man-made complex-valued signals encountered in wireless communication
and array signal-processing applications possess circular symmetry properties. More-
over, additive sensor noise present in the observed data is commonly modeled to be
complex, circular Gaussian distributed. There are, however, many signals of practical
interest that are not circular. For example, commonly used modulation schemes
such as binary phase shift keying (BPSK) and pulse-amplitude modulation (PAM)
lead to noncircular observation vectors in a conventional baseband signal model.
Transceiver imperfections or interference from other signal sources may also lead to
noncircular observed signals. This property may be exploited in the process of recover-
ing the desired signal and cancelling the interferences. Also by taking into account
the noncircularity of the signals, the performance of the estimators may improve, the
optimal estimators and theoretical performance bounds may differ from the circular
case, and the algorithms and signal models used in finding the estimates may be dif-
ferent as well. As an example, the signal models and algorithms for subspace estimation
in the case of noncircular or circular sources are significantly different. This awareness
of noncircularity has attained considerable research interest during the last decade,
see for example [1, 13, 18, 20, 22, 25, 38-41, 46, 47, 49, 50, 52-54, 57].

2.1.1 Signal Model
In many applications, the multichannel k -variate received signal z ¼ ( z 1 , ... , z k ) T
(sensor outputs) is modeled in terms of the transmitted source signals s 1 , ... , s d
possibly corrupted by additive noise vector n , that is

z ¼ Asþn
¼ a 1 s 1 þþa d s d þn (2 : 1)

where A ¼ ( a 1 , ... , a d ) is the k d system matrix and s ¼ ( s 1 , ... , s d ) T contains the
source signals. It is assumed that d k . In practice, the system matrix is used to
describe the array geometry in sensor array applications, multiple-input multiple-
output (MIMO) channel in wireless multiantenna communication systems and
mixing systems in the case of signal separation problems, for example. All the com-
ponents above are assumed to be complex-valued, and s and n are assumed to be
mutually statistically independent with zero mean. An example of a multiantenna sen-
sing system with uniform linear array (ULA) configuration is depicted in Figure 2.1.
The model (2.1) is indeed very general, and covers, for example, the following
important applications.
In narrowband array signal processing , each vector a i represents a point in known
array manifold (array transfer function, steering vector) a ( u ), that is a i ¼ a ( u i ), where
u i is an unknown parameter, typically the direction-of-arrival (DOA) u i of the i th
source, i ¼ 1, ... , d . Identifying A is then equivalent with the problem of identifying
u 1 , ... , u d . For example, in case of ULA with identical sensors,

T ,

e jv e j ( k 1) v

a ( u ) ¼ 1