COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 45

When the signal is circular, the augmented covariance matrix assumes the block
diagonal form

C 0
0 C

C circ ¼

and has eigenvalues that occur with even multiplicity. In this case, the conditioning of
the augmented covariance matrix C and C are the same. As the noncircularity of the
signal increases, the values of the entries of the pseudo covariance matrix moves away
from zero increasing the condition number of the augmented covariance matrix C , thus
the advantage of using a widely linear filter for noncircular signals comes at a cost
when the LMS algorithm is used when estimating the widely linear MSE solution.
An update scheme such as recursive least squares algorithm [43] which is less sensi-
tive to the eigenvalue spread can be more desirable in such cases. In the next example,
we demonstrate the impact of noncircularity on the convergence of LMS algorithm
using a simple input model.

B EXAMPLE 1.5

Define a random process
X ( n ) ¼

p

1 r 2 X r ( n ) þ jrX i ( n ) (1 : 42)

where X r ( n ) and X i ( n ) are two uncorrelated real-valued random processes, both
Gaussian distributed with zero mean and unit variance. By changing the value
of r [ [0, 1], we can change the degree of noncircularity of X ( n ) and for
r¼ 1 = p , the random process X ( n ) becomes circular. Note that since second-

order circularity implies strict-sense circularity for Gaussian signals, this model
lets us to generate a circular signal as well.
If we define the random vector X ( n ) ¼ [ X ( n ) X ( n 1) X ( nN þ 1)] T ,we
can show that the covariance matrix of X ( n ) is given by C¼ I , and the pseudo
covariance matrix as P ¼ (1 2 r 2 ) I . The eigenvalues of the augmented covari-
ance matrix C can be shown to be 2 r 2 and 2(1 r 2 ), each with multiplicity N .
Hence, the condition number is given by

1
r 2 1

k ( C ) ¼

if r [ [0, 1 = p , 1].

In Figure 1.7, we show the convergence behavior of a linear and a widely linear
LMS filter with input generated using the model in (1.42) for identification of a
system with coefficients w opt, n ¼ a [1 þ cos(2 p ( n 3) = 5] j [1 þ cos(2 p ( n 3) =
10)]), n ¼ 1, ... ,5,and a is chosen so that the weight norm is unity (in this case,
a¼ 0 : 432). The input signal to noise ratio is 20 dB and the step size is fixed at

p ] and by its inverse if r [ [1 =