COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 40

The complex Wolfe condition [82] can be easily obtained from the real Wolfe
condition using (1.25). It should be noted that the complex CG algorithm is a linear
version. It is straightforward to obtain a nonlinear version based on the linear version
as shown in [82] for the real case.
Other Newton Variant Updates As shown for the derivation of complex
gradient and Newton update rules, we can easily obtain complex versions of other real
Newton variant methods using (1.25) and (1.26). In [70], this is demonstrated for the
real-valued scaled conjugate gradient (SCG) method [79]. SCG belongs to the class of
CG methods and shows superlinear convergence in many optimization problems.
When the cost function takes a least-squares form, a complex version of the Gauss-
Newton algorithm can be developed as in [64]. In the Gauss-Newton algorithm, the
original Hessian matrix in the Newton update is replaced with a Gauss-Newton
Hessian matrix, which has better numerical properties hence providing better perform-
ance. For more general cost functions, BFGS is a popular and efficient Newton variant
method [82] and can be extended to the complex domain similarly.
1.4 WIDELY LINEAR ADAPTIVE FILTERING
As discussed in Section 1.2.5, in order to completely characterize the second-order
statistics of a complex random process, we need to specify both the covariance and
the pseudo-covariance functions. Only when the process is circular, the covariance
function is sufficient since the pseudo-covariance in this case is zero. A fundamental
result in this context, introduced in [94], states that a widely linear filter rather than the
typically used linear one provides significant advantages in minimizing the mean-
square error when the traditional circularity assumptions on the data do not hold.
A widely linear filter augments the data vector with the conjugate of the data, thus
providing both the covariance and pseudo-covariance information for a filter designed
using a second-order error criterion.
The assumption of circularity is a limiting assumption as, in practice, the real and
imaginary parts of a signal typically will have correlations and / or different variances.
One of the reasons for the prevalence of the circularity assumption in signal processing
has been due to the inherent assumption of stationarity of signals. Since the complex
envelope of a stationary signal is second-order circular [91], circularity is directly
implied in this case. However many signals are not stationary, and a good number
of complex-valued signals such as fMRI and wind data as shown in Section 1.2.5,
do not necessarily have circular distributions. Thus, the importance of widely linear
filters started to be noted and widely linear filters have been proposed for applications
such as interference cancelation, demodulation, and equalization for direct sequence
code-division-multiple-access systems and array receivers [23, 56, 99] implemented
either in direct form, or computed adaptively using the least-mean-square (LMS)
[99] or recursive least squares (RLS) algorithms [55]. Next, we present the widely
linear mean-square error filter and discuss its properties, in particular when computed
using LMS updates as discussed in [5]. We use the vector notation introduced in