Adaptive Equalization for Wireless Channels - Adaptive Signal Processing in Wireless Communications - page 236

them for performance improvement or complexity reduction. Notable extensions for
trained equalizers include application of nonlinearities to the data and error functions
in the LMS algorithm [17], modifications to exploit sparsity in the equalizer [18], and
consideration of alternate cost functions, such as the fourth power of the error instead
of the square [19, 20], or hybrids of the two. Notable extensions for blind equalizers
include the use of multiple moduli in CMA for higher-order constellations [21], and
hybrid cost functions that retain the benefits of CMA but provide improved convergence
speed [22].
In the past few years, there has been a surge of interest in algorithms that exploit the
idea of CMA, but with a different contour in the constellation space. CMA tries to force
the constellation points back onto a circular contour in the complex plane. Similarly, the
square contour algorithm (SCA) [23], also called the constant square algorithm (CQA)
[24], uses a square contour, which has a constant infinity norm (as opposed to the con-
stant 2-norm of a circle). This idea was generalized to include all possible norms on the
complex plane in [24] and [25], leading to a plethora of possible algorithms, includ-
ing extended CMA (ECMA) [25] and the constant norm algorithm with an L6 norm
(CNA-6) [24]. However, not all constellations are built on square or circle patterns;
hence, [26] proposed the use of a more complicated cross-shaped contour, leading to the
constant cross algorithm (CXA).
Further information on equalization of slowly time-varying channels can be obtained
from a variety of survey papers. Qureshi's paper [16] surveys early work in trained adap-
tive equalizers (pre-1985). Reviews of blind adaptive equalization algorithms can be
found in [27] and [28]. One of the more popular “encyclopedias” of adaptive filter algo-
rithms is Haykin's book [29].
Up until around 2000, almost all adaptive equalizers were designed for the case in
which the channel was frequency selective but quasi-static. That is, the channel impulse
response was assumed to be relatively constant and to approximately obey the model of
(9.2), but it could drift over time. This allows a gradient-descent type of equalizer to keep
up. In the past few years, interest has risen in equalization of channels that vary rapidly
with respect to the symbol period [1, 30-33]. In such a case, the channel is both time and
frequency selective, or doubly selective, and is modeled more appropriately by (9.1) than
by (9.2). Such channels cannot be equalized by gradient-descent equalizers (trained or
blind) since the update rule cannot keep up with the speed of the channel variations, and
more complex methods of equalization are required. One possibility is to take the two-
dimensional Fourier transform of the channel convolution matrix, and then perform
the equalization in this frequency domain. This is most appropriate for systems such
as orthogonal frequency division multiplexing (OFDM), in which the data are already
encoded in the frequency domain. For small amounts of time variation, MMSE symbol
estimation can be performed by considering the interference from adjacent frequency
bins due to the Doppler spread. The complexity can be kept manageable by ignoring the
small interference coefficients at large Doppler spreads [31] or by windowing in the time
domain to restrict the Doppler spread [1]. However, [1] and [31] were derived specifically
in the context of OFDM, for which the frequency-domain input signal is discrete, so this
approach is not applicable to other modulation schemes. A more general approach is to
model both the channel and the equalizer in the time domain as matrices rather than