Adaptive Equalization for Wireless Channels - Adaptive Signal Processing in Wireless Communications - page 237

as vectors [30]. This explicitly accounts for the time variations in the model, and allows
for far more degrees of freedom in the equalizer. The challenge then becomes accurately
determining all of the channel coefficients from limited observable data, and of comput-
ing the equalizer with a moderate amount of complexity. These are difficult goals, and
are the primary drawbacks of this approach [1], although this can be mitigated some-
what by explicitly exploiting the sparse structure of the channel matrix [33]. Another
approach that requires less degrees of freedom is to use a basis expansion model (BEM)
[32]. A BEM models the doubly selective channel using basis signals that allow for both
time and frequency selectivity, by substituting

Q / 2

∑

() ( )

{ }

p

    =

pqk ,,

j 2 π nN /

hnk , e h , k ∈ 0 , , L (9.4)

qQ =− / 2

into (9.1), where q indexes the Doppler spread and h ( p , q , k ) is the strength of Doppler com-
ponent q at lag k for receive antenna p . For each time instant, the channel is modeled
as a sum of complex exponentials with different Doppler spreads; hence, the amount of
Doppler that is modeled is controlled (as opposed to the matrix model of [30], which can
in principle allow arbitrarily large Doppler). The equalizer can also be modeled using the
BEM, and hence it is also effectively a time-varying filter.
This chapter covers adaptive equalizers, and the equalizers discussed above for doubly
selective channels are not adaptive in the classical sense of a recursive update rule. How-
ever, they do perform equalization by explicitly accounting for the time variations in
the channel model, which is the essence of adaptive equalization. As mobility becomes
more pervasive in communications and as data rates are pushed higher, many adaptive
equalizers may need to be recast in more complex forms, such as BEMs.

9.2 Adaptive Equalizer Algorithm Formulation

This section covers adaptive algorithm design approaches, in the context of traditional
digital communication systems (later sections discuss emerging digital modulation for-
mats). We begin with a discussion of methods of trained and blind equalizer design,
with a focus on the design philosophy rather than a specific algorithm. We then cover
methods of improving algorithm performance, by either accelerating convergence or
exploiting the sparse structure in wireless channels.

9.2.1 Trained Adaptive Algorithm Design Methodologies
Most trained adaptive equalizers take the form of a stochastic gradient descent of a cost
function. By far, the most popular choice of cost function is the mean squared error
(MSE), where the error is the difference of a desired signal d [ n ] and the filter output.
However, the point at which the error is measured can vary from application to applica-
tion, leading to different algorithms.
In a traditional single-carrier communication system, the input-output relation is
adequately modeled by (9.2) with no additional preprocessing at the transmitter. The