Adaptive Equalization for Wireless Channels - Adaptive Signal Processing in Wireless Communications - page 238

data sequence x [ n ] is in general complex valued and from a discrete signal constellation.
When training (knowledge of x [ n ] for short, intermittent time intervals) is available,
the desired signal in wireless communications is almost always a delayed version of the
input, d [ n ] = x [ n - Δ]. Then the error signal is

    =−     −    

T

en xn ∆ fy n .

(9.5)

 

      

dn xn ˆ − ∆



The cost function to be minimized is


 


  ,

JEen q =     (9.6)

with q = 2 leading to the MSE cost function, although q = 1 and q = 4 are sometimes
encountered in the literature, leading to the mean absolute error (MAE) and mean
fourth error (MFE) cost functions, respectively.
Computing a stochastic gradient descent of (9.6) requires computing the gradient
with respect to f and removing the expectation. Then the new equalizer is additively
adjusted in the direction of the negative gradient by a small step size μ,

    =     −∇

f n + 1 f n µ . J (9.7)

f

For q = 2, this leads to the LMS algorithm,

(9.8)

    =     +     *    

f n + 1 f n µ en y n

Due to the vector structure of f and y [ n ], (9.8) accounts for the SIMO channel model.
Similarly, q = 1 leads to the error sign LMS algorithm [34], and q = 4 leads to the least
mean fourth (LMF) algorithm [19].
Almost all adaptive equalizers in the literature follow this same structure: a cost func-
tion is proposed (usually in the form of (9.6) or a hybrid of several such cost functions),
and a stochastic gradient descent update is computed. In sections 9.3 and 9.4, we will
talk about modifications that can be applied to most gradient descent algorithms to
improve performance.

9.2.2 Blind Adaptive Algorithm Design Methodologies

The biggest challenge in adaptive equalization comes when training is unavailable. Even
in standards that include training, there is typically a long interval of data between
training symbols. If this is the case, the error signal of (9.5) cannot be computed for use
in the cost function.
Blind algorithms often turn to known statistical properties of the source signal in lieu
of training. For example, when the source data comes from a finite constellation, then