Adaptive Equalization for Wireless Channels - Adaptive Signal Processing in Wireless Communications - page 253

9.3.2 Adaptive CSE Algorithms
Now we return to the problem of CSE adaptation in cyclic-prefixed systems. As men-
tioned in the discussion of FEQ adaptation, training in multicarrier systems may be
spread across time and frequency, and it may be allocated in the frequency domain. In
order to train based on the equalizer output, we first assume that time-domain training
is available, which is equivalent to assuming that training is available on all subcarriers
in a given symbol. In this case, the obvious choice is to use an LMS-like equalization
rule. However, this is complicated by the fact that the goal of the CSE is not to make
the effective channel simply a delay, so we cannot form an error signal by comparing to
the delayed training data as in (9.5). Instead, the goal of the CSE is to make the effective
channel short, but (to a first-order approximation) we do not care about what shape the
effective channel takes so long as it is short. Falconer and Magee [10] solved this problem
in the context of CSE design in conjunction with MLSE by forming an additional filter,
b , of length ν + 1, called the target impulse response (TIR). Then the CSE and TIR can
each be adapted in turn by an LMS algorithm.
Mathematically, the error function and cost are

    =     −    

T T

en bx n − ∆ fy n ,

(9.20)


 


 

q

=     JEen

where Falconer and Magee chose q = 2, corresponding to an LMS algorithm, yet others
have considered q = 4, corresponding to an LMF algorithm [51]. Since the trivial setting
b = 0 , f = 0 permits a zero-cost solution, a constraint must be enforced, such as a unit-tap
or unit-norm filter. Iterating through the adaptation of each filter and maintenance of
the constraint, the MMSE adaptive CSE algorithm is

    =     +        

*

f n + 1 f n µ en y n

ˆ

    = 

   −    

   

*

b n + 1 b n µ en x n − ∆ (9. 21)

ˆ ˆ

    =+         .

b n + 1 b n 1 b n + 1

2

The choice of a larger q will add factors in the first two lines, and the choice of an alter-
nate constraint will require a different projection in the third line of (9.21). For ν = 0, b
becomes a scalar, and (9.21) reduces to the LMS algorithm for f alone.
The coupling of b and f , along with the constraint on b , makes formulation of an RLS
adaptation rule more complicated than for a traditional equalization problem. Use of
the unit-norm constraint leads to an optimal solution for b as an eigenvector rather than
a least squares solution, and hence RLS does not apply. However, if a unit-tap constraint
is used, the optimal b can be written as the solution to either a generalized eigen problem
[52] or a least squares problem [50]. Although this constraint does lead to a suboptimal