Stochastic Gradient Adaptive Algorithms - A Rapid Introduction to Adaptive Filtering - page 50

The use of spectrum partitioning techniques (where the channel is divided into
narrowband sections, which present very limited ISI), as Orthogonal Frequency
Division Multiplexing (OFDM) in wireless communications [ 17 ], permits to allevi-
ate considerably the problem of ISI on those applications. Here, we will consider
a solution based on equalization using adaptive filtering. In applications such as
single carrier wired communications, adaptive equalization techniques have been
successfully applied [ 16 ]. Adaptive equalization solutions can also be applied for
the reduction of ISI on high speed digital circuits [ 18 ]. The general idea of using
equalization can be summarized in the choice of an appropriate impulse response
w = w − L + 1 ,..., w L − 1 to be applied to the channel output y

w 0 ,..., ( n ) ,

L − 1

x ˆ ( n ) = w i y ( n − i ), (4.44)

i =− L + 1

( ) = ( ) −ˆ ( )

in such a way that the error e should be small in some appropriate

sense. For example, we could consider obtaining w as the solution to 6 :

n x n x n

⎡

2 ⎤
⎦

L − 1

⎣

w = arg min

w ∈ R

2 L + 1 E

x ( n ) − w i y ( n − i ) (4.45)

i =− L + 1

Notice the fact that the filtering is done over L samples before and after y .This

is a consequence of the existence of a natural delay, introduced by the channel h ,
which affects the transmitted symbols. The solution of ( 4.45 ) can be easily computed
using the tools presented in Chap. 2 , and require the knowledge of the statistics of
x

( n )

( ) ( )

and h . This clearly precludes the use of the solution of ( 4.45 ) in real world
applications. Therefore, an adaptive filtering solution is proposed next.

n , v n

4.4.3.1 Adaptive Solution

We will refer to the system schematized in Fig. 4.3 . As explained above, the adaptive
filter w

2 L + 1 acts on the channel outputs as

( n ) ∈ R

L − 1

x ˆ ( n ) = w i ( n − 1 ) y ( n − i ). (4.46)

i =− L + 1

6 This is not the only possible cost function that could be considered. Another popular solution is to
obtain a filter w that completely inverts the channel h , without taking into account the noise v .

This solution called zero forcing (ZF) [ 15 ] completely eliminates the ISI at the cost of possibly
increasing the influence of the noise. However, when the noise is sufficiently small and the channel
h does not present nulls in its frequency response, ZF offers a good performance.

( n )