Digital Signal Processing Reference
In-Depth Information
Thus, the following iteration
g
m
ν
=
(4.60)
ν
||
ν
||
g
=
(4.61)
converges asymptotically to the ideal ZF response, when initialized by a
unit-norm response
having a unique dominant element.
Since the combined response depends on the unavailable channel
impulse response
h
, the iterative procedure is not practical
∗
and must be
adjusted in order to work in the equalizer domain. To do so, let us consider
that both the channel and the equalizer can be represented by finite impulse
responses, whose coefficients form vectors
h
and
w
, respectively. In this case,
the combined response in terms of the channel and equalizer coefficients is
(
g
)
g
=
H
w
(4.62)
where
⎡
⎣
⎤
⎦
h
0
h
1
···
h
M
−
1
0
···
0
0
h
0
h
1
···
h
M
−
1
0
0
H
=
(4.63)
.
.
.
0
.
.
.
.
.
.
.
.
.
.
.
.
0
0
···
0
h
0
h
1
···
h
M
−
1
is the so-called channel convolution matrix.
The aim of the adaptive method would then be to yield
w
such that
=
H
=[
...
...
]
g
. However, it is
important to notice that since we are assuming an FIR equalizer, there may
not exist a ZF equalizer for a given channel, which leads us to the concept of
the attainable set of a channel. The attainable set
w
be an ideal solution, i.e.,
g
0,
,0,1,0,
,0
T
is given by
g
:
g
L
T
=
H
w
,
∀
w
∈
(4.64)
where
L
is the length of the equalizer. Thus, for a given global response,
we may obtain its projection onto the attainable set by applying a projection
operator, given by
−
1
H
H
P
T
=
H
H
H
H
(4.65)
∗
We refer the interested reader to [70,270], which contain a more detailed discussion regarding
the convergence of this iterative procedure.
