Digital Signal Processing Reference
In-Depth Information
Table 9.2
Proportionate Adaptation (PA) and Partial-Update (PU) Algorithms that
Can Be Written in the Form
f
[
n
] =
f
[
n
] + μ
e
[
n
]
g
[
n
]
y
*[
n
], with
g
[
n
] = diag[
g
[
n
]]
Type
Name
Value of
g
[
n
]
Reference
T
=
NLMS
g
n
11
,,...,
1
[29]
N
+
1
of them
{
}
{ }
=
γγ
k
n
max
f
n
,
ρ δ
0
max,
f
n
PA
PNLMS
[18]
k
∞
γγ
n
k
∈
{ }
g
n
=
,
k
0
,
,
N
k
∑
N
1
γγ
n
l
N
+
1
l
=
0
−
( )
+
( )
=
1
α
+
1
α
f
γγ
k
n
1
f
n
n
,
−≤≤
1
α
1
PA
IPNLMS
[42]
k
2
2
N
+
1
1
γγ
n
=
k
∈
{ }
g
k
n
,
k
0
,
,
N
,
∑
N
1
γγ
n
l
N
+
1
l
=
0
( )
( )
ln
f
n
ε
k
∈
{ }
=
PA
MPNLMS
g
n
,
k
0
,
,
N
[39]
k
∑
N
1
ln
f
n
ε
l
N
+
1
l
=
0
=
sequential
sequential
PU
Sequential
PU NLMS
g
n
g
n
−
1
[43]
( )
( )
j
j
1
mod
N
1
=
{
}
g
0
user-defined vector of
M
ones
NM
- 1zeros
1
0
,
j
=
arg max
y
n
=
k
maxLMS
PU
Max-LMS
g
n
[44]
0
≤≤
kN
j
,
else
{
}
0,
1,
j
∈
indicesof
M
largest
y
n
=
k
MmaxLMS
PU
M-max-LMS
g
n
[45]
j
else
Note:
The table lists the composition of the vector
g
[
n
] for each algorithm.
and its variants were primarily motivated by channel identification for acoustic echo
cancellation, although they have been applied to wireless channels as well. If we wish to
apply them to equalization of wireless channels, it makes sense to use an equalizer struc-
ture that preserves the sparsity of the channel in its own impulse response. Motivated
primarily by complexity reduction, the PFE and several related equalizer structures have
been recently proposed that satisfy this goal [36]. In the PFE, as shown in
Figure 9.2
, a
partial feedback filter (PFBF) cancels the sparse channel coefficients before the feedfor-
ward filter (FFF) smears them out, which cannot be done in a conventional DFE. Then
the FFF and the feedback filter (FBF) operate as a normal DFE would. The PFBF can be
long and sparse, allowing the other filters to be short.
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