Digital Signal Processing Reference
In-Depth Information
(
)
=
1,
select
M
values of
j
by some method
g
j
n
0
,
else
=
g
n
diag
g
n
(9.15)
n
=
+
*
f
+
1
f
n
µ
g
n
y
nen
,
where
g
[
n
] is a diagonal matrix of ones and zeros that turn on or of the updates to each
coefficient. (For applications such as echo cancellation, an NLMS normalization term as
in the denominator of (9.12) can be explicitly included, or it can be included implicitly
by dividing the step size
μ
by the normalization term.) With this structure, the partial
update of (9.15) has the same form as the proportionate update of (9.12), and in either
case, μ
g
[
n
] effectively forms a vector step size.
Specific tap selection rules for (9.15) have the mathematical form
=
sequential
sequential
g
n
g
n
−
1
( )
j
j
1
( )
mod
N
1
1
,
j
=
arg max
y
n
=
k
maxLMS
g
n
(9.16)
0
≤≤
kN
j
0
,
else
0,
{
}
1
,
j
∈
indicesof
M
largest
y
k
=
g
j
MmaxLMS
n
.
else
Observe that max-NLMS and M-max-NLMS are better suited for equalization than
channel identification when the application is digital communications. This is because
digital communication channels often have inputs drawn from a constant modulus
source (e.g., ±1 ±
j
), in which case all of the inputs to the channel model would have the
same magnitude, and no selection could be made, whereas the equalizer inputs contain
intersymbol interference and noise, with a wide range of magnitudes.
Other recent work has investigated extensions to M-max-NLMS that further reduce
the complexity. Selective-block NLMS [46] is like M-max NLMS, but in terms of blocks of
taps rather than individual taps. Similarly, the short-sort algorithm [47] recognizes that
in many cases, the significant taps are grouped together. The algorithm identifies a con-
tiguous block of significant taps and always updates those, and performs M-max NLMS
on the remaining taps, which reduces the sorting overhead. A slightly more sophisti-
cated approach is to monitor an activity measure for each tap [48] (instead of simply
comparing input magnitudes), and then favor those taps when updating.
Table 9.2
lists
popular proportionate adaptation and partial update algorithms.
Proportionate adaptation rules and partial-update rules were explicitly designed to
exploit sparsity in the filter impulse response. PNLMS and its variants and M-max-LMS
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