Digital Signal Processing Reference
In-Depth Information
9.4.2 Blind Adaptation
The simplest blind equalizer is DD-LMS. However, even DD-LMS has a slight twist in
the case of PPM. The DD cost function and update rule are
2
−
{ }
JE
=
z
b
Qz
b
2
)
.
(9.33)
(
−
{ }
=
−
f
b
+
1
f
b
µ
Yz
bb
Qz
b
The twist is that the decision function
Q
{·} :
K
K
operates in vector form, rather
than element-wise. The vector argument is considered in its entirety and is mapped to
the nearest vector in the signal constellation [57, 58].
A variant of CMA can also be derived for PPM, which will be called linear transversal
equalizer adaptation for biorthogonal modulation, blindly (LTBOMB). Like DD-LMS
for PPM, the distinction from traditional adaptive algorithms lies in the block structure
of PPM. The cost function and algorithm are [57, 58]
2
2
−
JE
=
z
b
1
2
(9.34)
(
)
=
−
T
f
b
+
1
f
b
−
µ
z
bb
z
1
Yz
bb
.
However, as implied by the time indices, the algorithm only updates once per block,
i.e., once every
K
chips. Again, note the matrix-vector structure of the update rule, as
opposed to standard CMA, which uses a scalar times vector update rule.
An alternative to CMA for blind, adaptive equalization of traditional communication
systems is the Shalvi-Weinstein algorithm (SWA) [61]. SWA is similar to CMA insofar as
it looks at higher-order statistics of the equalizer output, but SWA attempts to maximize
the magnitude of the kurtosis. In [58] and [62], the SWA philosophy is used to create a
blind, adaptive equalizer for PPM, called the recovery of
M
-ary biorthogonal signals via
p
-norm equivalence (TROMBONE). The cost function and algorithm are
4
4
2
−
JE
=
z
b
z
b
, uchthat
z
b
=
1
2
4
2
( )
2
f
=
−
T
b
+
1
f
b
µ
Yz
b
b
zI
b
−
diag
z
b
z
b
(9.35)
K
ˆ
f
b
+
1
=
f
b
+
1
,
ˆ
f
b
+
1
2
where
I
K
is the
K
×
K
identity matrix. The last line implements the constraint (similar
to the MMSE, MERRY, and CNA algorithms for multicarrier systems) to constrain the
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