Group
S
Chen
Sto
chastic
Least-Symb
ol-Erro
r-Rate
Adaptive
Equalization
fo
r
Pulse-Amplitude
Mo
dulation
Sheng
Chen
y
,
Berna
rd
Mulgrew
z
and
Ljaos
Hanzo
y
y
Depa
rtment
of
Electronics
and
Computer
Science
Universit
y
of
Southampton,
Southampton
SO17
1BJ,
U.K.
z
Depa
rtment
of
Electronics
and
Electrical
Engineering
Universit
y
of
Edinburgh,
Edinburgh
EH9
3JL,
U.K.
Presented
at
ICASSP2002,
Ma
y
13-17,
2002,
Orlando,
USA
o
rt
of
U.K.
Ro
y
al
Academy
of
Engineering
under
an
international
travel
grant
02-011)
is
gratefully
ackno
Group
Motivations
topic
is
w
ell
resea
rched,
and
a
va
riet
y
of
solutions
exists.
F
o
r
high-level
mo
dulation,
MAP/MLSE
sequence
detecto
r
to
o
complex
Even
MAP
o
r
Ba
y
esian
symb
ol-detecto
r
to
o
complex
Ao
rdable:
linea
r
equalizer
and
decision
feedback
equalizer
Classically
,
MMSE
solution
is
rega
rded
as
optimum
MMSE
w
ould
be
optimum
only
if
equalizer
soft
output
w
ere
Gaussian
Adopting
to
non-Gaussian
nature
leads
to
optimal
MSER
solution
r
equalizer
and
Group
S
Chen
Some
Previous
W
o
rks
Y
ao,
IEEE
T
r
ans.
Information
The
ory
1972
Shamash
&
Y
ao,
ICC'74
Chen
et
al.
,
ICC'96
,
IEE
Pr
o
c.
Communic
ations
1998
Y
eh
&
Ba
rry
,
ICC'97
,
ICC'98
?
,
IEEE
T
r
ans.
Communic
ations
2000
Chen
&
Mulgrew,
IEE
Pr
o
c.
Communic
ations
1999
?
Mulgrew
&
Chen,
IEEE
Symp.
ASSPCC
2000,
Signal
Pr
o
c
essing
2001
:
fo
r
multi-level
P
AM
Two-tap channel 1:0 + 0:5z 1
with 4-PAM and SNR= 35 dB
Two-tap m = 2 linear equaliser
with decision delay d = 0
Normalized MMSE:
w T
MMSE
= [0:9285 0:3713]
with log
10
(SER) = 2:7593
MSER ( > 0):
w T
MSER
= [0:8957 0:4447]
with log
10
(SER) = 7:1566
-0.2
0
0.2
0.4
0.6
0.8
1
w[0]
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
w[1]
-8
-7
-6
-5
-4
-3
-2
-1
0
log10(SER)
Group Exp ress equaliser output y ( k ) = w T ( r ( k ) + n ( k )) = y ( k ) + e ( k ) ? e ( k ) : Gaussian with zero mean and va riance w T w 2 n ? y ( k ) 2 Y 4 = f y q = w T r q ; 1 q N s g , which can be divided into M subsets Y l 4 = f y q 2 Yj s ( k d ) = s l g ; 1 l L Let combined impulse resp onse c T = w T H = [ c 0 c 1 c m + n h 2 ] . Then y ( k ) = c d s ( k d ) + X i 6= d c i s ( k i ) + e ( k ) Optimal decision making ^s ( k d ) =
8 > > < > > :
Two Useful Properties
Shifting: Y
l +1
= Y
l
+ 2c
d
Symmetry: distribution of Y
l
is symmetric around c
d s
l .
y
...
...
c s
d l
d
c (s −1)
l
c s
d
c s
d
c (s +1)
d
l
1
L
? For linear equaliser to work, Y
l
, 1 l L, must be linearly separable
This is not guaranteed
Group
SER
Exp
ression
of
y
(
k
)
p
y
(
x
)
=
1
p
2
n
p
w
T
w
1
N
s
L
X
l
=1
N
sb
X
i
=1
exp
0 B @
x
y
(
l
)
i
2
2
2
n
w
T
w
1 C A
N
sb
=
N
s
=L
is
numb
er
of
po
i
n
t
s
in
Y
l
and
y
(
l
)
i
2
Y
l
.
shifting
and
symmetric
p
rop
e
rties,
SER
of
equaliser
w
is:
P
E
(
w
)
=
N
sb
N
sb
X
i
=1
Q
(
g
l;
i
(
w
))
Q
is
usual
Q
-function,
=
2(
L
1)
=L
,
and
g
l;
i
(
w
)
=
y
(
l
)
i
c
d
(
s
l
1)
n
p
w
T
Group
S
Chen
MSER
Solution
solution
is
dened
as:
w
MSER
=
arg
m
in
w
P
E
(
w
)
of
P
E
(
w
)
r
P
E
(
w
)
=
p
2
n
p
w
T
w
1
N
sb
N
sb
X
i
=1
exp
(
y
(
l
)
i
c
d
(
s
l
1))
2
2
2
n
w
T
w
!
(
y
(
l
)
i
c
d
(
s
l
1))
w
T
w
w
r
(
l
)
i
+
(
s
l
1)
h
d
!
Computation
is
on
single
subset
Y
l
,
and
further
simplication
b
y
using
Y
l
with
s
l
=
1
Use
simplied
conjugated
gradient
algo
rithm
with
reseting
sea
rch
direction
to
negative
every
I
iterations
As
SER
is
inva
riant
to
a
p
o
sitive
scaling
of
w
,
it
is
computationally
advantageous
to
rmalize
w
eight
vecto
r
to
w
T
w
=
1
Group
Blo
c
k
Adaptation
Identify
channel
!
P
E
(
w
)
!
optimisation
Alternatively
,
k
ernel
densit
y
o
r
P
a
rzen
windo
w
estimate
app
roach
estimated
of
p
y
(
x
)
^
p
y
(
x
)
=
1
p
2
n
p
w
T
w
1
K
K
X
k
=1
exp
(
x
y
(
k
))
2
2
2 n
w
T
w
!
:
sample
length,
and
n
:
radius
pa
rameter.
F
rom
^
p
y
(
x
)
,
estimated
SER
^
P
E
(
w
)
=
K
K
X
k
=1
Q
(^
g
k
(
w
))
^g
k
(
w
)
=
y
(
k
)
^c
d
(
s
(
k
d
)
1)
n
p
w
T
w
d
=
w
T
^ h
d
,
and
^
h
d
an
estimate
fo
r
the
d
-th
column
h
d
of
Group
Sample-b
y-Sample
Adaptation:
ALSER
estimate
of
p
y
(
x
)
~
p
y
(
x;
k
)
=
1
p
2
n
exp
(
x
y
(
k
))
2
2
2 n
!
instantaneous
sto
chastic
gradient,
!
ALSER:
w
(
k
+
1)
=
w
(
k
)
+
p
2
n
exp
(
y
(
k
)
^c
d
(
s
(
k
d
)
1))
2
2
2 n
!
r
(
k
)
(
s
(
k
d
)
1)
^
h
d
(
k
)
No
need
fo
r
no
rmalization
to
simplify
complexit
y
Although
using
n
rather
than
n
p
w
T
w
app
ea
rs
to
involve
mo
re
app
ro
ximation,
to
w
o
rk
w
ell
|
not
restrict
to
unit
length
mak
es
it
easier
to
converge
to
An 8-PAM DFE Example
Lower-Bound SER
Comparison
Channel:
0:3 + 1:0z
1
0:3z 2
with 8-PAM
DFE:
m = 3, d = 2, n
b
= 2
-16
-14
-12
-10
-8
-6
-4
-2
0
20
25
30
35
40
45
log10(Symbol Error Rate)
SNR (dB)
MMSE
Weight vector has been normalized to a unit length, a point plotted as a unit impulse.
(a) MMSE
0
0.5
1
1.5
2
0
0.5
1
1.5
2
Distribution
y(k)
decision
threshold
(b) MSER
0
0.5
1
1.5
2
0
0.5
1
1.5
2
Distribution
Conditional PDF given s(k d) = 1, SNR=34 dB
normalized w
T
MMSE
= [ 0:05780:2085 0:9763], w
T
MSER
= [ 0:23650:7946 0:5592]
(a) MMSE
0
0.5
1
1.5
2
2.5
3
-0.5
0
0.5
1
1.5
2
2.5
conditional pdf
y
decision
threshold
decision
threshold
(b) MSER
0
0.2
0.4
0.6
0.8
1
-0.5
0
0.5
1
1.5
2
2.5
conditional pdf
y
Initial weight: (a) w
MMSE
, (b) [ 0:01 0:01 0:01]
T
Weight normalization applied
1e-9
1e-8
1e-7
1e-6
1e-5
1e-4
0
2000
4000
6000
8000
10000
Symbol Error Rate
Sample
MMSE
MSER
1e-9
1e-8
1e-7
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
0
200
400
600
800
1000
Symbol Error Rate
Sample
MMSE
MSER
training
decision direct
(a) (b)
In (a) training and decision directed indistinguishable, in (b) dashed curve: after 200-sample
training, switched to decision-directed with s(k^ d) substituting s(k d)
Learning Curves of ALSER Averaged Over 100 Runs, SNR=34 dB
Initial weight: (a) w
MMSE
, (b) [ 0:01 0:01 0:01]
T
Weight normalization not applied
1e-9
1e-8
1e-7
1e-6
1e-5
1e-4
0
2000
4000
6000
8000
10000
Symbol Error Rate
Sample
MMSE
MSER
1e-9
1e-8
1e-7
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1e+0
0
200
400
600
800
1000
Symbol Error Rate
Sample
MMSE
MSER
training
decision direct
(a) (b)
In (a) training and decision directed indistinguishable, in (b) dashed curve: after 200-sample
training, switched to decision-directed with s(k^ d) substituting s(k d)
Group
Conclusions
MMSE
views
equaliser
output
as
Gaussian
and
tries
to
t
pa
rameters
true
non-Gaussian
in
a
w
a
y
so
that
it
lo
oks
as
closely
as
p
ossible
a
Gaussian
one
(without
making
noise
sky
high)
Non-Gaussian
app
roach
leads
naturally
to
MSER
F
o
r
high-level
P
AM
mo
dulation
schemes,
MSER
equalisation
solution
be
i
n
g
derived
Eective
sample-b
y-sample
adaptation
has
b
een
develop
ed
Unlik
e
MSE
surface
which
is
quadratic,
SER
surface
is
highly
complex
Initial
equaliser
w
eight
values
can
critically
inuence
convergence
ALSER
is
pa
rticula
r
p
romising:
simpler