¯ ¯ ¯

Communiation Group
S Chen
Communiation Group
S Chen
Stohasti Least-Symbol-Error-Rate Adaptive
Motivations
Equalization for Pulse-Amplitude Modulation
Sheng Cheny, Bernard Mulgrewz and Ljaos Hanzoy
Equalization topi is well researhed, and a variety of solutions exists. BUT
y Department of Eletronis and Computer Siene
University of Southampton, Southampton SO17 1BJ, U.K.
sqes.soton.a.uk
lhes.soton.a.uk
For high-level modulation, MAP/MLSE sequene detetor too omplex
Even MAP or Bayesian symbol-detetor too omplex
z Department of Eletronis and Eletrial Engineering
University of Edinburgh, Edinburgh EH9 3JL, U.K.
Bernie.Mulgrewee.ed.a.uk
Aordable: linear equalizer and deision feedbak equalizer
Classially, MMSE solution is regarded as optimum
MMSE would be optimum only if equalizer soft output were Gaussian
Presented at ICASSP2002, May 13-17, 2002, Orlando, USA
Generally, equalizer soft output has a non-Gaussian distribution
Support of U.K. Royal Aademy of Engineering under an international travel grant
(IJB/AH/ITG 02-011) is gratefully aknowledged
? Adopting to non-Gaussian nature leads to optimal MSER solution for
linear equalizer and DFE
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Communiation Group
S Chen
Communiation Group
Some Previous Works
A Toy Example
MBER equalization almost as old as adaptive equalizer
Yao, IEEE
Shamash & Yao, ICC'74
Chen et
Yeh & Barry, ICC'97, ICC'98?, IEEE
Trans. Information Theory
,
al. ICC'96
, IEE
Chen & Mulgrew, IEE
S Chen
Two-tap hannel 1:0 + 0:5z 1
with 4-PAM and SNR= 35 dB
1972
Two-tap m = 2 linear equaliser
with deision delay d = 0
Pro. Communiations
Trans. Communiations
Pro. Communiations
Mulgrew & Chen, IEEE Symp.
1998
ASSPCC
Normalized MMSE:
w
2000
T
MMSE
= [0:9285
0:3713℄
with log10(SER) = 2:7593
log10(SER)
0
-1
-2
-3
-4
-5
-6
-7
-8
0.2
0
1999?
MSER ( > 0):
2000, Signal Proessing 2001
w
= [0:8957
0:4447℄
with log10(SER) = 7:1566
T
MSER
-0.2
-0.2
0
-0.4
0.2
0.4
w[0]
-0.6
0.6
w[1]
-0.8
0.8
1 -1
MSER solutions form a half line, origin is singular point
?: for multi-level PAM shemes
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Communiation Group
S Chen
PAM Channel Model
Communiation Group
S Chen
Express equaliser output
y(k) = w (r(k) + n(k)) = y(k) + e(k)
Channel of length n
T
h
X1
n
r(k) =
h
? e(k): Gaussian with zero mean and variane w w2
4
? y(k) 2 Y = fy = w r ; 1 q N g, whih an be divided into L subsets
T
h s(k
n
i) + n(k)
i
q
i=0
4
s(k) 2 S = fs = 2l
l
L
Y
1; 1 l Lg
q
s
s
q
l
T
d) +
d
X
0
1
m+n
s(k
2
i) + e(k)
i
6
h
i =d
m
1
Optimal deision making
℄T , and deision delay d
r(k) = r(k) + n(k) = Hs(k) + n(k)
As s(k) 2 fs ; 1 q N g where N = L
,
4
r(k) 2 R = fr = Hs ; 1 q N g
m+n
h
4
= fyq
y(k) = s(k
T
T
s
T
y(k) = w r(k)
m + 1)℄ , w = [w0 w
q
2 Yjs(k d) = s g; 1 l L
Let ombined impulse response = w H = [ ℄. Then
l
Linear equaliser of order m
r(k) = [r(k) r(k
T
s^(k
1
q
8
s;
>
>
< s1;
d) =
>
>
:
if y (k) (s1 + 1)d ;
if (sl 1)d < y (k) (sl + 1)d
for l = 2;
L 1;
if y (k) > (sL 1)d :
l
s ;
L
s
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Communiation Group
S Chen
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Communiation Group
S Chen
Two Useful Properties
Shifting: Y +1 = Y
l
l
PDF of y (k)
+ 2d
Symmetry: distribution of Y
l
cd s1
SER Expression
p (x) = p
is symmetri around dsl.
...
...
cd sl
cd (sl −1)
y
cd sL
y
1
2n
p
0 2 1
( )
1 XX
C
B x y
exp A
N
2 2 w w
ww
T
s
l
Nsb
L
i
n
l=1 i=1
where Nsb = Ns=L is number of points in Yl and yi(l) 2 Yl.
Utilizing shifting and symmetri properties, SER of equaliser w is:
X
Q(g
P (w) =
N =1
Nsb
cd (sl+1)
l;i
E
sb
? For linear equaliser to work, Y , 1 l L, must be linearly separable
This is not guaranteed
? In DFE, linear separability is guaranteed
g
7
l;i
w
( )=
w
( ))
i
where Q is usual Q-funtion, = 2(L
l
T
1)=L, and
y( ) (s
p
w w
l
d
i
n
l
1)
T
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Communiation Group
S Chen
Communiation Group
S Chen
MSER Solution
Blok Adaptation
MSER solution is dened as:
w
MSER
Identify hannel ! P (w) ! optimisation
Alternatively, kernel density or Parzen window estimate approah
E
w
= arg min PE ( )
w
w
An estimated PDF of py (x)
Gradient of PE ( )
1
p
2 w w N
rP (w) = p
E
T
n
(
yi
(s
(l )
d
sb
X
N
(
yi
exp
i=1
1))
l
sb
(l)
(s 1))
2 2 w w
d
n
w r
(l )
!
2
1)hd
+ (sl
l
T
p^ (x) =
y
T
n
X
exp
k=1
n
l
(x y (k))2
22nwT w
K
!
K : sample length, and : radius parameter. From p^ (x), estimated SER
!
ww
Computation is on single subset Y , and further simpliation by using Y with s = 1
Use simplied onjugated gradient algorithm with reseting searh diretion to negative
i
T
p2 pw w K1
1
l
l
P^ (w) =
E
T
X
Q(^
g (w))
K =1
K
k
k
where
g^ (w) =
gradient every I iterations
As SER is invariant to a positive saling of w, it is omputationally advantageous to
normalize weight vetor to w w = 1.
y
y(k)
^ (s(k
p
w
d
w
^ , and h
^ an estimate for the d-th olumn h
=w h
k
^
d
T
d
d)
1)
T
n
d
d
of
H
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Communiation Group
S Chen
Sample-by-Sample Adaptation: LSER
Single-sample estimate of py (x)
p^ (x; k) =
y
2
T
n
n
(y (k)
n
r(k)
(y (k)
^ (k)(s(k
d
^ (s(k
22
d
d)
1))2
!
1))w(k)
(s(k
d)
(x
1
exp
p~y (x; k) = p
2n
T
n
d)
S Chen
Single-sample estimate of py (x)
!
T
w(k + 1) = w(k) + p2 exp
Communiation Group
Sample-by-Sample Adaptation: ALSER
p2 1pw w exp (x2 wy(kw))
With a re-saling after eah update to ensure w w = 1, and using instantaneous stohasti
gradient, ! LSER:
2
10
Using instantaneous stohasti gradient,
w(k + 1) = w(k) + p2
r(k)
^ d (k)
1)h
y(k))2
22
!
n
! ALSER:
(y (k)
exp
n
^ (s(k
22
d
n
(s(k
d)
^ d(k)
1)h
d)
1))2
!
? No need for normalization to simplify omplexity
w(k + 1) = pw (kw(+k 1)+w1)(k + 1)
? Although using rather than n
T
n
pw w appears to involve more approximation, ALSER
T
seems to work well | not restrit to unit length makes it easier to onverge to a MSER
Adaptive gain and width n need to be set appropriately
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Communiation Group
S Chen
Communiation Group
? Feedbak lter oeÆients do not disappear. They have been set to their optimal values.
As s (k) 2 fs ; 1 q N g with N = L +1
Extension to DFE
f
\Linear-ombiner" DFE:
T
b
h
h
b
n
s
b
1
2
f
q
f
l
q
l
Lower bound SER for DFE
T
w under assumption of orret symbol feedbak
P (w) =
b
E
Dene translated observation spae
r0 (k) =4 r(k) H ^s (k) = ~r(k) + n(k)
2
l
? Y~ are always linearly separable. All results of linear equaliser are appliable.
n )℄ ,
b
X
Q(~
g (w))
N
=1
N
g~ (w) =
b
DFE beomes a \linear equaliser":
T
0
r (k) = y~(k) + e(k)
y~
(l)
i
2 Y~ , and N
l
f sb
f sb
l;i
f sb
l;i
y(k) = w
f
g whih an be partitioned into L subsets
~ s(k d) = s g; 1 l L
Y~ =4 fy~ 2 Yj
T
q
h
s (k) = [s(k d 1) s(k d
r(k) = H s (k) + H ^s (k) + n(k)
N g
4
y~(k) 2 Y~ = fy~ = w ~r ; 1 q N
T
b
T
Under assumption ^b (k) =
d
f
4
d 1) s^(k d n )℄ and b = [b1 b ℄
Choose d = n 1, m = n and n = n 1
Dene s (k) = [s(k) s(k d)℄ and partition H = [H1 j H2℄
where ^b (k) = [^
s(k
f
f
T
b
s
f;q
~ = f~rq = H1sf;q ; 1 q
~r(k) 2 R
y(k) = w r(k) + b ^s (k)
T
S Chen
y~( )
l
i
i
(s
p
w w
d
l
1)
T
n
= Nf =L = Ld is number of points in Y~l
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Communiation Group
S Chen
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Communiation Group
S Chen
Distribution of Subset Y~5 (s5 = 1), 64 points, SNR=34 dB
An 8-PAM DFE Example
Weight vetor has been normalized to a unit length, a point plotted as a unit impulse.
0
0 :3 + 1 :0 z
1
with 8-PAM
0 :3 z
DFE:
m = 3, d = 2, n
b
2
=2
decision
threshold
1.5
(a) MMSE
-4
Distribution
Channel:
log10(Symbol Error Rate)
-2
Comparison
2
MMSE
MSER
1
0.5
-6
0
0
-8
0.5
1
1.5
2
1.5
2
y(k)
-10
2
-12
1.5
decision
threshold
(b) MSER
-14
Distribution
Lower-Bound SER
1
0.5
-16
20
25
30
35
SNR (dB)
40
45
15
0
0
0.5
1
y(k)
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Communiation Group
S Chen
Conditional PDF given s(k d) = 1, SNR=34 dB
normalized w
= [ 0:0578 0:2085 0:9763℄, w
=[
T
0:2365 0:7946 0:5592℄
T
MMSE
MSER
Communiation Group
Learning Curves of LSER Averaged Over 100 Runs, SNR=34 dB
Initial weight: (a)
decision
threshold
1
0.5
0
0.5
1
1
y
1.5
2
2.5
conditional pdf
, (b) [ 0:01 0:01 0:01℄T
Weight normalization applied
1e+0
1e-1
MMSE
1e-5
1e-6
1e-7
1e-8
MSER
decision threshold
decision direct
1e-2
1e-3
training
1e-4
1e-6
1e-7
MSER
1e-9
0
0.6
0.4
2000
4000 6000
Sample
8000
10000
0
(a)
0.2
0
-0.5
0
0.5
1
1.5
2
200
400
600
Sample
Initial value is ritial for onvergene, MMSE not neessarily good initial hoie
Communiation Group
S Chen
Learning Curves of ALSER Averaged Over 100 Runs, SNR=34 dB
MMSE
, (b) [ 0:01 0:01 0:01℄T
1e-4
Weight normalization not applied
1e-1
Symbol Error Rate
Symbol Error Rate
MMSE
1e-6
1e-7
1e-8
decision direct
1e-2
1e-9
training
1e-4
2000
4000 6000
Sample
8000
10000
MMSE
1e-5
1e-6
1e-7
MSER
1e-9
0
Communiation Group
0
200
(a)
400
600
Sample
S Chen
Only ZF, equaliser output is Gaussian with noise enhanement
MMSE generally non-optimal and tries to t parameters to non-Gaussian
PDF in a way so that it looks as losely as possible to a Gaussian one
Non-Gaussian approah leads naturally to MSER
For high-level PAM modulation shemes, MSER equalisation solution has
being derived
Eetive sample-by-sample adaptation has been developed
Unlike MSE surfae whih is quadrati, SER surfae is highly omplex
Initial equaliser weight values an ritially inuene onvergene speed
ALSER is partiular promising: simpler omputation
1e-3
1e-8
MSER
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Conlusions
1e+0
1e-5
1000
(b)
17
w
800
In (a) training and deision direted indistinguishable, in (b) dashed urve: after 200-sample
training, swithed to deision-direted with s^(k d) substituting s(k d)
2.5
y
Initial weight: (a)
MMSE
1e-5
1e-8
1e-9
0.8
(b) MSER
MMSE
1e-4
1.5
0
-0.5
w
Symbol Error Rate
(a) MMSE
decision
threshold
2
Symbol Error Rate
conditional pdf
3
2.5
S Chen
800
1000
(b)
In (a) training and deision direted indistinguishable, in (b) dashed urve: after 200-sample
training, swithed to deision-direted with s^(k d) substituting s(k d)
Compared with LSER, no performane degradation, muh simpler
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