ISIT 2000, Sorrento, Italy, June 25-30,2000
Optimal 4- and 8-State Across-the-subchannels
TCM Encoders for DMT Systems
Abstract - We give t h e o p t i m a l 4- a n d 8-state trellises for across-the-subchannels T C M for D M T systems.
The upper bound on the convergence length of a trellis is given
by PI
Lmaz
I. INTRODUCTION
TCM can be performed for DMT systems in two ways :
coding parallely and coding across the subchannels. The decoding delay in the latter case is M times less than that in
the former case, where A4 is the number of subchannels [l].
We refer the latter as across-the-subchannels TCM for DMT
systems.
At the receiever input, the SNR's in different subchannels
are different due to the channel impulse response. Thus, the
minimum weighted Euclidean distance becomes the decision
criteria for ML decoding, and hence we use weighted Viterbi
decoding. Due to this weighting, the best trellis known for
single carrier systems need not be the best in our case.
11. CLASSIFICATION
OF
V .Umapathi Reddy'
Department of Electrical
Communication Engineering
Indian Institute of Science
Bangalore 560 012
India
vurQece.iisc.ernet.in
B. Sundar Rajanl
Department of Electrical
Communication Engineering
Indian Institute of Science
Bangalore 560 012
India
bsrajanOece.iisc.ernet.in
V . Shashidhar
Department of Electrical
Communication Engineering
Indian Institute of Science
Bangalore 560 012
India
shashidhar@protocol.ece.iisc.ernet.in
TRELLISES
We classify all the S-state trellises into y classes (where y
= log, S ) as { S ( 2 2 , p:) 1 5 z 5 y } , where S(2=3p)
denotes an
S-state trellis with a node at a level connected to 2= nodes in
the next level and having 2p parallel transitions. We label the
top most node as SO and the last node as s27-1.
Definition 1 : A cyclic trellis is a trellis in which the branches
diverging from a node s, at any level connect to 2b-p nodes
of the next level, beginning from s(, 2 6 - p ) mod 2'1 and ending
at s ( ( , + ~ ) z b - - p - - 1 ) mod 2 7 , where b is the number of input bits
per symbol.
=
Y
L,J
+1
where bl refers to that part of the input bits which affects the
. denotes the largest integer less than
state of encoder and J1
or equal to I.
T h e o r e m 1 : The convergence length of a cyclic trellis is
equal to L,,, , i.e., cyclic trellises achieve the upper bound on
the convergence length.
111.
O P T I M A L 4- A N D %STATE
min
1E[O,M-11
{2s1w,
+
>
S , ~ I W , ~ ~ 4stwt
}
2 2.
else the 4(2;bm*n-1)
cyclic trellis, for b,,
T h e o r e m 3 : The best trellis for %state across-thesubchannels TCM is
(a) the 8 @ ; O ) cyclic trellis, for b,,, = 1,
(b) the 8(4;0)cyclic trellis, for b,,
= 2,
(c) the 8(8;bm1n-3)
cyclic trellis if
SkWk}
> 8~twt
cyclic trellis, forb,,,
else the 8(4'bm*n-2)
(a)
(hl
(Cl
TRELLISES
Let b,,,
= min,e[O,~--l~{b,},
where b, is the number of input
where
bits in a t h subchannel and stwt = minlE~o,~-ll{s,w,},
s, and wl are the squared miniumum Euclidean distance of
the z t h subchannel symbol constellation and weighting factor
for that subchannel, respectively.
T h e o r e m 2 : The best trellis for 4-state across-thesubchannels TCM is
(a) the 4(*s0)cyclic trellis, for b,,, = 1,
(b) the 4(4vbmln
-')
cyclic trellis if
2 3.
(d)
ACKNOWLEDGMENTS
Figure 1: Some possible 4-state trellises : (a) 4(2;0)non-cyclic
( b ) 4";')
cyclic (c) 4(';') cyclic (d) 4(4;0)cyclic
Definition 2 : The Convergence length of a trellis is defined
the minimum of all lengths of pairs of paths that diverge
from a node, excepting the parallel transitions, and converge
at. another node.
as
'This work was partly supported by CSIR, India, through Research Grants (No:25(0086)/97/EMR1-11) and (22(0298)/99/EMR11) t,o B.S.Rajan
?-This work was supported in part by DARPA Grant F49620-951-0525-PO0005 during his stay at Stanford university.
0-7803-5857-0/00/$10.00 02000 IEEE.
B S R a j a n gratefully acknowledges IBM India Research Lab,
for the travel support to present this paper.
REFERENCES
[l) J . A. C. Bingham, "MulticarrierModulation for Data Transmission: An Idea Whose Time Has Come," IEEE Communication
Magazine, pp.5-14, May 1990.
[2] S. Hamidreza Jamali and Tho Le-Ngoc, "Coded-modulation
Techniques for Fading Channels," Kluwer Academic Publishers. 1994.
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