ADAPTIVE NESTED TRELLIS DECODING FOR BLOCK CODES Bahram Honary, Garegin Markarian and Michael Darnel1
Bahram Honary
,
Professor, Lancaster Universi ty, Lancaster, UKGaregin Markarian, Lecturer, LancasterUniversity; Lancaster,UK Michael Darnell, Professor, Hull University, Hull, UK
ABSTRACT. A new adaptive trellis decoding technique for block codes is proposed. The technique is based on a recently introduced encoding technique which allows t h e design of almost all known linear codes together with their minimal trellises. The proposed adaptive
trellis decoding technique allows t h e
implementation of a single s o f t maximum likelihood trellis decoder for decoding of different codes. I t has been shown t h a t both t h e encoding and decoding techniques can be implemented with a low complexity and cost.
1. INTRODUCTION.
Trellis decoding technique for linear block codes has received sufficient attention for both practical and theoretical reasons [l-5). On t h e practical side, a minimal trellis s t r u c t u r e of a block code c a n b e used for performing low- complexity maximum likelihood s o f t decision decoding. From t h e theoretical viewpoint, such trellises can b e used
for
constructing other codes. In this paper we introduce a new simple encoding technique which allows t h e design of almost all known linear block codes together with their trellises. The trellises of t h e designed codes a r e similar t o t h e trellises of t h e coset codes introduced by Forney [2]. However, t h e encoding procedure is simpler and allows t h e design of linear block codes with any arbitrary code length.2. CODE DESIGN TECHNIQUE.
The technique is based on generalised array
codes (GACs) and their trellis structures [6]. A generalised (n,k,dmin) array code is an array code in which t h e column and row subcodes may
0-7803-1820-Xl94 $4.00 0 1994 IEEE 682
have different numbers of information and parity check symbols; t h e code length n=n1n2 and t h e t o t a l number of information digits, k= k l + k2+ ...+ kn2, where n1 and n2 a r e t h e number of columns and rows respectively, and kp is t h e number of information digits in t h e pth row 161. GACs may b e considered as being formed from t h e union of coset(s) of a basis conventional array code, where t h e cosets
are
construct,ed such t h a t t h e union of cosets has t h e s a m e minimum distance as t h a t of t h e parent array code. Using t h e concept of t h e GAC, almost all known linear block codes together with their low complexity trellises can b e designed. The encoding procedure can b e also described. as an array representation of coset codes introduced by Forney 121. To illustrate t h e proposed technique an example is given here for an (8,4,4) GAC which is equivalent t o t h e well known (&,4,4) Reed-Muller (RM) code. The procedure for code design is given below: (i) Design t h e basic, (8,3,4), (n=4 x 2, nl=2, n2=4) array code, C1, with t h e single-parity check (4,3,2) column and repetition row codes R1=(2,1,2):
P 4 p4
where xi, i=l,2,3, represent information digits and p-, j=Y,
...
4, represent parity check symbols. ( i i ) Design an additional array code, C2, with t h e following structure:0 x4
0 x 4
c,
=where x4 is an additional information digit. (iii)
modulo-2 basis a s follows:
Add t h e two codes C 1 and C2 on a
P4 (X4@P4)
The designed code is a linear non-systematic code and has t h e following parameters: (n0=8,k0=4,d0=4). I t has been shown [71 t h a t by deleting t h e symbol which is located in t h e second column and 4th row, t h e designed (8,4,4) code will b e transformed t o t h e (7,4,3) Hamming code.
T h e proposed technique has been
investigated for t h e design of other known codes, such as (15,5,7) BCH, (16,5,8) RM, (15,7,5) BCH, (17,9,5) optimum, (24,12,8)
extended Golay, (23,12,7) Golay, etc. For
example, following t h e procedure outlined above, one can design t h e (32,6,16) RM binary code by choosing a
(4
x 8) ( n l = 4 , n2=8) basic array code, C1, with an (8,1,8) row code R I , and (4,3,2) single parity check column code. The first additional code, C2, should consist of a(4
x 5) all zero matrix and a(4
x 3) matrix of repetition column codes. The (31,6,15) code can b e derived easily by deleting t h e parity check symbol which is located in t h e 4th row and 8 t h column. Figure 1 illustrates t h e trellis diagrams of a few examples of t h e above codes. I t isapparent from this Figure t h a t t h e proposed code design technique provides codes w i t h v e r y low-complexity trellis structures.
3. NESTED TRELLIS DECODING.
In many practical applications, such as time- varying communication channels, i t is vital t o have an adaptive coding scheme in which different elements of t h e overall coding format have different level of resilience t o noise and
distortion. The conventional adaptive block
coding techniques
use
a number of differentencoders/decoders e a c h for different channel conditions. Although t h e error performance of such systems is improved, their complexity and cost a r e increased dramatically. To reduce t h e
cost
of sucha
system i t is vitalt o
havea
singles o f t maximum likelihood trellis decoder
(SMLTD) which could deal with different codes being used for information transmission; however, known encoding/decoding techniques do not posses such properties.
In this paper a novel nested s o f t maximum likelihood trellis decoding (SMLTD) SMLTD technique is introduced [9]. The technique can b e employed for decoding a number of different block codes in single decoder, and provides low complexity implementation in comparison with other techniques
.
A new adaptive trellis decoding technique for product codes based on GACs has been introduced recently [61. This adaptive coding
and SMLTD scheme allows
us
t ouse
oneencoder/decoder for different codes. This may b e done either by designing a set of codes t h a t have equal number of s t a t e s and columns in their trellises and only differ in labelling of trellis branches [7-91 or alternately by designing a set of different codes with trellises which a r e embedded into each other. The trellises of such codes will b e nested into e a c h other and t h e procedure for t h e design of such an adaptive encoder and its SMLTD is a s follows.
(i) L e t (n,k,d) b e a block code with maximal code length and information rate.
(ii) Represent n=m x n2 where both m and n2
a r e integers and define t h e size of array ( m is number of columns). The following condition
d
2n,
( 4 )must b e satisfied in order t o design t h e best code, where d is a minimum Hamming distance of t h e code.
Design a basic ( m x "2, k,d) GAC [6];
The trellis diagram of t h e designed GAC
will have Ns=2{maxkp) number of s t a t e s and Nc=m+l number of columns, where kp is number of information digits in p-th row, and each branch will b e labelled with n2 digits according t o [6].
(iii) (iv)
Delete t h e parity check symbol located in m t h row and n p d column. A new code will have t h e following parameters (n- l,k,d-1) and a similar trellis. The only difference between two trellises will b e in t h e number of digits being used for labelling branches a t final depth of t h e trellis.
Delete t h e pth column in t h e basic GAC. A new code will have t h e following parameters (n-nz,k-l,d'), where d ' r 9 - 1 . The trellis diagram for t h e new code will have t h e s a m e number of columns Nc, but numb r of s t a t e s will b e reduced t o Nc=2 maxrkp-ll. The labelling of t h e branches will require n2-1 digits and t h e new trellis is a p a r t of t h e basic trellis. Delete t h e parity check symbol located in m t h row and n2-1 column. The new code will have t h e following parameters (n-n2-l,k-l,d'-l) and a similar trellis. The only difference between two trellises will b e in t h e number of digits being used for labelling branches a t final depth of t h e trellis.
(viii) Repeat steps (vi) and (vii) until only one information digit will b e left (repetition code). In this c a s e Ns=2 and Nc=m+l.
An example of such a novel technique is shown in Fig.2. From this Figure i t follows t h a t trellises of different codes ((4,1,4) repetition [2], (4,3,2) single parity check [2], (7,4,3) Hamming [7], (8,4,4) Reed-Muller [71, (1 5,5,7) BCH [91, (16,5,8) Reed-Muller [9], (31,6,15) BCH [9]) a r e nested into t h e trellis diagram of t h e (32,6,16) Reed-Muller code [9], represented in t h e form of GAC. This means, t h a t t h e presented trellis diagram can b e used for s o f t maximum likelihood decoding of any of t h e above listed codes. The trellis diagram shown in Figure 2 c a n b e nested itself into a trellis of block codes with higher block lengths or information rates. For example, all t h e trellises shown in Figure 2 can b e nested into t h e trellis
diagram of (32,20,8) block code which has t h e same number of columns and 64 states.
The information throughput of t h e system can b e optimised by an appropriate choice of component codes. This can b e achieved since t h e size of nested trellis (number of s t a t e s and columns) depends on t h e component codes.
4. CONCLUSION
A novel
low
complexity adaptive nested SMLTDtechnique is introduced. The technique
is
applicable in all communication systems aiming a t providing low-complexity encoding and s o f t maximum likelihood trellis decoding of block binary error control codes. The main benefit. would b e lower complexity (and hence lower cost) and maximum likelihood performance,
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
Wolf J.K. "Efficient maximum likelihood decoding of linear block codes using a t r e l l i s " . IEEE T r a n s a c t i o n s on Information Theory, vol.IT-24, No 1,
Forney G.D. "Coset codes-Part 2: Binary
lattices and related codes". IEEE
Transactions on Information Theory,
Muder D. J. "Minimal trellises for block codes". IEEE Transactions on Information Theory, vol.IT-34, No 5 , 1988, pp.lG49- 1053.
Kasami T., Takata T., Fujiwara T., S.Lin.
"On, complexity of trellis structure of linear block codes". Proc. of IEEE International Symposium on Information Theory, Budapest, Hungary, June 1991. Berger Y., Be'ery Y. Bounds of t h e Trellis Size of Linear Block Codes. -
"IEEE Transactions on Information
Theory", vo1.39, No 1, 1993, pp.203-209. Honary B., Markarian G., Farrell P. "Generalised array codes and their trellis structure". Electronics Letters, vo1.29,
Horiary B., Markarian G. "Low
complexity trellis decoding of Hamming codes''. Electronics Letters, vo1.29, No
Markarian G., Honary B., Soyjaudah K.M.
"Trellis decoding technique of adaptive product codes".- Proceedings of 2nd I n t e r n a t i o n a l S y m p o s i u m o n C o m m u n i c a t i o n T h e o r y a n d Applications", July 1993, Lake District,
UK, p.96.
Honary B., Markarian G, Darnel1 M.
Nested Trellis Decoder For Block Codes. UK Patent Application No.9414Z6-7, 14.07.94. 1978, pp.76-80.
vol.IT-34, NO 5, 1988, pp. 1152-1187.
NO
6,
1993, pp.541-542.Fig.1.
Trellises of Different
Block
Codes
oo/oo o / o o o/oo
0/01 0/01
Trellis Diagram of t h e (R,4.4) R M Code
Trellis D i a g r a m of t h e ( 1 5 , 7 , 5 ) GAC
Trellis Diagram of t h e (7,4,3) H a m m i n g Code
kzc:Fx!
T r e l l i s Diagram of t h e ( 3 1 , 6 , 1 5 ) Code
Trellis No 1, for ( 4 , 3 , 2 ) code.
Trellis No 2 together with trellis No 1, for both
( 7 , 4 , 3 . ) Hamming and (8,4,4) Reed-Muller codes.
_ _ _ _ ~ _ . . . _ _ _
Trellis No 3, together with NoNo 1 and 2 for both
(15.5,7) BCA and (16,5,8) RM codes.
-
TreeEis No 4 , together with NoNo 1.2,and 3 for both (31.6.15) BCH and (32.6.16) RM codes.
