535 SOFT-DECSON THRESHOLD DECODNG OF CONVOLUTONAL CODES R.M.F. Goodman*, B.Sc., Ph.D. W.H. Ng*, M.S.E.E. Sunnnary Exising majoriy-decision hreshold decoders have so far been limied o his paper a new mehod for implemening sof-decision majoriy hreshold decoding of convoluional codes is inroduced and explained. The mehod is illusraed by describing sof-decision decoders for a simple random conrol properies under boh Gaussian and burs noise condiions. n he ~isappoining in comparison wih convoluional code sysems designed specifically for burs-error correcion. n his paper we presen a sof-decision majoriy hreshold decoding scheme achieved by us"ing sof-decision is abou 2 db for infinie-level quanisquanisaion (as asummed in his paper) is only.2 db. ~ve herefore expec a maximum improvemen of abou 1.8 db for sof-decision majoriy n his paper we firsly ouline hard-decision majoriy hreshold decoding lengh 2 code as an example. Nex we describe our general mehod for
536 c' = x CHANNEL - X X X X-l one 1 delay C = X EB X-l Fig. 1 simple convoluional encoder where g = 11 1 2. Hard-decision majoriy hreshold decodi?g pu rae of l/v), he firs of which is he unchanged informaion digi. g(l) g(2) g(4)... resric our discussion in his paper o rae one-half codes. K=2 segmens, o review he basic hard-decision majoriy hreshold decoding of only a single one-bi delay elemen and a single modulo-2 adder (exclusive-or gae). Given a sequence of informaion digis x =... X+l..., where denoes he ime uni of he informaion digi X' each previous informaion digis. For serial ransmission he coded digis are sen o he channel in order c' c" by appro priae acion of he swich. (hard-decision) single-error-correcion capabiliy of he code. The decoding acion is explained wih reference o he six poins, a, b, c, 81' 82' and fi~-l. The six poins are inerpreed as follows: a = x.. b = $ n'
c = -l (X-l 537 2= (x l l 5 - - - - - - - nl = 1 if 5 = 5 = 1 -l 1 2 oherwise l 2 @nl 2 )@(x 1 ) x 2 @nll 1) 5 = nl 6} nl 6} nl 1 -l 5 2 -l -l -2 orhogonal on he noise digi nl 1. Thus if a single error occurs - anywhere in he 5 digi span covered by he orhogonal check sums, he only case when 51 = 52 = 1 is when n~-l = 1. gae sends an esimae n~-l of n~-l o cancel he noise digi n~-l from he rece.ived digi (X-l 6) n~-l)' anp hus produce an esimae x 1 of he ransmied digi x 1. From equaion (1) i can be seen - - ha if more han one error occurs in he 5 digi span covered by {51' 52}' hen he error correcion capabili~y of he code is exceeded and he decoded The decoder described above can be improved by he use of feedback. This n',- - - - - i - c' a b DECODER X-1, 1, X X-11 CHANNEL FEEDBACK 1,-- --- S 1 2 c", c -- -- 1 Fig.2 n" "AND" Gae
539 can be simplified by feeding ba~k n~-2 o cancel n~-2 in equaion (1). We may hen replace S2. w~ h S2 = S 2 "" ~ n-2 = n-l '1" w n-l w 1' n-2 "'-, w n-2. f he esimae n~-2 is correc, ha is n~-2 = n~-2' hen S2 = n" This means ha provided he previously decoded digi was correc, - he decoder check sums {Sl' S2} only span 4 digis, and can herefore correc a single error anywhere in 4 digis as opposed o 5 digis in he previous case. decoder ha makes use of pas decisions o simplify feedback decoder, whils a decoder ha does no use pas decisions is called a de inie decoder. n general, if i is possible o form a se of 2e pariy check equaions which are orhogonal on a specified noise digi, hen i is possible o build a hard-decision majoriy hreshold decoder which can correc any combinaions of e or fewer errors over one consrain span. Figure 3 shows he encoder/decoder arrangemen for a riple error-correcing rae one-half (24,12) majoriy decoder which has K=12, and an effecive consrain lengh of 22 digis wihin which 3 or fewer errors can be correc~ed. This decoder can achieve a bi error rae of 1-5 on he binary symmeric channel (which is comparable o he (23,12) perfec Golay code), and can be buil wih only 16 sandard inegraed circuis (which is much less han ha required o decode he Golay code). 3. Sof-decision majoriy hreshold deocding n his secion we inroduce our new mehod for sof-decision majoriy hreshold decoding. Our basic approach is o derive a modified se of, orhogonal check sums {Si} which can be used o esimae each noise digi in he sof-decision sense. BCD equivalen. For example, [] =, [1] = 1, [1] = 2,... [111] = 7. The X are herefore expressed as [] when X =, or [111) when x = 1, in he sof-decision sense. The noise digis are expressed in a similar manner bu can ake any inermediae value beween and 7, ha is, = [) < [n'.] < [111] = 7, where he square brackes indicae - -J - a quanised or sof-decision noise digi. Noe ha he mos significan digi of a quanised digi is he hard decision digi iself. For example, and [n'.] = [11] implies n '. = 1. Le us define dh o be he hard-decision minimum disance beween he wo halves of he iniial code ree. The guaraneed error-correcing capabilsecion 2 has dh = 3, and is herefore a single error-correcing code.
54.-i ') ',f" r ~ J~ +::> ",><.-i -+::>,~ c..-j.-i.-i ~ ~ r i :>;, r--f -+::> ~ ~ '"-'.-i *N (/}.-i + -+::> *.-i ~ (/}.. Q) +::> Q) +::> ' H 'OH ~ Q) Po ro N ' > +::> ;j H ~ ~~ a ro. ~ s.a ~ r--. 4-j.-i ~ ' QJ 1::1 - +::> ~,..4 ~ ~ 4-j rx:j ' rx:j ~ - ~ ~ H ' - +::> rx:j, -,., :>;, >, --r- ro-.j ~ Z ~ r--f,..4 ~, Q) ' ' '" QJ '
digis herefore have o be obained by a process of esimaion as follows. Refer o Fig. 2. The sof-decision received digi a poin a is [x ~ n']. he esimaes of X d = (Q-l) x ' dh 541 and is error correcion capabiliy is e sof-decision s s digis, where es is he larges ineger saisfying es ~ (ds - 1)/2. The simple example code herefore has d = (8-1) x d = 21, and e = 1. [(d -1)/2]/(7x4) =.36 for sof decision. s S * = [n'] + [nl] + [n' ] 1 -l S*= 2 [n' -l ]+[nll] -l ' (2) where feedback has been used o produce S2*. We may hen le he or le a~-l = 1 if Sl* + S2* - [n~_~] > 1. Tha is, we assume ha he is ha we canno direcly obain he quanised noise digis, because any As we use 8-level quanisaion he informaion digi x is esimaed as: esimaed noise digi [a~] is derived from he following equaions: ~ 4. n a similar manner, he received digi [x - - esimae. From he receiv~d digi [X ~ X-l ~ n~] a poin c, and from
no acceped ary previous decoding error. Noe ha as n~-l = n~-l = 1, Using he following sof-decision procedure, however, X-1 can be 8 * = [11'] + [11' 1 - and 8 2 * = [11' 1 ] + [11"] = [1] + [1] + [111] = 1, 1 ] + [11" 1 ] = [1] + [all] = 5 - - (6) The value of 81* + 82* - [11~-1] is hen 1 + 5-2 = 13 which is Thus x = - - - l is 1 ] = 1, here is no conradicion in 11', indicaes direcly wheher or no a conradicion exiss in he original assumpion X-1. orhogonal check sums and i is no convenien o direcly derive he 1 542 Fig. 4 shows he sof-decision hreshold decoder for he rae one-half code used above, and he following example illusraes is operaion. Le us assume h"a X = X-1 =, ha he noisedigis are [n~-l] = [11], [n~-l] = [1], [n~] = [1], [n~] = [], and ha he decoder has a hard-decision decoder would decode X-1 = 1 hus giving a decoding error. decoded correcly. (1) Because he received digi [X ~ n~] = [1], we le X = a and [11~] = [X ~ n~] = [1]. (2) Because he received digi [X-1 ~ n~-l] = [11], we le X-1 = 1 and [11~-1] = [X-1] ~ [X-1 ~ n~-l] = [1] (3) Because he received digi [X ~ X-1 ~ n~] = [], we le [11~] = [x ] ~ [x ] ~ [x ~ x 1 - - = [111]. [X-1] ~ [X-2] ~ [X-1 ~ X-2 ~ n~-l] = [X-1] ~ [X-1 ~ n~-l] = [111] ~ [1 ] = [all], and all he noise digis are (5) By using ordinary addiion: greaer han (ds - 1)/2 = 1, and herefore indicaes 11~-1 = 1. This however conradics our assumpion of X-1 = a and he hard-decision received digi x 1 correced by he modu1o-2 addiion of 11~-1. Noe ha by assumming, and recalculaing seps 2,3, and 4, we have [11~-1] = [11] [11~] = [], and [11~-1] = [1]. Thus 81* = 6, 82* = 9, and 81* + 82* - -- is correcly decoded as X-1 = o. and herefore X For a simple sof-decision decoder described above he value of 81* + 82* - [11' 1] can be calculaed and he esimae of 11' 1 - - Wih a muliple error-correcing code here are more han wo value 81* - X-1 = value he of X-1. of n + hese 82* cases [11~-1] our for mehod boh involves cases he and 1. decoder compuing X-1 =
more 543 value of Sl* + S2* - [n~-l]' A complex decoder illusraing his poin is oulined in he nex secion. 4. Sof-decision diffuse hre~hold decodi?g n general, mos real channels are ime-varying and subjec o burss of errors. Diffuse convoluional coding is one mehod of providing bursand-random error-correcion capabiliy, and is mos suiable for a channel densiy burss occur on a noise background whose random error rae is relaively high. majoriy-decision hreshold decoding. The sysem is rae one-half., b-difdecisions were correc, i is possible o form four check-sums orhogonal on he noise digi n~-3b-l as follows: - -l -b-l -l -3b-l -l -b-l This exra modificaions. Our basic approach is o calculae he algebraic sum of four sof-decision noise sums which are orhogonal on he informaion digi X-3b-l' for boh he assumpions X-3b-l =, and X-3b-l = 1. S. = [fll] + [f" Q) nl] + [n ] + [nl] -b-l -2b-l -3b-l ' if we le X-3~-1 =, and i = 1 if X-3~-1 = 1. The decoding decision depends on he value decode if So ~ Sl or X-3~-1 = 1 if So~ Sl'
544 <] (Y"'\ +>,.c: -'-" (, C- Pi ' "... r.. " (,) or-i or-i ',:q Q) Q).c: r.x: r.x: S Q),j cd +> :> ;:1 Q), PiQ)Q) ;::!.::: cd a +>.::: s QJ r.x: (/) (,) r.x: c: CJ co c: or-! -.J ;j > c: ::r: 4-1 (,) 4-1 - -- - - - - or-! c: or-! ><: QJ ldue ~La'~ :>:. cd 'U Q) oj:: <J <J ' < +> r.x: (,) <J z +> r.x: ><: p or-! +>., ><: r!:;!j! :'()~o,b
. 545,
~ X-3~-1 ~ X-4~-1 $ X-5~-2 $ n~-2~-1] S(4) = [r-3~-1] ~ [X-4~-1 $ X-5~-1 ~ X-6~-2] = [n~-3~-1] 546 $ n '], [x 1 $ n' 1], [x $ n' ] - - -~ -~ and [X-2~ $ n~-2~] X' X-l' X-~' and X-2~' by aking he hard-decision esimae of each mos significan bi. (2) The value of So is found as follows. From he four received digi [r] [r-l] = n~-~-l] [r-2~-1] = [X-2~-1 [r-3~-1] = [X-3~-1 The previously esimaed informaion digis x, x- 1 -u - 2 u and he previously decoded digis X-4~-1' X-5~-1' X-5~-2' X-6~-2: S(l) = S (2) = [r ] $ [x ] = [n" $ n" ] -l -l -l -~-l S(3) = [r-2~-1] Hence, S = L S(j), for j = 1 o 4. (3) The value of Sl is given by Sl = L S(j) $ [ll], for j = 1 o 4. 5. Conclusions 3 1 = if S < Sl 3 1 =1 if S > Sl. - u- - - u- hreshold decoding of convoluional codes. The mehod consiss of enables he advanage of increased coding gain o be realised wihou deermined, as opposed o one wih hard decision, he increase in he esimaed oupu digi. 6. References 1. Massey, J.L.: Threshold Decoding., M..T. Press, Cambridge, Mass., 1963. 2. Wozencraf, J.M., and Jacobs,.M.: Principles of Communicaion Jnr.: Convoluional coding for channels wih memory, EEE Trans., T-14, 1968.