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References1 CHRISTMAS, w.J., KITLER, J., and PETROU, M : Non-iterativecontextual correspondence matching. Computer Vision -ECCV94, 1994, (Springer-Verlag, Stockholm, Sweden),2, pp. 137-1422 CHRISTMAS, W. J , KITTLER, J., and PETROU, M : Matching incomputer vision using probabilistic relaxation, IEEE Trans.Pattern Anal. Mach. Intell. , 1995, 11, S), pp. 149-164

softdemodulator- -

Algorithm for continuous decoding of turbocodesS. Benedetto, D. Divsalar, G. Montorsi and F. Pollara

softdecoder *

Indexing terms: Turbo codes, Convolutional codes, DecodingA new continuous version OT the maxinium U posteriori algorithmis described and applied to sequence oriented decoding of parallelconcatenated convolutional codes.

Introduction: Optimum symbol decision algorithms for digitaltransmission must base their decisions on the maximum a pos ter i-or i (MAP) probability. They have been known since the early sev-enties [l], although they are much less popular than the Viterbialgorithm and almost never applied in practical systems. Only recently interest in these algorithms has been revived, in connectionwith the problem of decoding concatenated coding schemes, alsoknown as turbo codes.

The decoding algorithm for turbo codes uses an iterative pro-cedure whose heart is an algorithm which computes the sequenceof the a pos ter ior i probability (APP) distributions of the informa-tion symbols. There have been several attempts to achieve, or atleast to approach, this goal. The algorithm originally proposed [2]is based on a modification of the soft-output Viterbi algorithm(SOVA), whereas other approaches consisted of revisiting the orig-inal symbol MAP decoding algorithm and simplifying it to a formsuitable for implementation [3, 41. These algorithms require thetransformation of the parallel concatenated convolutional codingscheme (PCCC) into an equivalent block code. This in turnrequires trellis termination of all constituent encoders (thus reduc-ing the code rate) and block synchronisation, and leads to a nonu-niform reliability of the decoded bits.

In this Letter we propose two new sliding-window versions ofthe MAP algorithm (the optimum one and its suboptimum ver-sion) which allow a continuous decoding of the PCCC, avoidingthe problem of trellis termination and leading to a more flexiblesystem solution.rnemoryless

source

Description o the algorithms: We describe the sliding-windowMAP algorithms with reference to the system whose block dia-gram is shown in Fig. l. The information sequence U emitted bythe source enters an encoder which generates code sequences c.The code symbols c enter the modulator, which performs a one toone mapping of the symbols with the modulator signals x,. Thesignals x,~ re transmitted over a stationary memoryless channelwith output symbolsy k .The channel is characterised by the transi-tions probability distribution PCylx). The channel output sequenceis fed to the symbol by symbol soft output demodulator, whichproduces a sequence of probability distributions y(x) according tothe memoryless transformation

a(1)Y k ( 2 ) =P ( Z k =Z , Y k )=P ( Y k / Z k=5 ) P ( Z k =Z)

The sequence of y,(x) is then supplied to the soft output symboldecoder, which processes them to obtain the probability distribu-tions Pk(uIy),defied as

Pk(UlY) 6 ( U k =UlY) (2)The probability distributions Pk(uIy)are the symbol by symbol apos ter ior i probabilities (APP) and represent the optimum symbolby symbol soft output.

The original MAP decoding algorithm [l],which is referred tohere as the BCJR algorithm (from the authors initials), requiresthe whole sequence to have been received before the decodingprocess starts. To apply it in a PCCC we need to subdivide theinformation sequence into blocks, encode them by terminating thetrellises of both constituent codes, and then decode the receivedsequence block by block.

51*II

SI*II

SIeII

* eSN S N Nk- 1 k k+l

j 67znFig. 2 Meaning of notationsA more flexible decoding strategy is offered by a moditicationof the BCJR algorithm in which the decoder operates on a fixedmemory span, and decisions are forced with a given delay D . Wecall this new algorithm the sliding window BCJR algorithm (SW-BCJR). Its description will be based on time-invariant couvolu-

tional codes and will make use of the notations pictorially shownin Fig. 2, where:(i) Si s the value assumed by s the generic state at time k,belonging to the set S ={S,,.., SN}.(ii) S;(u) is one of the precursors of Si, and precisely the onede fied by the information symbolU emitted during the transitionS&) +Se(iii) &+(U) is one of the successors of S,, and precisely the onedefined by the information symbol U emitted during the transitions;4 S,(u).(iv) A signal x is associated to each transition in the trellis, thissignal depends on the state where the transition originates fromand on the information symbol U determining that transition.When necessary, we will explicitly state this dependence writingx(u,Si) hen the transition ends in S,, and x(S,,u)when the transi-tion originates from S i n e SW-BCJR algorithm will be describedin steps. It is composed of the following steps, gwen withoutproofInitialisation of the orward recursion: Initialise the quanti tiesq(S,)according to

1 if S O =S,C2O(Sz) ={0 otherwise ( 3 )

Forward recursion (k > 0 ) : Compute the quantities a,-B-,(SJaccording toa k ( S z ) =ha ~ ~ k - l [ s ; ( U ) I Y k [ ~ ( U , s z ) l (4)

U

beingh, a constant determined through the constraintC%(SZ)1S,

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Initialisation of the backward recursion ( k 1 0):nitialise thequantities & ( S i ) according toP k ( S j ) =1/N, QSj ( 5 )

Backward recursion ( k >0): ompute the quantities Pk-&,(SI)from time k - back to time k - D according to

kbeing h p a normalisation constant determined through the sameconstraint as h,.Computat ion o a posteriori transition probabilities: Compute the aposteriori transition probabilities ok-,(S1,u) t time k -D accordingto

fl k -D (st U ) =hc k - D- 1 (st Y k - D [x st U ) ] k- [s?U ) ](7)

Final step: Compute the AFP at time k - D asS,The main features of the algorithm are the initialisation of thebackward recursion assuming a uniform probability for all thestates, which makes it suitable for continuous decoding, and thefact that it does not need the storage of the N x D values of thequantities as they are updated with a delay of D steps.

SimpliJications o the algor i thm: As already proposed for theblock-decoding originalMAP algorithm, the SW-BCJR algorithmcan also be simplified by passing to the logarithms. This leads tothe problem of evaluating the logarithm of a sum of exponentials

r 1

L z JTwo solutions are possible. If we extract the term with the highestexponential

AM =max A,1so that

(9)the first solution computes the second term in the right-hand sideusing look-up tables, as proposed in [4] for the block-decoding.We call this algorithm SW-Log-BCJR. The second neglects thesecond term and leads to the simpler, suboptimum SWAL-BCJR,where A stands fo r approximate. A pleasing feature of the sec-ond algorithm is that it does not require the estimate of the noisevariance.

10

1oihaJ- -3nD l

18-510 1 3 5 7 9 11 13 15 17 19numberof iterations /6721?1

Fig. 3 Convergence o turbo-decoding. Bit error probability againstnumber of iterations using SW-Log-BCJR algorithmCurves differ for signal-to-noise ratio(i) E,,/No = -0.05, (ii) EJN, = 0.00, (iii) E b / N o= 0.05, (iv) E,/No =0.10, (v) E,,/No =0.15, (vi) Eb/No=0.20, (vii) & / N o =0.25, (viii) E,,/N o =0.35, (ix) E,/No =0.45, (x) E b / N o=0.50

Simulation results: We have simulated the two sliding-windowSW-Log-BCJR and SWAL-BCJR algorithms embedded into theiterative PCCC decoding procedure. All results refer to a rate 113PCCC with two equal, recursive convolutional constituent encod-ers with 16 states and a random convolutional interleaver oflength 16,384.

In Fig. 3we show the obtained bit error probabilities againstthe number of iterations of the decoding procedure using the SW-Log-BCJR algorithm for various values of the signal-to-noiseratio. The decoding algorithm converges down to signal-to-noiseratios of 0.05dB. Since for rate 113 codes the Shannon capacitylimit is -0.55dB, our result is, at 0.60dB from it, the closest resultever published.From the simulation of the simpler SWAL-BCJR algorithm, wehave found that the penalty incurred by it amounts to -0.60dB.0 EE 1996Electronics Letters Onine No: 19960217S. Benedetto and G. Montorsi (Dipartimento di Elettronica, Politecnicodi Torino, C. Duca degli Abruzzi 24, 10129 Torino, Italy)D. Divsalar and F. Pollara (Jet Propulsion Laboratory, 4800 OakGrove Drive, Pasadena, CA 91109, USA)

4 December I995

References1 BAHL, L.R., COCKE, J., JELINEK, F., and RAVIV, . Optimal decodingof linear codes for minimizing symbol error rate, ZEEE Trans.Info. Theory, 1974, pp. 284-2872 BERROU, c., ADDE, P., ETTIBOUA, A. , and FAUDEIL, s.: A low

complexity soft-output Viterbi decoder architecture. Proc. ICC93,Geneve, Switzerland, May 19933 PIETROBON, s.s.:Implementation and performance of a serial MAPdecoder for use in an iterative turbo decoder. Proc. 1995 IEEEInt. Symp. Info. Theory, Whistler, British Columbia, Canada,September 1995, pp. 4714 ROBERTSON, P., VILLEBRUN, E., and HOEHER, P.: A comparison ofoptimal and sub-optimal MAP decoding algorithms operating inthe log domain. Proc. ICC95, Seattle, Washington, June 1995, pp.1009-1 0 13

Design of stuck-open fault testableCMOScomplex gatesY. Tsiatouhas, Th. Haniotakis, C. Halatsis andA. Arapoyanni

Indexing terms: CM OS integrated circuits, Built-in self testTests that detect transistor stuck-open (TSOP) faults independentof timing skews in input changes may not exist for al TSOPfaults in CMOS complex gates. A new design method is presentedwhich does not require any extra hardware or test inputs toimprove the testability of a CMOS complex gate.

Introduction: CMOS logic circuits exhibit sequential behaviour inthe presence of TSOP faults. A test that detects a single TSOPfault in the p(n)-network of a CMOS gate requires an initialisinginput vector which forces the output to the logic value 0(1), fol-lowed by a test input vector which activates one or more conduct-ing paths from V,,(V,) to the output, all coming through thefaulty transistor. All such conducting paths in the fault-free opera-tion become nonconducting due to the presence of the fault. Con-sequently, the output goes to a high-impedance state keeping thelogic value 0(1), which is determined by the charge retained in thecapacitance of the output node placed on by the initial vector,instead of the expected logic value l(0). The initialising nput vec-tor may be invalidated due to time-skews in input changes. Teststhat prevent test invalidation are defmed as robust tests [l]. Insuch robust test sequences, the Hamming distance between the ini-tialising and the test vector is kept at unity in order to avoid apossible intermediate state that may cause a test invalidation.Many testable designs for CMOS complex gates have been pro-posed in the open literature. A review can be found in [2] .All

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