Forchhammer, Søren

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Analysis of bit-stuffing codes and lower bounds on capacity for 2-D constrained arrays using quasi-stationary measures

A method for designing quasistationary probability measures for two-dimensional (2-D) constraints is presented. This measure is derived from a modified bit-stuff coding scheme and it gives the capacity of the coding scheme. This provides a constructive lower bound on the capacity of the 2-D constraint. The main examples are checkerboard codes with binary elements. The capacity for one instance of the modified bit-stuffing for the 2-D runlength-limited RLL(2,∞) constraint is calculated to be 0.4414 bits/symbol. For the constraint given by a minimum (1-norm) distance of 3 between 1s a code with capacity 0.3497 bits/symbol is given

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Analysis of bit-stuffing codes and lower bounds on capacity for 2-D constrained arraysusing quasistationary measuresForchhammer, SørenPublished in:Proceedings. International Symposium on Information Theory, 2004. ISIT 2004.Link to article, DOI:10.1109/ISIT.2004.1365199Publication date:2004Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Forchhammer, S. (2004). Analysis of bit-stuffing codes and lower bounds on capacity for 2-D constrained arraysusing quasistationary measures. In Proceedings. International Symposium on Information Theory, 2004. ISIT2004. IEEE. DOI: 10.1109/ISIT.2004.1365199Analysis of Bit-Stuﬃng Codes and Lower Bounds on Capacityfor 2-D Constrained Arrays using Quasi-Stationary MeasuresSøren ForchhammerResearch Center COM, 345vTechnical University of DenmarkDK-2800 Lyngby, Denmarke-mail: sf@com.dtu.dkAbstract — A method for designing quasi-stationaryprobability measures for two-dimensional (2-D) con-straints is presented. This measure is derived froma modiﬁed bit-stuﬀ coding scheme and it gives thecapacity of the coding scheme. This provides a con-structive lower bound on the capacity of the 2-D con-straint. The main examples are checkerboard codeswith binary elements. The capacity for one instanceof the modiﬁed bit-stuﬃng for the 2-D runlength-limited RLL(2,∞) constraint is calculated to be 0.4414bits/symbol. For the constraint given by a minimum(1-norm) distance of 3 between 1s a code with capac-ity 0.3497 bits/symbol is given.I. IntroductionWe present a method for designing two-dimensional (2-D) con-strained codes based on bit-stuﬃng. We consider 2-D arrayswith elements taken from a ﬁnite alphabet. The constraint isspeciﬁed by the set of admissible conﬁgurations on an N byM rectangle. For 2-D RLL(1,∞), bit-stuﬃng provides eﬃ-cient coding [1]. In [2] bit-stuﬃng for 2-D RLL(d,∞), d ≥ 2were considered. We shall take a slightly diﬀerent approach tobit-stuﬃng in order to facilitate analysis e.g. providing a con-structive lower bound on the capacity of the constraint. Themethod is generally applicable to checkerboard constraints,where a 1 must be surrounded be a certain pattern of 0s,meaning that a 0 is always admissible.II. Quasi-stationary measuresA quasi stationary measure may be introduced by concate-nating arrays. Given a constraint, let W denote a stochasticvariable deﬁned on an n by m array, which may take on anyconﬁguration admissible by the constraint. Let X and Z de-note variables representing the ﬁrst and last M − 1 columns(with n elements). Let Y denote a variable representing themiddle m − 2M + 2 columns. We assume that the measureson the boundaries, X and Z are identical for the measures,W to be considered. Thus, starting with X0Y0Z0, arraysYiZi may repeatedly be added to form X0{YjZj}K0 , suchthat Zi−1YiZi has the same measure as W. The entropy(per symbol) is given by the conditional entropy of YiZi givenZi−1 which isHW (m)−HX(M − 1)m−M + 1 . (1)where HW (m) is the entropy of W (per row) and HX(M−1) isthe entropy of X (per row). A simple way to specify W in (1)is to assign probabilities to the bit-stuﬃng scheme below. Theboundaries X and Z are speciﬁed by identical but independentbit-stuﬃng schemes. The middle columns Y are speciﬁed bybit-stuﬃng conditional on the boundaries X and Z.III. Numerical resultsTwo examples with binary elements and constraint size N =M = 3 are considered. For the RLL(2,∞) constraint, anal-ysis of the modiﬁed bit-stuﬃng was carried out calcultatingcapacities, C for m = 12. The transition probabilities for anew row of W were determined by the products of (condi-tional) probabilities adressing and bit-stuﬃng the elements ofthe new line of X and Z before Y and using the same condi-tional probabilities for the corresponding elements of X andZ. Thus the prerequisites for (1) is satisﬁed. Let p1 denotethe probabilities of writing a 1 when this is admissible. Simplebit-stuﬃng writing an unbiased sequence with p1 = 1/2 gaveC = 0.388. Using a single biased sequence gave C = 0.437 foroptimal choice of p1. Finally the values of p1 may be chosenindependently for each column of X and Y. (The p1 valuesof Z are given by X.) This gave a best value of C = 0.44149,also providing a lower bound for the constraint. This is a fairimprovement on the lower bound of 0.4267 on the capacity of(diagonal) bit-stuﬃng in [2].Capacities were also calculated for applying the modiﬁedbit-stuﬃng scheme to the constraint given by a min. (1-norm)distance of 3 between 1s. The results obtained for m = 15were C = 0.276 when writing an unbiased sequence, C =0.344 for a single biased sequence and C = 0.3477 choosingdiﬀerent biased sequences for each column of X and Y. Forthis constraint the boundaries, Zi−1 and Zi must be at leastan additional row ahead in order to bit-stuﬀ the elements ofX and Z independently of past elements of Y. A more elab-orate scheme for specifying W in (1) was also devised. Theprobabilities p1 were made dependent on the other elementson the (N − 1 =) 2 previous rows. The next row of X (andZ) is speciﬁed by probabilities conditioned on the two pre-vious rows. The new row of Y is speciﬁed by probabilitiesconditioned on 3 rows of X and Z and 2 rows of Y. Theseconditional probabilities were obtained from the maxentropicsolution [3] for W (with two rows forming the states). Thisgave a capacity of C = 0.3497, which also provides a new lowerbound for the constraint.References[1] R.M. Roth, P.H. Siegel and J.K. Wolf, “Eﬃcient codingschemes for the hard-square model,” IEEE Trans. Inform. The-ory, vol. 47, pp. 1166-76, March, 2001.[2] S. Halevy, J. Chen, R.M. Roth, P.H. Siegel, J.K. Wolf, “Im-proved bit-stuﬃng bounds on two-dimensional constraints,”Proc. ISIT’02, Lausanne, Switzerland, p. 385, 2002.[3] S. Forchhammer and J. Justesen, “Entropy bounds for con-strained two-dimensional random ﬁelds,” IEEE Trans. Inform.Theory, vol. 45, no. 1, pp. 118–127, Jan. 1999.ISIT 2004, Chicago, USA, June 27 – July 2, 2004,(((Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on February 17,2010 at 07:05:18 EST from IEEE Xplore.  Restrictions apply. 

Analysis of bit-stuffing codes and lower bounds on capacity for 2-D constrained arrays using quasistationary measures

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Analysis of bit-stuffing codes and lower bounds on capacity for 2-D constrained arrays using quasistationary measures

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