A Deliberate Bit Flipping Coding Scheme for Data-Dependent Two-Dimensional Channels
TL;DR: A deliberate bit flipping (DBF) coding scheme for binary two-dimensional channels, where specific patterns in channel inputs are the significant cause of errors, and devise a constrained combinatorial formulation for minimizing the number of flipped bits for a given set of harmful patterns.
Abstract: In this paper, we present a deliberate bit flipping (DBF) coding scheme for binary two-dimensional (2-D) channels, where specific patterns in channel inputs are the significant cause of errors. The idea is to eliminate a constrained encoder and, instead, embed a constraint into an error correction codeword that is arranged into a 2-D array by deliberately flipping the bits that violate the constraint. The DBF method relies on the error correction capability of the code being used so that it should be able to correct both deliberate errors and channel errors. Therefore, it is crucial to flip minimum number of bits in order not to overburden the error correction decoder. We devise a constrained combinatorial formulation for minimizing the number of flipped bits for a given set of harmful patterns. The generalized belief propagation algorithm is used to find an approximate solution for the problem. We evaluate the performance gain of our proposed approach on a data-dependent 2-D channel, where 2-D isolated-bits patterns are the harmful patterns for the channel. Furthermore, the performance of the DBF method is compared with classical 2-D constrained coding schemes for the 2-D no isolated-bits constraint on a memoryless binary symmetric channel.
Summary (3 min read)
- In principle, the ultimate coding approach for such datadependent channels is to design a set of sufficiently distinct error correction codewords that also satisfy channel constraints , .
- The theory of 1-D constrained coding is mature as well as practical aspects of 1-D code and decoder design.
- (a) (b) Fig. Paper Organization: Section II presents the notations and definitions used throughout the paper.
- In Section IV, the problem of minimizing the number of flipped bits in the DBF method is formulated.
II. NOTATIONS AND DEFINITIONS
- The authors denote a discrete random variable with an upper case letter (e.g., X) and its realization by the lower case letter (e.g., x).
- Throughout the paper, white squares denote zero bits and black squares represent 1. Consider a k-ominoe P and the set of all 2k binary configurations of that shape XP .
III. CHANNEL MODEL
- The authors call this set of P-shaped configurations the set of harmful configurations.
- Fig. 4. 2-D isolated-bits patterns containing the bit xi,j .
- The authors use the concept of polyominoes to just demonstrate the effect of harmful configurations on its neighboring bits over a 2-D binary pattern, also known as Remark 1.
- The probability that the channel is in the bad state (or, in the good state) depends on the input probability distribution, also known as Remark 3.
- The authors assume that the set of harmful patterns for the channel is the set of 2-D isolated-bits patterns, which are given in Fig.
IV. PROBLEM FORMULATION
- The authors want to send the pattern x over the communication channel in Section III, with the list of harmful configurations XBPi,j .
- The theory of constrained coding began with Claude Shannon’s classical 1948 paper , “A Mathematical Theory of Communications.”, also known as Remark 4.
- Finding the error pattern which removes a given set of 2-D configurations from a 2-D pattern and has the minimum Hamming weight via an exhaustive search among all admissible error patterns can be computationally prohibitive for large patterns, also known as Remark 5.
V. A PROBABILISTIC GRAPHICAL FORMULTION FOR MINIMZING BIT FLIPS
- The authors devise a probabilistic graphical formulation for the problem of minimizing the number of bit flips in the DBF method.
- In the following, the authors present a probabilistic formulation using a graphical model to find approximate solution for this problem using the GBP algorithm.
- For each bit xi,j ∈ Am,n, the distortion now is defined as the probability of having a distorted pattern xPi,j which has the Hamming distance wH(x̂Pi,j⊕xPi,j ) with x̂Pi,j 6∈ XBPi,j .
- In  and , it is shown that the region-based approximation (RBA) method provides an approximate solution for the partition function by minimizing the region-based free energy (as an approximation to the variational free energy).
- The authors first define a factor graph representation for the problem (maximizing p (x̂|x) in (30) for a given input pattern x subject to the constraint that x̂ ∈ S) and then formulate the RBA scheme for finding an approximate solution for this constrained maximization problem.
VI. NUMERICAL RESULTS
- The authors present numerical analyses of the GBPbased DBF method for removing harmful patterns.
- Without loss of generality, the authors focus on the 2-D isolated-bits configurations in all their experiments.
- The authors first present the analysis on statistics of the number of flipped bits for removing 2-D isolated-bits patterns from random 2-D patterns.
- To illustrate the usefulness of DBF method, the authors investigate its performance over the data-dependent channel in Section III under different scenarios in terms of the probability of uncorrectable bit errors, where the harmful configurations for the channel are the 2-D isolated-bits patterns.
- Finally, the authors compare the performance of the DBF method on a memoryless BSC with the row-by-row and bit-stuffing constrained coding schemes for the 2-D n.i.b. constraint, presented in  and  respectively.
A. Statistics of The Number of Bit Flips for Removing 2-D Isolated-Bits Patterns
- The performance of the DBF method relies on the error correction capability of the code being used, and of course the number of deliberate bit errors.
- Therefore, it is necessary to find how many bits in average are flipped within a codeword, and how this number compares to the error correction capability of the code.
- The authors have extracted the statistics of the number of bit flips for removing 2-D isolated-bits patterns from random 2-D patterns by the DBF method.
- Using the flipping probabilities in Fig. 8 and (32), the UBER is calculated for BCH codes of length 1024 with different rates (and consequently dmin).
- The choice of λ in the probabilistic formulation of problem, (28), depends on the constraint and the underlying method for solving the minimization problem.
B. Performance Evaluation of The GBP-Guided DBF Method
- The authors investigate the usefulness of DBF method for data-dependent 2-D channels, where specific patterns in channel inputs are the main cause of errors.
- For different values of αb and αg , the authors compare the average probability of error with and without incorporating the DBF method.
- Prior to transmission over the channel, the 2-D isolatedbits patterns are removed from the input pattern by flipping minimum number of bits.
- The transmitted pattern and channel output without DBF are x(m) and x(m)⊕ êCH, respectively.
C. Comparison Results on BSC
- The authors compare the proposed scheme of imposing the 2-D n.i.b. constraint by deliberate errors against the row-by-row and the bit-stuffing coding schemes on a BSC.
- The encoder first generates two sequences with different statistics, Bernoulli(1/2) and Bernoulli(1/3), from the sequence of information bits using a probability transformer.
- The redundancy for imposing the constraint is now used in their scheme to strengthen the ECC (BCH code), resulting in a gain over the other schemes.
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