A Deliberate Bit Flipping Coding Scheme for Data-Dependent Two-Dimensional Channels
Summary (3 min read)
Introduction
- 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 [15], [16].
- 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 [11], “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 [39] and [47], 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 [40] and [10] 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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"A Deliberate Bit Flipping Coding Sc..." refers background in this paper
...RECENT advances in magnetic recording systems [3], [4], optical recording devices [5] and flash memory drives [6] necessitate to study two-dimensional (2-D) coding techniques for reliable storage/retrieval of user data....
[...]