Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characterization of Blind Code Rate Recovery in Linear Block Codes

Atreya Vedantam, Radha Krishna Ganti

TL;DR

A metric is presented to characterize the quality of the code rate recovery which uses any rank based estimation technique and an expression is derived for a better code rate estimate in high noise conditions and compared with existing estimates.

Abstract

Forward Error Correction (FEC) is used ubiquitously in the communication pipeline. We explore noncooperative decoding where we aim to recover the code rate of a linear block code. We present a metric to characterize the quality of the code rate recovery which uses any rank based estimation technique. We derive a closed form expression for this metric in terms of the algorithmic and the environmental parameters and assert that it should be low for good recovery. We use this metric to derive an expression for a better code rate estimate in high noise conditions and compare it with existing estimates. Finally we validate the derived expression for the metric and the improved performance in the code rate estimate by simulating the recovery of a Low Density Parity Check (LDPC) code. This also enables us to derive the optimal algorithmic parameters for recovery.

Characterization of Blind Code Rate Recovery in Linear Block Codes

TL;DR

A metric is presented to characterize the quality of the code rate recovery which uses any rank based estimation technique and an expression is derived for a better code rate estimate in high noise conditions and compared with existing estimates.

Abstract

Forward Error Correction (FEC) is used ubiquitously in the communication pipeline. We explore noncooperative decoding where we aim to recover the code rate of a linear block code. We present a metric to characterize the quality of the code rate recovery which uses any rank based estimation technique. We derive a closed form expression for this metric in terms of the algorithmic and the environmental parameters and assert that it should be low for good recovery. We use this metric to derive an expression for a better code rate estimate in high noise conditions and compare it with existing estimates. Finally we validate the derived expression for the metric and the improved performance in the code rate estimate by simulating the recovery of a Low Density Parity Check (LDPC) code. This also enables us to derive the optimal algorithmic parameters for recovery.
Paper Structure (8 sections, 2 theorems, 32 equations, 5 figures)

This paper contains 8 sections, 2 theorems, 32 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical Model
  3. Algorithm
  4. Analysis
  5. Simulation Results
  6. Conclusion
  7. Proof of Theorem 1
  8. Proof of Theorem 2

Key Result

Theorem 1

Suppose $p, m_1, m_2, \ldots, m_d \in \{0, 1\}^{M_s}$ are columns of the word matrix such that $p = \bigoplus_{i=1}^d m_i$. Let $p_e'$ be the probability of a bit error in the code word matrix. Suppose $M_s >> d$ and $p_e'$ was small such that $M_sp_e'$ is large. Then the probability that the rank i

Figures (5)

  • Figure 1: Contour plot of the algorithmic error versus $t_1$ and $t_2$ for a code of length $n = 136$ received at an SNR of $10$ dB. $t_1^* = 0.99$ and $t_2^* = 1$ and the minimum $f(t_1^*, t_2^*) = 0.0153$. Black lines represent constant $M$ curves (see equation \ref{['contours']}).
  • Figure 2: Theoretical and Simulated $\mathbb{E}[C]$ for three cases: $t_1 = 0.3, t_2=n/2$, $t_1 = 0.3, t_2 = n/8$, $t_1 = 0.8, t_2 = n/2$ for $1,000$ messages of a $544$ length LDPC code from $5$ to $20$ dB SNR. The close match demonstrates the correctness of the theoretical result over a wide range of $t_1$ and $t_2$.
  • Figure 3: Error between the theoretical $\mathbb{E}[C]$ and observed number of columns in error normalized to the code length ($544$). The error goes to $0$ for high SNR.
  • Figure 4: The naive code rate estimate $k'/n$ (solid lines) and improved estimate $(k'-\mathbb{E}[C])/(n - \mathbb{E}[C])$ (dashed lines) versus SNR for three different cases of $t_1$ and $t_2$. Simulation is done for $1000$ messages of a $544$ length LDPC code with a true code rate of $0.3235$.
  • Figure 5: The naive code rate estimate $k'/n$ (solid lines) and improved estimate $(k'-\mathbb{E}[C])/(n - \mathbb{E}[C])$ (dashed lines) versus SNR for three different cases of $t_1$ and $t_2$. Simulation is done for $2000$ messages of a $1088$ length LDPC code with a true code rate of $0.3235$.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • proof
  • proof