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General BER Expression for One-Dimensional Constellations

Mikhail Ivanov, Fredrik Brännström, Alex Alvarado, Erik Agrell

TL;DR

The paper addresses uncoded BER analysis for one-dimensional constellations, notably M-PAM, by introducing a general ready-to-use BER expression derived under the ABD demodulator and the equivalence of ABD with SD for BER. It develops a universal PBER formula in terms of bit-patterns and decision thresholds, and shows that pattern sets can be grouped into a finite number of BER classes, with a closed-form count Q for these classes in M-PAM. For equally spaced M-PAM, the BER can be written as P_C = (1/(mM)) Σ_{n=1}^{M−1} α_n Q((2n−1)d√(2ρ)), where α encodes pattern-dependent weights linked to the differential distance spectrum; high-SNR behavior collapses to M−1 groups determined by a1. The work provides practical tools for evaluating labeling choices and their BER impact, and identifies future directions to generalize the approach to arbitrary constellations and to formulate a unifying rule for combining patterns into labelings.

Abstract

A novel general ready-to-use bit-error rate (BER) expression for one-dimensional constellations is developed. The BER analysis is performed for bit patterns that form a labeling. The number of patterns for equally spaced M-PAM constellations with different BER is analyzed.

General BER Expression for One-Dimensional Constellations

TL;DR

The paper addresses uncoded BER analysis for one-dimensional constellations, notably M-PAM, by introducing a general ready-to-use BER expression derived under the ABD demodulator and the equivalence of ABD with SD for BER. It develops a universal PBER formula in terms of bit-patterns and decision thresholds, and shows that pattern sets can be grouped into a finite number of BER classes, with a closed-form count Q for these classes in M-PAM. For equally spaced M-PAM, the BER can be written as P_C = (1/(mM)) Σ_{n=1}^{M−1} α_n Q((2n−1)d√(2ρ)), where α encodes pattern-dependent weights linked to the differential distance spectrum; high-SNR behavior collapses to M−1 groups determined by a1. The work provides practical tools for evaluating labeling choices and their BER impact, and identifies future directions to generalize the approach to arbitrary constellations and to formulate a unifying rule for combining patterns into labelings.

Abstract

A novel general ready-to-use bit-error rate (BER) expression for one-dimensional constellations is developed. The BER analysis is performed for bit patterns that form a labeling. The number of patterns for equally spaced M-PAM constellations with different BER is analyzed.

Paper Structure

This paper contains 8 sections, 3 theorems, 20 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction and Motivation
  2. Preliminaries
  3. Notation Convention
  4. System Model
  5. Demodulators
  6. BER for One-Dimensional Constellations
  7. BER for $M$-PAM
  8. Conclusions

Key Result

Theorem 1

For any ${\rho}$, $\mathcal{X}$, and $\mathbb{C}$, $\hat{b}_j^{\mathrm{SD}} = \hat{b}_j^{\mathrm{ABD}}$ for all $j=1,\dots,m$.

Figures (4)

  • Figure 1: Thresholds for 8-PAM with different patterns vs. SNR. Due to the symmetry of the patterns the thresholds are symmetric with respect to zero. Only positive thresholds are shown. Squares represent the constellation points.
  • Figure 2: The BER for 8-PAM with patterns $\boldsymbol{p}_{15}$, $\boldsymbol{p}_{60}$, $\boldsymbol{p}_{102}$, and the BRGC. Solid lines correspond to the BD and dashed lines correspond to the ABD.
  • Figure 3: The PBER for the patterns for 8-PAM and 16-PAM. All the curves merge into $M-1$ groups at high SNR as predicted by Remark \ref{['rem:high_snr_pattern']}.
  • Figure 4: The BER for all the 460 labelings with different BER for 8-PAM.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Remark 2
  • Remark 3