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Over-the-Air Majority Vote Computation with Modulation on Conjugate-Reciprocal Zeros

Alphan Sahin

TL;DR

This study proposes a new approach to compute the majority vote (MV) function based on modulation on conjugate-reciprocal zeros (MOCZ) and introduces three different methods, which provide robustness against phase and time synchronization errors and theoretically analyze the CERs.

Abstract

In this study, we propose a new approach to compute the majority vote (MV) function based on modulation on conjugate-reciprocal zeros (MOCZ) and introduce three different methods. In these methods, each transmitter maps the votes to the zeros of a Huffman polynomial, and the corresponding polynomial coefficients are transmitted. The receiver evaluates the polynomial constructed by the elements of the superposed sequence at conjugate-reciprocal zero pairs and detects the MV with a direct zero-testing (DiZeT) decoder. With differential and index-based encoders, we eliminate the need for power-delay information at the receiver while improving the computation error rate (CER) performance. The proposed methods do not use instantaneous channel state information at the transmitters and receiver. Thus, they provide robustness against phase and time synchronization errors. We theoretically analyze the CERs of the proposed methods. Finally, we demonstrate their efficacy in a distributed median computation scenario.

Over-the-Air Majority Vote Computation with Modulation on Conjugate-Reciprocal Zeros

TL;DR

This study proposes a new approach to compute the majority vote (MV) function based on modulation on conjugate-reciprocal zeros (MOCZ) and introduces three different methods, which provide robustness against phase and time synchronization errors and theoretically analyze the CERs.

Abstract

In this study, we propose a new approach to compute the majority vote (MV) function based on modulation on conjugate-reciprocal zeros (MOCZ) and introduce three different methods. In these methods, each transmitter maps the votes to the zeros of a Huffman polynomial, and the corresponding polynomial coefficients are transmitted. The receiver evaluates the polynomial constructed by the elements of the superposed sequence at conjugate-reciprocal zero pairs and detects the MV with a direct zero-testing (DiZeT) decoder. With differential and index-based encoders, we eliminate the need for power-delay information at the receiver while improving the computation error rate (CER) performance. The proposed methods do not use instantaneous channel state information at the transmitters and receiver. Thus, they provide robustness against phase and time synchronization errors. We theoretically analyze the CERs of the proposed methods. Finally, we demonstrate their efficacy in a distributed median computation scenario.
Paper Structure (17 sections, 9 theorems, 57 equations, 7 figures, 1 table)

This paper contains 17 sections, 9 theorems, 57 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Model
  3. Preliminaries on Huffman Sequences
  4. Problem Statement
  5. Proposed Methods
  6. Method 1: Uncoded MV Computation
  7. Method 2: Differential MV Computation
  8. Method 3: Index-based MV Computation
  9. Complexity Analysis and Numerical Stability
  10. Computation-error rate Analysis
  11. Numerical Results
  12. Concluding Remarks
  13. Proof of Lemma \ref{['lemma:exp']}
  14. Proof of Lemma \ref{['lemma:exp2']}
  15. Proof of Lemma \ref{['lemma:exp3']}
  16. ...and 2 more sections

Key Result

Lemma 1

Let $U^{+}_{\ell}$ and $U^{-}_{\ell}$ denote the number of transmitters with positive and negative votes for the $\ell$th MV computation. For the mapping in eq:schemeOne with $\mathrm{Pr}({v^{(u)}_{\ell'}=1})=\mathrm{Pr}({v^{(u)}_{\ell'}=-1})=1/2$, $\forall\ell'$, $\ell'\neq\ell$, where where the expectation in eq:expectedPlus is over the distributions of channels and votes.

Figures (7)

  • Figure 1: Example zero placements for Methods 1-3. The star and circle markers indicate the chosen zeros and the possible zero locations for a Huffman polynomial, respectively.
  • Figure 2: Transmitter and receiver block diagrams.
  • Figure 3: CER for a given $U^{+}_{\ell}$ ($U=25$, ${\tt{SNR}}=10$ dB).
  • Figure 4: CER for a given SNR ($U=25$, $U^{+}_{\ell}=22$, $K=16$).
  • Figure 5: PMEPR distribution.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 1: Huffman sequence and Huffman polynomial
  • Lemma 1
  • Corollary 1
  • Lemma 2
  • Corollary 2
  • Lemma 3
  • Lemma 4: sahin2023reliable
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • ...and 5 more