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Generating gaussian pseudorandom noise with binary sequences

Francisco-Javier Soto, Ana I. Gómez, Domingo Gómez-Pérez

TL;DR

This work addresses the hardware-efficient generation of Gaussian random numbers by leveraging the Central Limit Theorem on sums of pseudorandom binary sequences. It derives explicit bounds that link the output moments to the input sequences' combined correlation measures $\theta_k(s,N)$ and formalizes a block-sum construction $S(i)=M^{-1/2}\sum_{n=1}^{M}s(i+Mn)$ to attenuate term dependence. The analysis shows Gold codes with a Mersenne-prime period attenuate higher-order correlation peaks (up to $k=4$) better than $m$-sequences, which is corroborated by computational experiments comparing moment statistics and histograms. The findings support practical, low-resource GRNG designs and provide guidance on code choice and parameter tuning for reliable Gaussian approximations in hardware-constrained settings.

Abstract

Gaussian random number generators attract a widespread interest due to their applications in several fields. Important requirements include easy implementation, tail accuracy, and, finally, a flat spectrum. In this work, we study the applicability of uniform pseudorandom binary generators in combination with the Central Limit Theorem to propose an easy to implement, efficient and flexible algorithm that leverages the properties of the pseudorandom binary generator used as an input, specially with respect to the correlation measure of higher order, to guarantee the quality of the generated samples. Our main result provides a relationship between the pseudorandomness of the input and the statistical moments of the output. We propose a design based on the combination of pseudonoise sequences commonly used on wireless communications with known hardware implementation, which can generate sequences with guaranteed statistical distribution properties sufficient for many real life applications and simple machinery. Initial computer simulations on this construction show promising results in the quality of the output and the computational resources in terms of required memory and complexity.

Generating gaussian pseudorandom noise with binary sequences

TL;DR

This work addresses the hardware-efficient generation of Gaussian random numbers by leveraging the Central Limit Theorem on sums of pseudorandom binary sequences. It derives explicit bounds that link the output moments to the input sequences' combined correlation measures and formalizes a block-sum construction to attenuate term dependence. The analysis shows Gold codes with a Mersenne-prime period attenuate higher-order correlation peaks (up to ) better than -sequences, which is corroborated by computational experiments comparing moment statistics and histograms. The findings support practical, low-resource GRNG designs and provide guidance on code choice and parameter tuning for reliable Gaussian approximations in hardware-constrained settings.

Abstract

Gaussian random number generators attract a widespread interest due to their applications in several fields. Important requirements include easy implementation, tail accuracy, and, finally, a flat spectrum. In this work, we study the applicability of uniform pseudorandom binary generators in combination with the Central Limit Theorem to propose an easy to implement, efficient and flexible algorithm that leverages the properties of the pseudorandom binary generator used as an input, specially with respect to the correlation measure of higher order, to guarantee the quality of the generated samples. Our main result provides a relationship between the pseudorandomness of the input and the statistical moments of the output. We propose a design based on the combination of pseudonoise sequences commonly used on wireless communications with known hardware implementation, which can generate sequences with guaranteed statistical distribution properties sufficient for many real life applications and simple machinery. Initial computer simulations on this construction show promising results in the quality of the output and the computational resources in terms of required memory and complexity.
Paper Structure (5 sections, 3 theorems, 16 equations, 2 figures, 1 table)

This paper contains 5 sections, 3 theorems, 16 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Gaussian Random Number Generation from Binary Sequences
  3. Correlation properties of Gold codes
  4. Computational Experiments
  5. Conclusions and Future Work

Key Result

Theorem 1

Let $s(i)$ be a binary sequences of period $N$, $M$ a positive integer with $M \ll N$, and $k$ a non-negative integer. Then the following holds, where if $k$ is odd, the first term in the right of the inequality disappears.

Figures (2)

  • Figure 1: Comparison of histograms with 100 bins generated by GRNGs using the $m$-sequence (left) and the Gold code (right). The histograms for binary sequence model are shown in the top row, while the Tauworthe model is shown in the bottom row.
  • Figure 2: Comparison of the triple product moments (absolute value) generated by GRNGs using the $m$-sequence (left) and the Gold code (right). The calculations for the binary sequence model are shown in the top row, while the Tauworthe model is shown in the bottom row.

Theorems & Definitions (8)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 2
  • Theorem 3
  • Definition 3