Moments of Autocorrelation Demerit Factors of Binary Sequences

Core Concepts

The paper analyzes the distribution of autocorrelation demerit factors of binary sequences, showing that the central moments are quasi-polynomial functions of the sequence length and that the standardized moments converge to those of the standard normal distribution as the sequence length tends to infinity.

Abstract

The paper focuses on the autocorrelation properties of binary sequences, which are doubly infinite and finitely supported. The autocorrelation demerit factor is a measure of the lowness of autocorrelation, defined as the sum of the squares of the autocorrelation values at all nonzero shifts, normalized by the squared Euclidean norm of the sequence.
The authors investigate the distribution of demerit factors of the set of 2^ℓ binary sequences of length ℓ, endowed with a uniform probability measure. Previous work had determined the mean, variance, skewness, and kurtosis of this distribution as a function of ℓ.
This paper extends those results in two ways:

It shows that every pth central moment of the demerit factor distribution is a quasi-polynomial function of ℓ with rational coefficients.
It determines the asymptotic behavior of the standardized moments as ℓ tends to infinity, showing that they converge to the same values as those of the standard normal distribution.

The key techniques used include:

Combinatorial analysis of partitions and assignments associated with binary sequences
Mobius inversion on the poset of partitions
Identification of special "separable" and "principal" partitions that dominate the asymptotic behavior
The paper provides a comprehensive understanding of the statistical properties of autocorrelation demerit factors of binary sequences.

Stats

The mean demerit factor is 1 - 1/ℓ.
The variance of the demerit factor is (16ℓ^3 - 60ℓ^2 + 56ℓ) / (3ℓ^4) for even ℓ, and (16ℓ^3 - 60ℓ^2 + 56ℓ - 12) / (3ℓ^4) for odd ℓ.
The skewness of the demerit factor is a quasi-polynomial function of ℓ with rational coefficients.

Quotes

"An aperiodic binary sequence of length ℓ is a doubly infinite sequence f = . . . , f−1, f0, f1, . . . with fj ∈{−1, 1} when 0 ≤j < ℓ and and fj = 0 otherwise."
"The demerit factor of f is the sum of the squares of the autocorrelations at all nonzero shifts for the sequence obtained by normalizing f to unit Euclidean norm."
"We endow the 2^ℓ binary sequences of length ℓ with uniform probability measure and consider the distribution of their demerit factors."

Key Insights Distilled From

Limiting Moments of Autocorrelation Demerit Factors of Binary Sequences

by Daniel J. Ka... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2307.14566.pdf

Deeper Inquiries

How could the insights from this analysis of binary sequence autocorrelation be applied to improve the design of communication systems or other applications that rely on binary sequences

The insights gained from analyzing the autocorrelation properties of binary sequences can be directly applied to enhance the design of communication systems. By focusing on reducing the demerit factor of binary sequences, which indicates low self-similarity under translation, communication systems can be optimized for better synchronization and timing accuracy.
For instance, in wireless communication systems, where accurate timing is crucial for signal reception and decoding, utilizing binary sequences with low autocorrelation demerit factors can improve the overall performance. By selecting sequences that exhibit minimal self-similarity under translation, the systems can achieve better synchronization, leading to enhanced data transmission reliability and efficiency.
Furthermore, in radar and sonar applications, where precise timing is essential for target detection and tracking, employing binary sequences with low autocorrelation demerit factors can improve the accuracy and resolution of the measurements. This can result in more reliable identification of targets and better overall system performance.

What other statistical properties or applications of binary sequences could be explored using similar combinatorial and asymptotic techniques

The combinatorial and asymptotic techniques used in analyzing binary sequence autocorrelation properties can be extended to explore various other statistical properties and applications of binary sequences. Some potential areas of exploration include:

Spectral Analysis: Investigating the frequency domain characteristics of binary sequences through techniques such as Fourier analysis to understand their spectral properties and frequency components.

Error-Correcting Codes: Studying the relationship between binary sequences and error-correcting codes to develop efficient coding schemes for data transmission and storage applications.

Pattern Recognition: Exploring the use of binary sequences in pattern recognition algorithms and machine learning models to identify patterns and trends in data sets.

Cryptography: Analyzing the randomness and unpredictability of binary sequences to assess their suitability for cryptographic applications, such as generating secure encryption keys.

By applying similar combinatorial and asymptotic techniques to these areas, researchers can gain valuable insights into the statistical properties and potential applications of binary sequences in diverse fields.

Are there any connections between the asymptotic behavior of the demerit factor moments and the underlying mathematical structure of binary sequences or their autocorrelation properties

The asymptotic behavior of the demerit factor moments reveals important connections to the underlying mathematical structure of binary sequences and their autocorrelation properties.

Structural Properties: The dominance of principal partitions in contributing to the moments suggests that certain structural characteristics of binary sequences play a significant role in determining their autocorrelation properties. Understanding these structural properties can provide insights into the fundamental nature of binary sequences.

Autocorrelation Patterns: The relationship between the demerit factor moments and the autocorrelation patterns of binary sequences indicates a deep connection between statistical measures and signal processing characteristics. By studying the asymptotic behavior, researchers can uncover hidden patterns and correlations within binary sequences.

Algorithmic Design: The insights gained from analyzing the demerit factor moments can inform the development of algorithms for generating binary sequences with specific autocorrelation properties. This can lead to the creation of optimized sequences for various applications in communications, signal processing, and cryptography.

Overall, the asymptotic analysis of demerit factor moments provides a deeper understanding of the mathematical foundations of binary sequences and their implications for practical applications.