Research on perfect binary correlation sequences based on higher-order cumulant

Yi Zhong,Zheng Zhou,Ting Jiang

doi:10.1186/1687-1499-2013-123

Abstract

Current researches define the perfect signal by the inner product of sequence itself and its delay sequence. However, these traditional sequences based on second-order statistics cannot handle colored Gaussian measurement noise automatically. Because higher-order cumulant (HOC) is insensitive to the adding Gaussian noise and symmetrical non-Gaussian noise, a new kind of perfect binary signal with good periodic correlation function based on HOC is presented. This paper proposes a new concept of perfect binary-third-order cyclic autocorrelation sequences (PBTOCAS) and defines the quasi-perfect binary-third-order cyclic autocorrelation sequences (QPBTOCAS) and the almost perfect binary-third-order cyclic autocorrelation sequences (APBTOCAS). Then the properties of these binary sequences are studied, and we theoretically prove that binary-third-order cyclic autocorrelation sequences can effectively suppress colored Gaussian noise. Finally, some QPBTOCAS and APBTOCAS with short lengths by computer searching are listed. From the observation of the PBTOCAS, we can see that it can well suit engineering applications, remedying the defect of the conventional pseudo-noise code used in very low signal-noise-ratio environments.

Highlights

Higher-order statistics (HOS) is a mathematical tool to describe the higher-order statistical properties of the random process, including higher-order cumulants and moments
Since high-order cumulants are blind to any kind of Gaussian process, this paper introduces higher-order cumulant (HOC) into the field of perfect binary sequence, which breaks through the limitations of the perfect signal defined by second-order statistics, to fill out the blank of perfect signal in the area of research on Gaussian noise suppression
We have not found the perfect binary-third-order cyclic autocorrelation sequences (PBTOCAS) of the definition in (3) at present, the realization of QPBTOCAS and APBTOCAS could be further constructed according to the conjecture

Summary

Introduction

Higher-order statistics (HOS) is a mathematical tool to describe the higher-order statistical properties of the random process, including higher-order cumulants and moments. Current researches define the perfect signal by the inner product of sequence itself and its delay sequence These traditional sequences based on second-order statistics cannot handle colored Gaussian measurement noise automatically, affecting the accuracy of their properties in engineering application. Definition 3 Suppose a sequence x(n) = (x0, x1, ⋯, xN 1) with length N and its imbalance satisfies I ∈ {−1, 0, 1}, x(n) is considered as a zero-mean or a closely approximate zero-mean stationary random process, the third-order cumulant of cyclic autocorrelation binary sequence x(n) is defined as c3xðτ; τ2Þ. Definition 6 Suppose a binary sequence x(n) = (x0, x1, ⋅ ⋅⋅, xN − 1) with length N and its imbalance satisfies I ∈ {−1, 0, 1}, x(n) is defined as the APBTOCAS if there exists a value τ1 = u(0 ≤ u ≤ N − 1) and satisfies the following equation:. We have similar definition to the QPBTOCAS and APBTOCAS

Transformation features

Some searching results of QPBTOCAS

Conclusions