Large Families of “Grey” Arrays with Perfect Auto-correlation and Optimal Cross-Correlation

Abstract

Large sets of distinct 2D arrays of variable size that possess both strong auto-correlation and weak cross-correlation properties are highly valuable in many imaging and communications applications. We use the discrete Finite Radon Transform to construct $p \times p$ arrays with “perfect” correlation properties, for any prime p. Array elements are restricted to the integers $\{0,\pm 1, +2\}$. Each array exhibits perfect periodic auto-correlation, having peak correlation value $p^2$, with all off-peak values being exactly zero. Each array contains just $3(p-1)/2$ zero elements, the minimum number possible using this alphabet. Large families with size $M = p^2-1$ of such arrays can be constructed. Each of the $M(M-1)/2$ intra-family periodic cross-correlations is guaranteed to have one of the three lowest possible merit factors. These family size M can be extended to multiples of $p^2-1$ if we permit more than the three lowest cross-correlation levels.