On the Synthesis of Two-Dimensional Arrays with Desirable Correlation Properties

Considerable effort has been devoted in the literature to the synthesis of one-dimensional, periodic, binary and nonbinary sequences having small values for their out-of-phase autocorrelation functions. This paper considers the synthesis of two-dimensional, periodic, binary and nonbinary sequences (arrays) which exhibit similar properties for their two-dimensional autocorrelation functions. These arrays may have future application in the areas of optical signal processing, pattern recognition, etc. Various procedures are presented for the synthesis of such arrays. Two perfect binary arrays and an infinite class of perfect nonbinary arrays are given. A class of binary arrays is presented which are the two-dimensional analog of the quadratic residue sequences and are shown to have out-of-phase autocorrelation of −1, or to alternate between +1 and −3. The perfect maps of Gordon are shown to have all values of out-of-phase autocorrelation equal to −1. Other methods of constructing arrays based upon good one-dimensional sequences are also discussed. Synthesis procedures are given for constructing pairs of arrays such that their cross-correlation is identically zero for all shifts and, in addition, individually have good autocorrelation functions. Examples are given for the various synthesis procedures.