Two-dimensional patterns with distinct differences; constructions, bounds, and maximal anticodes

A two-dimensional (2-D) grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various 2-D shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid

Folding, Tiling, and Multidimensional Coding

Folding a sequence $S$ into a multidimensional box is a method that is used to construct multidimensional codes. The well known operation of folding is generalized in a way that the sequence $S$ can be folded into various shapes. The new definition of folding is based on lattice tiling and a direction in the $D$-dimensional grid. There are potentially $\frac{3^D-1}{2}$ different folding operations. Necessary and sufficient conditions that a lattice combined with a direction define a folding are given. The immediate and most impressive application is some new lower bounds on the number of dots in two-dimensional synchronization patterns. This can be also generalized for multidimensional synchronization patterns. We show how folding can be used to construct multidimensional error-correcting codes and to generate multidimensional pseudo-random arrays

Manx Arrays: Perfect Non-Redundant Interferometric Geometries

Interferometry applications (e.g., radio astronomy) often wish to optimize the placement of the interferometric elements. One such optimal criterion is a uniform distribution of non-redundant element spacings (in both distance and position angle). While large systems, with many elements, can rely on saturating the sample space, and disregard “wasted” sampling, small arrays with only a few elements are more critical, where a single element can represent a significant fraction of the overall cost. This paper defines a “perfect array” as a mathematical construct having uniform and complete element spacings within a circle of radius equal to the maximum element spacing. Additionally, the largest perfect non-redundant array, comprising six elements, is presented. The geometry is described, along with the properties of the layout and situations where it would be of significant benefit to array application and non-redundant masking designs

Manx Arrays: Perfect Non-Redundant Interferometric Geometries

Interferometry applications (e.g., radio astronomy) often wish to optimize the placement of the interferometric elements. One such optimal criterion is a uniform distribution of non-redundant element spacings (in both distance and position angle). While large systems, with many elements, can rely on saturating the sample space, and disregard “wasted” sampling, small arrays with only a few elements are more critical, where a single element can represent a significant fraction of the overall cost. This paper defines a “perfect array” as a mathematical construct having uniform and complete element spacings within a circle of radius equal to the maximum element spacing. Additionally, the largest perfect non-redundant array, comprising six elements, is presented. The geometry is described, along with the properties of the layout and situations where it would be of significant benefit to array application and non-redundant masking designs.</p

Thinned coprime arrays for DOA estimation

Sparse arrays can generate a larger aperture than traditional uniform linear arrays (ULA) and offer enhanced degrees-of-freedom (DOFs) which can be exploited in both beamforming and direction-of-arrival (DOA) estimation. One class of sparse arrays is the coprime array, composed of two uniform linear subarrays which yield an effective difference co-array with higher number of DOFs. In this work, we present a new coprime array structure termed thinned coprime array (TCA), which exploits the redundancy in the structure of the existing coprime array and achieves the same virtual aperture and DOFs as the conventional coprime array with much fewer number of sensors. An analysis of the DOFs provided by the new structure in comparison with other sparse arrays is provided and simulation results for DOA estimation using the compressive sensing based method are provided