Extended Closed-Form Expressions for the Robust Symmetrical Number System Dynamic Range and an Efficient Algorithm for Its Computation

• Published in: IEEE transactions on information theory Vol. 60; no. 3; pp. 1742 - 1752
• Main Authors: Pace, Phillip E., Stanica, Pantelimon, Luke, Brian L., Tedesso, Thomas W.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2014; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Applied sciences; Arrays; Closed-form solutions; Coding, codes; Complexity theory; congruences; Detection, estimation, filtering, equalization, prediction; Dynamic range; dynamic range algorithm; Equations; Exact sciences and technology; Information theory; Information, signal and communications theory; Integer Gray code; Numbers; Radiolocalization and radionavigation; robust symmetrical number systems (RSNS); Signal and communications theory; Signal, noise; Symmetry; Telecommunications; Telecommunications and information theory; Upper bound; Vectors; Dynamic response; Parameter estimation; Direction-of-arrival estimation; Gray code; dynamic range algorithm; Signal estimation; Radiolocalisation; Algorithm; Integer Gray code; AD converter; Algorithm performance; Radar; robust symmetrical number systems (RSNS); Error detection; congruences; Error correction; Localization; Number theory
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2014.2301820

The robust symmetrical number system (RSNS) is a number theoretic transform based on N ≥ 2 sequences that can extract the maximum amount of information from symmetrical folding waveforms. The sequences, based on coprime moduli, exhibit an integer gray code property making the RSNS well suited for many applications that benefit from an inherent error detection and correction capability, such as analog-to-digital converters, direction finding arrays, and radar waveform design. To use the RSNS, it is necessary to know the greatest length of combined sequences without ambiguities, called the dynamic range M̂, for which only a few closed-form expressions currently exist. In this paper, an efficient algorithm for computing M and its position within the combined set of sequences is presented and shown to be independent of the size of the moduli. The algorithm is used to generate the equations for several groups of additional moduli arrangements. Closed-form expressions for M are conjectured and proved using the obtained congruence equations that define the ambiguity locations.