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.

The robust symmetrical number system (RSNS) is a number theoretic transform formed using N ≥ 2 integer sequences and ensures that two successive RSNS vectors (paired terms from all N sequences) differ by only one integer - integer Gray code property. The dynamic range M of the RSNS is defined as the greatest length of combined sequences that contain no ambiguities or repeated paired terms. For all but a select few RSNS sequences there is no closed-form solution to compute the dynamic range and its position. This paper presents an efficient algorithm for computing the dynamic range and its position. The dynamic range is shown to satisfy Mfwhere Pfis the RSNS fundamental period Pf= 2N Πmi. It then follows that Miis the dynamic range of the residue number system. An example is presented to demonstrate the algorithm. The efficiency of the algorithm is examined by comparing the speed of computation to a naive search algorithm (using MATLAB on a PC).

The robust symmetrical number system (RSNS) is a modular system formed using$N \geq 2$integer sequences and ensures that two successive RSNS vectors (paired terms from all$N$sequences) differ by only one integer. This integer Gray-code property reduces the possibility of encoding errors and makes the RSNS useful in applications such as folding analog-to-digital converters (ADCs), direction finding antenna architectures, and photonic processors. This paper determines the length of combined sequences that contain no vector ambiguities. This length or longest run of distinct vectors we call the RSNS dynamic range$(\mathhat{M})$. The position of$\mathhat{M}$which is the starting point in the sequence is also derived. Computing$\mathhat{M}$and the position of$\mathhat{M}$allows the integer Gray-code properties of the RSNS to be used in practical applications. We first extend our two-sequence results to develop a closed-form expression for$\mathhat{M}$for a three-sequence RSNS with moduli of the form$2^r -1, 2^r, 2^r+1$. We then extend the results to solving the$N$-sequence RSNS ambiguity locations in general.

Without dispute, symmetrical folding waveforms are the most common type of waveform in engineering science (e.g., cosine, sine). symmetrical number systems (SNS) have been formed to extract the maximum amount of information from symmetrically folded waveforms. The robust symmetrical number system (RSNS) is formed using N ges 2 sequences and ensures that two successive RSNS vectors differ by only one digit. This Gray-code property reduces the possibility of encoding errors and makes the RSNS useful in applications such as folding analog-to-digital converters, direction finding antenna architectures and photonic processors. This paper determines the length of combined sequences that contain no ambiguities (Mcirc) which we call the RSNS dynamic range. The position of M is also derived. We first extend our two-sequence results to develop Mcirc for a three-sequence RSNS with moduli of the form 2r- 1, 2r, 2r+ 1. We then extend the results to solving the N-channel RSNS redundancy locations in general