$N$-Sequence RSNS Ambiguity Analysis

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.