An optimum SNS-to-binary conversion algorithm and pipelined field-programmable logic design

The Optimum Symmetrical Number System (OSNS) formulation is a direct consequence of the need to extract the maximum amount of information from a symmetrically folded waveform, and has found use in applications such as folding analog-to-digital converters and phase-sampled direction finding antenna architectures. One of the key problems in an OSNS hardware realization is recombining the OSNS symmetrical residues (S₁,S₂,...S₃) to determine the unknown incoming value. The symmetrical residues cannot be converted (e.g., using the Chinese Remainder Theorem) in a straightforward manner, since the integers within each modulus are ambiguous. This paper presents an OSNS-to-binary conversion algorithm for N=3 moduli of the form m₁=2^k+1, m₂=2^k, and m₃=2^k-1. The algorithm consists of three main steps: 1) conversion of the symmetrical residues into complete residues; 2) solving the resulting congruences in binary; and 3) determining the unknown incoming value. A B=14-bit pipelined field-programmable logic design (FPLD) using Fe=6 is also presented to illustrate the algorithm. The number of bits throughout the FPLD are quantified and an example calculation is worked out to numerically demonstrate the efficiency of the design