This paper presents a general architecture for designing efficient reverse converters based on the moduli set {2α, 22β+1-1, 2β-1}, where β < α ≤ 2β, by using a parallel implementation of mixed-radix conversion (MRC) algorithm. The moduli set {2α, 22β+1-1, 2β-1} is free from modulo (2k+1)-type which can result in an efficient arithmetic unit for residue number system (RNS). The values of α and β can be selected to provide the required dynamic range (DR) and also to adjust the desired equilibrium between moduli bit-width. The simple multiplicative inverses of the proposed moduli set and also using novel techniques to simplify conversion equations lead to a low-complexity and high-performance general reverse converter architecture that can be used to support different DRs. Moreover, due to the current importance of the 5n-bit DR moduli sets, we also introduced the moduli set {22n, 22n+1-1, 2n-1} which is a special case of the general set {2α, 22β+1-1, 2β-1}, where α=2n and β=n. The converter for this special set is derived from the presented general architecture with higher speed than the fastest state-of-the-art reverse converter which has been designed for the 5n-bit DR moduli set {22n, 22n+1-1, 2n-1}. Furthermore, theoretical and FPGA implementation results show that the proposed reverse converter for moduli set {22n, 22n+1-1, 2n-1} results in considerable improvement in conversion delay with less hardware requirements compared to other works with similar DR.

In this paper, the new residue number system (RNS) moduli sets {22n, 2n -1, 2n+1 -1} and {22n, 2n -1, 2n-1 -1} are introduced. These moduli sets have 4n-bit dynamic range and well-formed moduli which can result in high-performance residue to binary converters as well as efficient RNS arithmetic unit. Next, efficient residue to binary converters for the proposed moduli sets based on mixed-radix conversion (MRC) algorithm are presented. The converters are ROM-free and they are realized using carry-save adders and modulo adders. Comparison with the other residue to binary converters for 4n-bit dynamic range moduli sets shown that the presented designs based on new moduli sets {22n, 2n -1, 2n+1 -1} and {22n, 2n -1, 2n-1 -1} are improved the conversion delay and result in hardware savings. Also, the proposed moduli sets can lead to efficient binary to residue converters, and they can speed-up internal RNS arithmetic processing, compared with the other 4n-bit dynamic range moduli sets.