An O(n) Residue Number System to Mixed Radix Conversion technique

This paper investigates the conversion of residue number system (RNS) operands to decimal, which is an important issue concerning the utilization of RNS numbers in digital signal processing applications. In this line of reasoning, we introduce an RNS to mixed radix conversion (MRC) technique, which addresses the computation of mixed radix (MR) digits in such a way that enables the MRC parallelization. Given an RNS with the set of relatively prime integer moduli {m_i}_i=1;n, the key idea behind the proposed technique is to maximize the utilization of the modulo-m_i adders and multipliers present in the RNS processor functional units. For an n-digit RNS number X = (x₁; x₂; x₃;_hellip; x_n) the method requires n iterations. However, at iteration i, the modulo-m_i units are utilized for the calculation of the MR digit a_i, while the other modulo units are calculating intermediate results required in further iterations. Our approach results in an RNS to MRC with an asymptotic complexity, in terms of arithmetic operations, in the order of O(n), while state of the art MRCs exhibit an asymptotic complexity in the order of O(n²). More in particular, when compared with the best state of the art MRC, our technique reduces the number of arithmetic operations by 5:26% and 38:64% for moduli set of length four and ten, respectively.