Residue Number System Operands to Decimal Conversion for 3-Moduli Sets

This paper investigates the conversion of 3-moduli Residue Number System (RNS) operands to decimal. First we assume a general {m(i)}(i)=1,3 moduli set with the dynamic range M = Pi(3)(i=1) m(i) and introduce a modified Chinese Remainder Theorem (CRT) that requires mod-m(3) instead of mod-M calculations. Subsequently, we further simplify the conversion process by focussing on {2n + 2, 2n + 1, 2n} moduli set, which has a common factor of 2. We introduce in a formal way a CRT based approach for this case, which requires the conversion of {2n + 2, 2n + 1, 2n} set into moduli set with relatively prime moduli, i.e., {m(1)/2, m(2), m(3)}, when n is even, n >= 2 and {m(1), m(2), m(3)/2}, when n is odd, n >= 3. We demonstrate that such a conversion can be easily done and doesn't require the computation of any multiplicative inverses. Finally, we further simplify the 3-moduli CRT for the specific case of {2n + 2, 2n + 1, 2n} moduli set. For this case the propose CRT requires 4 additions, 4 multiplications and all the operations are mod-m(3) in case n is even and mod-m(3)/2 if n is odd. This outperforms state of the art converters in terms of required operations and due to the fact that the numbers involved in the calculations are smaller it results in less complex adders and multipliers.