Augmenting the balanced residue number system moduli-set $\{m_1=2^n,m_2=2^n-1,m_3=2^n+1\}$, with the co-prime modulo $m_4=2^{2n}+1$, increases the dynamic range (DR) by around 70%. The Mersenne form of product $m_2 m_3 m_4=2^{4n}-1$, in the moduli-set $\{m_1,m_2,m_3,m_4\}$, leads to a very efficient reverse convertor, based on the New Chinese remainder theorem. However, the double bit-width of the m_4 residue channel is counter-productive and jeopardizes the speed balance in $\{m_1,m_2,m_3\}$. Therefore, we decompose $m_4$ to two complex-number n-bit moduli $2^n\pm\sqrt{-1}$, which preserves the DR and the co-primality across the augmented moduli set. The required forward modulo-$(2^{2n}+1)$ to moduli-$(2^n\pm\sqrt{-1}) $conversion, and the reverse are immediate and cost-free. The proposed unified moduli-$(2^n\pm\sqrt{-1})$ adder and multiplier, are tested and synthesized using Spartan 7S100 FPGA. The 6-bit look-up tables (LUT), therein, promote the LUT realizations of adders and multipliers, for $n=5$, where the DR equals $2^{25}-2^5$. However, the undertaken experiments show that to cover all the 32-bit numbers, the power-of-two channel $m_1$ can be as wide as 12 bits with no harm to the speed balance across the five moduli. The results also show that the moduli-$(2^5\pm\sqrt{-1})$ add and multiply operations are advantageous vs. moduli-$(2^5\pm1)$ in speed, cost, and energy measures and collectively better than those of modulo-$(2^{10}+1)$.

翻译：在平衡余数系统模集$\{m_1=2^n,m_2=2^n-1,m_3=2^n+1\}$中增加互素模$m_4=2^{2n}+1$，可将动态范围(DR)提升约70%。在模集$\{m_1,m_2,m_3,m_4\}$中，乘积$m_2 m_3 m_4=2^{4n}-1$的梅森形式使得基于新孙子定理的反向转换器极为高效。然而，$m_4$余数通道的双倍位宽会产生不利影响，并破坏$\{m_1,m_2,m_3\}$间的速度平衡。为此，我们将$m_4$分解为两个复数域n位模$2^n\pm\sqrt{-1}$，这既保持了动态范围，又维持了扩展模集间的互素性。所需的前向模$(2^{2n}+1)$到模$(2^n\pm\sqrt{-1})$转换及其逆转换均可即时无成本实现。所提出的统一模$(2^n\pm\sqrt{-1})$加法器与乘法器已在Spartan 7S100 FPGA上完成测试与综合。其中6位查找表(LUT)促进了$n=5$（此时动态范围为$2^{25}-2^5$）情况下加法器与乘法器的LUT实现。实验表明，为覆盖全部32位数，二次幂通道$m_1$的位宽可扩展至12位而不影响五个模间的速度平衡。结果同时显示，模$(2^5\pm\sqrt{-1})$的加法与乘法运算在速度、成本及能耗指标上均优于模$(2^5\pm1)$运算，且整体性能优于模$(2^{10}+1)$运算。