Balanced Modular Addition for the Moduli Set $ \{2^{q},2^{q}\mp 1,2^{2q}+1\}${2q,2q∓1,22q+1} via Moduli-($ 2^{q}\mp \sqrt{-1}$2q∓-1) Adders

Moduli-set <inline-formula><tex-math notation="LaTeX">$ \mathbf{\tau}=\{2^{\boldsymbol{q}},2^{\boldsymbol{q}}\pm 1\}$</tex-math></inline-formula> is often the base of choice for realization of digital computations via residue number systems. The optimum arithmetic performance in parallel residue channels, is generally achieved via equal bit-width residues (e.g., <inline-formula><tex-math notation="LaTeX">$ \boldsymbol{q}~ \mathbf{i}\mathbf{n}~ \mathbf{\tau}$</tex-math></inline-formula>) that usually leads to equal computation speed within all the residue channels. However, the commonly difficult and costly task of reverse conversion (RC) is often eased in the existence of conjugate moduli. For example, <inline-formula><tex-math notation="LaTeX">$ 2^{\boldsymbol{q}}\mp 1\in \mathbf{\tau}$</tex-math></inline-formula>, lead to the efficient modulo-(<inline-formula><tex-math notation="LaTeX">$ 2^{2\boldsymbol{q}}-1$</tex-math></inline-formula>) addition, as the bulk of <inline-formula><tex-math notation="LaTeX">$ \mathbf{\tau}$</tex-math></inline-formula>-RC, via the New-CRT reverse conversion method. Nevertheless, for additional dynamic range, <inline-formula><tex-math notation="LaTeX">$ \mathbf{\tau}$</tex-math></inline-formula> is augmented with other moduli. In particular, <inline-formula><tex-math notation="LaTeX">$ \mathbf{\phi}=\mathbf{\tau}\cup \{2^{2\boldsymbol{q}}+1\}$</tex-math></inline-formula>, leads to efficient RC, where the added modulo is conjugate with the product <inline-formula><tex-math notation="LaTeX">$ 2^{2\boldsymbol{q}}-1$</tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">$ 2^{\boldsymbol{q}}\mp 1\in \mathbf{\tau}$</tex-math></inline-formula>. Therefore, the final step of <inline-formula><tex-math notation="LaTeX">$ \mathbf{\phi}$</tex-math></inline-formula>-RC would be fast and low cost/power modulo-(<inline-formula><tex-math notation="LaTeX">$ 2^{4\boldsymbol{q}}-1$</tex-math></inline-formula>) addition. However, the <inline-formula><tex-math notation="LaTeX">$ 2\boldsymbol{q}$</tex-math></inline-formula>-bit channel-width jeopardizes the existing delay-balance in <inline-formula><tex-math notation="LaTeX">$ \mathbf{\tau}$</tex-math></inline-formula>. As a remedial solution, given that <inline-formula><tex-math notation="LaTeX">$ 2^{2\boldsymbol{q}}+1=\left(2^{\boldsymbol{q}}-\boldsymbol{j}\right)\left(2^{\boldsymbol{q}}+\boldsymbol{j}\right)$</tex-math></inline-formula>, with <inline-formula><tex-math notation="LaTeX">$ \boldsymbol{j}=\sqrt{-1}$</tex-math></inline-formula>, we design and implement modulo-(<inline-formula><tex-math notation="LaTeX">$ 2^{2\boldsymbol{q}}+1$</tex-math></inline-formula>) adders via two parallel <inline-formula><tex-math notation="LaTeX">$ \boldsymbol{q}$</tex-math></inline-formula>-bit moduli-(<inline-formula><tex-math notation="LaTeX">$ 2^{\boldsymbol{q}}\mp \boldsymbol{j}$</tex-math></inline-formula>) adders. The analytical and synthesis based evaluations of the proposed modulo-(<inline-formula><tex-math notation="LaTeX">$ 2^{\boldsymbol{q}}\mp \boldsymbol{j}$</tex-math></inline-formula>) adders show that the delay-balance of <inline-formula><tex-math notation="LaTeX">$ \mathbf{\tau}$</tex-math></inline-formula> is preserved with no cost overhead vs. <inline-formula><tex-math notation="LaTeX">$ \mathbf{\phi}$</tex-math></inline-formula>. In particular, the binary-to-complex and complex-to-binary convertors are merely cost-free and immediate.