Forward and Reverse Converters for the Moduli-Set $\{2^{2q+1},2^q+2^{q-1}\pm 1\}$

Modulo-$(2^q+2^{q-1}\pm 1)$ adders have recently been implemented using the regular parallel prefix (RPP) architecture, matching the speed of the widely used modulo-$(2^q\pm 1)$ RPP adders. Consequently, we introduce a new moduli set ${\tau }^{+}=\{2^{2q+1},2^q+2^{q-1}\pm 1\}$, with over $(2^{q+2})\times$ dynamic range and adder speeds comparable to the conventional $\tau =\{2^q,2^q\pm 1\}$ set. However, to fully leverage ${\tau }^{+}$ in residue number system applications, a complete set of circuitries is necessary. This work focuses on the design and implementation of the forward and reverse converters for ${\tau }^{+}$. These converters consist of four and seven levels of carry-save addition units, culminating in a final modulo-$(2^q+2^{q-1}\pm 1)$ and modulo-$(2^{2q+1}+2^{2q-2}-1)$ adder, respectively. Through analytical evaluations and circuit simulations, we demonstrate that the overall performance of a sequence of operations including residue generation – including residue generation, $k$ additions, and reverse conversion – using ${\tau }^{+}$ surpasses that of $\tau$ when $k$ exceeds a certain practical threshold.