Forward and Reverse Converters for the Moduli-Set $\{2^{2q+1},2^q+2^{q-1}\pm1\}$

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.