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

TL;DR

This paper introduces a new moduli set $ au^+$ for residue number systems, along with the design of forward and reverse converters, demonstrating improved performance over traditional sets in certain operation sequences.

Contribution

It presents the design and implementation of forward and reverse converters for the novel moduli set $ au^+$, enabling efficient residue computations.

Findings

Converters use carry-save addition units for efficiency.

Performance surpasses traditional sets when multiple operations are performed.

Analytical and simulation results validate the proposed converters' effectiveness.

Abstract

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.