Scaling In Residue Number System Research Articles

Zero-Aware Low-Precision RNS Scaling Scheme

Scaling is one of the complex operations in the Residue Number System (RNS). This operation is necessary for RNS-based implementations of deep neural networks (DNNs) to prevent overflow. However, the state-of-the-art RNS scalers for special moduli sets consider the 2k modulo as the scaling factor, which results in a high-precision output with a high area and delay. Therefore, low-precision scaling based on multi-moduli scaling factors should be used to improve performance. However, low-precision scaling for numbers less than the scale factor results in zero output, which makes the subsequent operation result faulty. This paper first presents the formulation and hardware architecture of low-precision RNS scaling for four-moduli sets using new Chinese remainder theorem 2 (New CRT-II) based on a two-moduli scaling factor. Next, the low-precision scaler circuits are reused to achieve a high-precision scaler with the minimum overhead. Therefore, the proposed scaler can detect the zero output after low-precision scaling and then transform low-precision scaled residues to high precision to prevent zero output when the input number is not zero.

A 2 n scaling scheme for signed RNS integers and its VLSI implementation

High efficient implementation of scaling in residue number system (RNS) is one of the critical issues for the applications of RNS in digital signal processing (DSP) systems. In this paper, an efficient scaling algorithm for signed integers in RNS is proposed firstly through introducing a correction constant in negative integers scaling procedure. Based on the proposed scaling algorithm, an efficient RNS 2n scaling implementation method is presented, in which Chinese remainder theorem (CRT) and a redundant modulus are used to perform the base extension to obtain the least significant n bits of RNS integers. With the redundant modulus, the RNS sign detection can be achieved by the parity detection. And then, an approach to update the residue digit of the redundant channel is also proposed. Meanwhile, this paper provides a method of computing the correction constant of the redundant channel in negative integers scaling. The analysis results indicate that the complexity of the proposed scaling algorithm grows linearly with the word-length of the RNS dynamic range without using Look-up Table (LUT). Furthermore, the proposed algorithm is employed for a specific moduli set 2n scaling. The synthesis results show that the critical path of the proposed algorithm is shortened by 12%, the area and power consumption performance is improved by about 35%, compared to the existing cascading 2n scaling method for very large scale integration (VLSI) implementation under the same restriction. Besides, the VLSI layout indicates that the parallel structure is simpler.

New power-of-2 RNS scaling scheme for cell-based IC design

Previous scaling schemes are based on the conversion of the unpositional residue number system (RNS) digits into a positional number system via Chinese remainder theorem (CRT) or mixed-radix-conversion (MRC) and the back conversion into RNS with an associated size and speed penalty in cell-based integrated circuit (CBIC) designs. This paper presents a new scaling approach, which allows faster and more efficient schemes, because the scaling uses only RNS operations within the small word length channels.

A fast and accurate RNS scaling technique for high speed signal processing

A fast and accurate magnitude scaling technique in the residue number system (RNS) is proposed. This technique obtains the residues of the scaled integer, when scaled by a product of a subset of the moduli, in approximately log n cycles, where n is the total number of moduli in the RNS. The scaled integer has an error of at most unity. The technique is based on a judicious decomposition of the Chinese remainder theorem (CRT) and the use of a redundant channel which carries (at the least) the odd-even information about the integer being scaled. >