The Residue Number System (RNS) has been considered as an efficient parallel tool to increase the performance of computational systems. However, RNS consists of several parts which lead to complexity of RNS. In this paper, a new method to the design of RNS systems based on unification of the carry-propagate adders (CPAs) of the arithmetic units of RNS with reverse converter is presented. The proposed method eliminates the complex modular CPA for some of channels, and improves the current reverse converter designs by shifting some complex additions from arithmetic channel of RNS to the reverse converter. Experimental results shows a significant reduction in area, power consumption and the delay in compare with previous design.

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Since the past decade, the computing industry has intensively tried to decrease the power consumption of portable systems. While lots of works have been done in this area, Residue Number System (RNS) has emerged as a viable nontraditional approach to increase the performance. The main functionality of RNS is that it decomposes complex arithmetic circuits into simpler ones [1]. Instead of performing large computations, every arithmetic’s such as long bit multiplication is broken down into little parallel modulo multiplications. Despite the advantages of the RNS based arithmetic systems, the usage of this system has been rather restricted. The main limit is the additional hardware required especially for conversion from RNS to binary number. Therefore, every work that aims to make this component smaller, faster and less power consumer lead to make the RNS system more efficient and practical. So far, many reverse converters architecture has offered [1-6] but some drawbacks still can be seen in that designs. First, they are designed independently of arithmetic unit. Second, the New CRT-II [2], which is the one of the newest algorithms in reverse converter design, makes the reverse converter smaller than the previous designs, but slower due to use of many modular CPAs. This paper aims to address these challenges by considering some properties which have been ignored in overall design of RNS systems. The three-moduli set{ -1, , } has interesting attributes and consequently has been widely investigated. However, extensions of this moduli set have been introduced in order to increase the Dynamic Range, such as the , [4] or , , },, , }[5], and , , }and{, , } [6]. In this paper, by taking into account some properties of reverse converter, the last modular adder of the addition or multiplication in arithmetic unit are considered together with the reverse converter to increase the performance. The proposed method, eliminates the last adder which is slow in arithmetic unit and replace it with a faster adder in reverse converter. This strategy not only results in speed up, but also results in overall area reduction. The Proposed Approach The RNS is composed of the three basic components: forward converter, arithmetic unit and the reverse converter [1]. However, there are limited works in which all of these three components has been considered together. In this paper for the first time, the reverse converter design is done based on the final modular adder of the arithmetic unit. It means that, the final modular adder that has the same modulus as the modular adder in the beginning of the reverse converter is considered as a part of reverse converter instead of a part of arithmetic unit. There are the two common adders in computer arithmetic [1]: carry-propagate adder (CPA) and carry-save adder (CSA). The CSA has three input operands while CPA has two operands. Besides, CSA is faster and simpler than CPA adder which uses the complex parallel prefix structure to increase the speed. The issue of parallel prefix for increasing the speed has become more serious when it comes about the modular adder. However, CSA generates redundant output in the form of carry and sum binary vectors. It should be mentioned that, the proposed technique can be only applied on the reverse converters that uses the new CRT-II [2] formula since only in this type of formula the modulo of adder in input of the converter can be the same as the modulo of one of the arithmetic unit modulus. The main idea of this paper is that, in RNS system for some modulus, if the modulo of the final modular adder of the arithmetic unit is the same as the modulo of modular adder which is in the beginning of the reverse converter, this adder can be eliminated and then the Carry and Sum signals are entered to the reverse converter where they are added by an extra CSA which is placed to converter. To explain the proposed method, the main formulas of [4] can be written as follows: ; (1) ; (2) (3) Fig. 1.The previous design including a maultiplier which is connected to a reverse converter in the right and, the new design which is consisted of a multiplier without the last modular adder, and a modified reverse converter in the left Where , and are four residues in the moduli of respectively. However, in a full RNS system these residues are resulted from a modular multiplication or addition operation where each of them has a modular adder as the final adder. For example, can be written as: . (4) Where, s2 and c2 are the redundant results of CSA addition of corresponding partial products. As can be seen the conversion formulas in (1)-(3) are composed of the two terms and. In the term , the modulo of is the same as the result modulo . Therefore, as described before the term can be replaced by its equivalent from (4) and instead of two cascaded modular CPA additions, a CSA and a modular CPA addition can be performed. Hence, we have: = . (5) If the residue is considered as (6), in the same way, Y can be written as (7): (6) = (7) Where s4 and c4 are the redundant results of a CSA. Fig.1 compares the presented method in the left with previous method in right. It can be seen in the proposed method that the modular adder in the end of multiplier is specified by a dotted rectangular shape. This adder can be eliminated as shown in Fig. 1. The output of these adders are x2 and x4 which are inserted to the converter respectively. Instead of x2 and x4, the corresponding redundant CSA results (s, c) and (s1, c1) are entered directly to the converter. Performance Evaluation The proposed and previous designs are described in VHDL codes, synthesised and optimized by Synopsys Design Compiler. The 180-nm general-purpose standard cell library was used. Generally, the designs are divided in two groups: i) reverse converter of [4] plus modular adders in used its inputs and ii) the proposed reverse converter that eliminate the modular adder in input of reverse converts as described in before. For each group the following states were considered: - full adder (FA) based CPAs is used in reverse converter and modular adder; - parallel prefix based CPAs is used in reverse converter and modular adder. Tables 1 and 2 give information about experimental results which have been achieved based on the different designs. The implementations were done for the three circuit parameters: delay, power consumption and area. It must be noted that the comparison was done between the proposed designs and [4]. Tables 3 and 4 show that the proposed method of shifting modular adder from arithmetic unit to the reverse converter results in significant improvement in delay and Energy (Power-Delay Product). For example, the speed up in the first state for n = 12 is more than 26% rather than previous design. The proposed method has the best speed up when the FA-based CPA is used in adders. Besides, Tables 3 and 4 indicate the amount of PDP improvement of proposed customized adder plus reverse converter over previous design. For example, the PDP improvement has been about 22% for 10-bit in the state-2. Generally, two points can be extracted from experimental results: - although the power consumption is not improved in some cases, the gain which is obtained by speed up is so significant that totally makes the performance better and compensate the power consumption loss; - the best PDP is related to the state-2 where all CPAs are parallel prefix. Therefore, the proposed method has the best PDP when parallel prefix is used. In addition, as it was predictable, the efficiency of the proposed technique is contributed in the fact that the design area has been reduced by 13% for 12-bit in compare with previous design for the state-1. This improvement also can be seen in all states which has been better at least more than 5%. Table 1. Experimental results for proposed and previous design when FA-based CPAs is used in reverse converter and modular adder n Proposed design Previous design Delay (ns) Area (um2) Power (mw) Delay (ns) Area (um2) Power (mw) 10 8.39 28962. 23.123 10.6 30935. 21.95 12 10.1 35582. 28.667 12.8 37701. 26.10 Table 2. Experimental results for proposed and previous design when Parallel prefix based CPAs is used in reverse converter and modular adder n Proposed design Previous design Delay (ns) Area (um2) Power (mw) Delay (ns) Area (um2) Power (mw) 10 3.48 46283. 23.196 4.14 51469 23.127 12 3.68 57942. 28.703 4.51 61581 28.728 Table 3. The percentage of improvement (%) for proposed design over previous method for the four parameters: delay, area, power consumption and PDP when FA-based CPAs is used in reverse converter and modular adder n Delay (%) Area (%) Power (%) PDP (%) 10 25.73 5.95 -8.9438 14.49 12 26.91 6.26 -5.7 19.56 Table 4: The percentage of improvement (%) for proposed design over previous method for the four parameters: delay, area, power consumption and PDP when Parallel prefix based CPAs is used in reverse converter and modular adder n Delay (%) Area (%) Power (%) PDP (%) 10 22.55 6.28 0.0870 14.49 12 20.73 8.84 0.6149 21.47 As it can be seen the best performance for area is observed when the parallel prefix adder is used only in arithmetic unit. Conclusion In this paper, a new approach to design RNS systems has been introduced. This method is relied on the fact that the reverse converter design and arithmetic unit must be considered together not separately which is common in the previous methods. By taking the advantages of this view, the modular CPA of final arithmetic unit which is one of the main factor in increasing RNS overhead is replaced by a CSA. Experimental results show significant reduction in both speed and power consumption of RNS system that uses the proposed method.

About the authors

Mostafa Mirhosseini Seyed

Islamic Azad University

Email: s.m.mirhoseini@srbiau.ac.ir

Sabbagh Molahosseini Amir

Islamic Azad University

Email: sabbagh@iauk.ac.ir

Hosseinzadeh Mehdi

Iran University of Medical Sciences

Email: hosseinzadeh.m@Iums.ac.ir


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