Abstract—In this paper, radix-2 r arithmetic is applied to the multiple constant multiplication (MCM) problem. Given a number M of nonnegative constants with a bit-length N, we determine the analytic formulas for the maximum number of additions, the average number of additions, and the maximum number of cascaded additions forming the critical path. We get the first proved bounds known so far for MCM. In addition of being fully-predictable with respect to the problem size (M, N), the RADIX-2 r MCM heuristic exhibits a sublinear runtime- complexity O(M×N/r), where r is a function of (M, N). For high- complexity problems, it is most likely the only one that is even feasible to run. Another merit is that it has the shortest adder- depth in comparison to the best published MCM algorithms. Index Terms— High-Speed and Low-Power Design, Linear- Time-Invariant (LTI) Systems, Multiplierless Single/Mutiple Constant Multiplication (SCM/MCM), Radix-2 r Arithmetic. I. BACKGROUND AND MOTIVATION CM is an arithmetic operation that multiplies a set of fixed-point constants { } 0 1 2 1 , , , , M C CC C − L with the same fixed-point variable X. This operation dominates the complexity of many numeric systems such as FIR/IIR filters, DSP transforms (DCT, DFT, Walsh, …), LTI controllers, crypto-systems, etc. To be efficiently implemented, i.e., rapid, compact, and low-power, MCM must avoid costly multipliers. The hardware alternative will be multiplierless, i.e., using only additions, subtractions, and left-shifts. We assume that addition and subtraction have the same area/speed cost, and that the shift is costless since it can be realized without any gates, i.e., just by using hardwiring. Therefore, the MCM problem is defined as the process of finding the minimum number of addition/subtraction operations. The computational complexity of MCM is conjectured to be NP-hard [1]. But because the solution-space to explore is so huge, optimal solutions require excessive runtime and become impractical even for MCM operations of a medium complexity [1][2]. Only MCM heuristics can react in a reasonable amount of time, producing however, suboptimal solutions. Using the radix-2 r arithmetic, we developed in a previous work [3][4] a fully-predictable heuristic for SCM, denoted RADIX-2 r SCM. We obtained the lowest analytic bounds known so far for SCM in adder-cost (Upb), average (Avg), and adder-depth (Ath). Compared to the standard Canonical- Signed-Digit (CSD) representation [5] in the case of a serial implementation (adders connected in series), for an N-bit constant a saving of 50% is attained at N=1134, N=128, and N=64, in Avg, Upb, and Ath, respectively. The savings keep increasing as N is getting larger. In addition, RADIX-2 r SCM shows a sublinear runtime complexity with respect to N, and the memory space required is very small; for N=8192 corresponds a look-up table of 1024 entries only. These two features makes RADIX-2 r SCM very useful for huge constants, given that the lowest runtime complexity of non-digit-recoding algorithms (Bernstein [6], Lefèvre [7], BHM [8], Hcub [9], and MAG [10]) is O(N 3 ) [3]. A summary of the main features of RADIX-2 r SCM is given in Table I. The main idea of RADIX-2 r SCM is that the base number system (2 r ) is properly adapted to the bit-length (N) of the constant to achieve, either an optimal adder-cost (r 1 ) [3] or a lower adder-depth (r 2 ) [4]. To decide which expression of r (r 1 or r 2 ) to choose depends actually on the design requirements. If area is targeted, r 1 is used. But in case speed or power is a concern, r 2 is suitable. Note that intermediate values of r (r 1 <r<r 2 ) lead to a tradeoff between area and speed/power. The main purpose of this work is to first apply the radix-2 r arithmetic to the MCM problem and derive the analytic expressions for Upb, Ath, and Avg in the same way we did for SCM [3][4]. As a second step, we look at an actual circuit implementation through the application of RADIX-2 r MCM to the design optimization of a benchmark FIR filter. This paper is organized as follows. Section I gives an overview on RADIX-2 r SCM. In Section II we define RADIX-2 r MCM and determine the respective metrics. RADIX-2 r MCM is confronted in Section III to some of the best published MCM heuristics. Finally, Section IV provides some concluding remarks and suggestions for future work. II. RADIX-2 r MCM A nonnegative N-bit constant C is expressed in radix-2 r as ( ( ) ) rj r rj r r rj r r N j rj rj rj rj c c c c c c C 2 2 2 2 2 2 1 1 2 2 1 / 1 0 2 2 1 1 0 1 × − + ⋅ ⋅ ⋅ + + + + = − + − − + − − + = + + − ∑ ( ) ∑ − + = × = 1 / 1 0 2 r N j rj j Q , (1) where 0 1 = = − N c c and * Ν ∈ r . In (1), the two’s complement representation of C is split into ( ) ⎡ ⎤ r N / 1 + slices ( j Q ), each of r+1 bit length (see Fig.1.a). Each pair of two contiguous slices has one overlapping bit. A digit-set ( ) r DS 2 corresponds to (1), such as ( ) { } 1 1 1 1 2 , 1 2 , ... , 1 , 0 , 1 , . . . , 1 2 , 2 2 − − − − − − + − − = ∈ r r r r r j DS Q . The sign of the Q j term is given by the c rj+r–1 bit, and j k j m Q j × = 2 , with { } 1 2 1 0 − ∈ r , ... , , , k j and ( ) { } 1 , 0 2 U r j OM m ∈ , where ( ) { } 1 2 ,..., 7 , 5 , 3 2 1 − = − r r OM . ( ) r OM 2 is the set of odd positive digits in radix-2 r recoding, with ( ) 1 2 2 2 − = − r r OM . Multiple Constant Multiplication Algorithm for High Speed and Low Power Design Abdelkrim K. Oudjida, Ahmed Liacha, Mohammed Bakiri, and Nicolas Chaillet M A.K. Oudjida (a_oudjida@cdta.dz), A. Liacha, and M. Bakiri are with “Centre de Développement des Technologies Avancées”, CDTA, Cité du 20 août 1956, Baba-Hassen, Algiers, 16303, Algeria. N. Chaillet (nicolas.chaillet@femto-st.fr) and M. Bakiri are with FEMTO- ST Institute, UFC/CNRS/ENSMM/UTBM, 32 avenue de l'Observatoire, 25044 Besançon, Cedex, France. This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TCSII.2015.2469051 Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.