Multiplierless multiple constant multiplication

Abstract

A variable can be multiplied by a given set of fixed-point constants using a multiplier block that consists exclusively of additions, subtractions, and shifts. The generation of a multiplier block from the set of constants is known as the multiple constant multiplication (MCM) problem. Finding the optimal solution, i.e., the one with the fewest number of additions and subtractions is known to be NP-complete. We propose a new algorithm for the MCM problem, which produces solutions that require up to 20% less additions and subtractions than the best previously known algorithm. At the same time our algorithm, in contrast to the closest competing algorithm, is not limited by the constant bitwidths. We present our algorithm using a unifying formal framework for the best, graph-based, MCM algorithms and provide a detailed runtime analysis and experimental evaluation. We show that our algorithm can handle problem sizes as large as 100 32-bit constants in a time acceptable for most applications. The implementation of the new algorithm is available at www.spiral.net.

Sum-of-product architectures computing just right

“…Many other successful approaches exist, in particular those based on multiple constant multiplication (MCM) using the transpose form (where the registers are on the output path). [11], [12], [13], [14], [15]. A technique called Distributed Arithmetic, which predates FPGA [16], can be considered a generalization of the KCM technique to the MCM problem.…”

An Improved Non-CSD 2-Bit Recursive Common Subexpression Elimination Method to Implement FIR Filter

“…The algorithms in [15], [17], and [11] produce multipliers with large LDs which increase the delay of the multiplier substantially. The algorithms in [11], [16]- [18], consider one coefficient at a time and do not consider the effect on the rest of the coefficients while synthesizing them. A multiple adder graph algorithm (MAG) [19] minimizes the adder cost by considering the effect on the remaining coefficients.…”

“…They claimed that the algorithm produced the best reduction in number of LOs compared to the best known algorithm in the literature without increasing the LD requirement. The results in [1] (in terms of LOs and LD) have been compared with several CSE methods, such as Hartley [4], Pasko [5], BHM [17], C1 algorithm [22], MAG [19], NR-SCSE [6], HCUB [18], CSDC [21], and CRA-2 [15], [9]. The authors in [1] also presented design examples of several FIR filters using the proposed common subexpression algorithm.…”

Two-Rate Based Structures for Computationally Efficient Wide- Band FIR Systems

“…Efficient realization of constant multiplications is an active research area and much effort has been focused on the case where one input data is multiplied by several constant coefficients. This problem has mainly been motivated by single-rate FIR filters, where for a transposed direct form FIR filter the input is multiplied by several coefficients, see [45][46][47][48][49]. The resulting implementation of several multiplications is denoted multiplier block, as in [45].…”

“…9 a total of 33 adders are required for the multiplier block using the RAG-n algorithm in [45]. This is an optimal result since there are 33 different (odd) coefficients as discussed in [58], and, hence, there is no need to apply the slightly more efficient algorithms in [47][48][49]. Furthermore, 80 structural adders and 118 registers are required for the FIR subfilters.…”