Table-Based versus Shift-And-Add Constant Multipliers for FPGAs

Abstract

The multiplication by a constant is a frequently used operation. To implement it on Field Programmable Gate Arrays (FPGAs), the state of the art offers two completely different methods: one relying on bit shifts and additions/subtractions, and another one using look-up tables and additions. So far, it was unclear which method performs best for a given constant and input/output data types. The main contribution of this work is a thorough comparison of both methods in the main application contexts of constant multiplication: filters, signalprocessing transforms, and elementary functions. Most of the previous state of the art addresses multiplication by an integer constant. This work shows that, in most of these application contexts, a formulation of the problem as the multiplication by a real constant allows for more efficient architectures. Another contribution is a novel extension of the shift-and-add method to real constants. For that, an integer linear programming (ILP) formulation is proposed, which truncates each component in the shift-and-add network to a minimum necessary word size that is aligned with the approximation error of the coefficient. All methods are implemented within the open-source FloPoCo framework. 1 With FloPoCo version 4.1.2, try the command flopoco FPExp we=11 wf=53 and look in the produced VHDL for the signal absKLog2.

Lossless Differential Table Compression for Hardware Function Evaluation

“…Finally, table-based methods have been also used to implement multiplication by constants [9], [10], [11] which are themselves used in elementary function implementations [22,Fig. 2] [23,Fig.…”

“…Again the tables there are perfectly suited to LDTC. However, it is less obvious here that this potential can be exploited, as the tables are already finely tailored to the LUT-based logic of the FPGAs [10], [11].…”

Towards Arithmetic-Centered Filter Design

“…The first is the arithmetic to be used internally, in particular for constant multiplication. Two families of methods exist [5], one based on shift-and-add [6]- [9], the other one on precomputed tables [10], [11]. Each method offers the possibility of sharing intermediate results in multiple constant multiplication to improve area, performance, or power.…”

Application-Specific Arithmetic in High-Level Synthesis Tools

“…Multiplication by a constant was mentioned in Section 1. We just refer to the rich existing literature on the subject [4,8,11,14,23,30,41,[47][48][49]. These are mostly academic works, but back-end tools already embed some of it, so this optimization could be the first to be implemented.…”

“…Consider, for example, the integer multiplication by a constant. Optimization of hardware constant multiplication has been the subject of much research [4,14,23,30,41,47], some of which is specific to FPGAs [8,11,48,49]. Some of the early HLS tools could exploit this literature and generate an optimized architecture.…”