Multiplierless Algorithm for Multivariate Gaussian Random Number Generation in FPGAs

Abstract

The multivariate Gaussian distribution is used to model random processes with distinct pair-wise correlations, such as stock prices that tend to rise and fall together. Multivariate Gaussian vectors with length n are usually produced by first generating a vector of n independent Gaussian samples, then multiplying with a correlation inducing matrix requiring O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) multiplications. This paper presents a method of generating vectors directly from the uniform distribution, removing the need for an expensive scalar Gaussian generator, and eliminating the need for any multipliers. The method relies only on small read-only memories and adders, and so can be implemented using only logic resources (lookup-tables and registers), saving multipliers, and block-memory resources for the numerical simulation that the multivariate generator is driving. The new method provides a ten times increase in performance (vectors/second) over the fastest existing field-programmable gate array generation method, and also provides a five times improvement in performance per resource over the most efficient existing method. Using this method, a single 400-MHz Virtex-5 FPGA can generate vectors ten times faster than an optimized implementation on a 1.2-GHz graphics processing unit, and a hundred times faster than vectorized software on a general purpose quad core 2.2-GHz processor.