An FPGA-specific algorithm for direct generation of multi-variate Gaussian random numbers

The multi-variate Gaussian distribution is used to model random processes with distinct pair-wise correlations, such as stock prices that tend to rise and fall together. Multi-variate 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 0(n²) 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 ROMs and adders, and so can be implemented using just logic resources (LUTs and FFs), saving DSP and block-RAM resources for the numerical simulation that the multi-variate generator is driving. The new method provides a ten times increase in raw performance over the fastest existing FPGA generation method, and also provides a five times improvement in performance per resource over the most efficient existing method. Using this method a single 400MHz Virtex-5 FPGA can generate vectors ten times faster than an optimised CUDA implementation on a 1.2GHz GPU, and a hundred times faster than SIMD optimised software on a quad core 2.2GHz CPU.