Accumulation of Round-Off Error in Fast Fourier Transforms

Abstract

The fast Fourier transform (FFT) is an algorithm to compute the discrete Fourier coefficients with a substantial time saving over conventional methods. The finite word length used in the computer causes an error in computing the Fourier coefficients. This paper derives explicit expressions for the mean square error in the FFT when floating-point arithmetics are used. Upper and lower bounds for the total relative mean square error are given. The theoretical results are in good agreement with the actual error observed by taking the FFT of data sequences.

A Methodology for the Formal Verification of FFT Algorithms in HOL

“…Weinstein [29] presented a statistical model for roundoff errors of the floating-point FFT. Kaneko and Liu [19] presented a detailed analysis of roundoff error in the FFT decimation-in-frequency algorithm using floating-point arithmetic. This analysis is later extended by the same authors to the FFT decimation-in-time algorithm [21].…”

“…Our focus will be on the process of translating the hand proofs into equivalent proofs in HOL. The analysis we develop is mainly inspired by the work done by Kaneko and Liu [19], who proposed a general approach to the error analysis problem of the decimation-in-frequency FFT algorithm using floating-point arithmetic. Following a similar idea, we have extended this theoretical analysis for the decimation-in-time and fixed-point FFT algorithms.…”

“…The accumulation of roundoff error is determined by the recursive equations (19), (20), (21), (22), and (23), with initial conditions given by equation (18). In HOL, we first constructed complex numbers on reals similar to [18].…”

“…Thereafter, we proved lemmas to rewrite the errors as complex numbers using the real and imaginary parts according to the equations (15), (16), and (17). Finally, we proved lemmas to determine the accumulation of roundoff error in floating-and fixed-point FFT algorithms by recursive equations and initial conditions according to the equations (18), (19), (20), (21), (22), and (23).…”

“…Finally, based on these lemmas, we derive, for each of the two canonical forms of realization, expressions for the accumulation of roundoff error in floating-and fixed-point FFT algorithms using recursive definitions and initial conditions. While theoretical work on computing the errors due to finite precision effects in the realization of FFT algorithms with floating-and fixed-point arithmetics has been extensively studied since the late sixties [19], this paper contains the first formalization and proof of this analysis using a mechanical theorem prover, here HOL. The formal results are found to be in good agreement with the theoretical ones.…”