TY - JOUR

T1 - Turing Computability of Fourier Transforms of Bandlimited and Discrete Signals

AU - Boche, Holger

AU - Monich, Ullrich J.

N1 - Publisher Copyright:
© 1991-2012 IEEE.

PY - 2020

Y1 - 2020

N2 - The Fourier transform is an important operation in signal processing. However, its exact computation on digital computers can be problematic. In this paper we consider the computability of the Fourier transform and the discrete-time Fourier transform (DTFT). We construct a computable bandlimited absolutely integrable signal that has a continuous Fourier transform, which is, however, not Turing computable. Further, we also construct a computable sequence such that the DTFT is not Turing computable. Turing computability models what is theoretically implementable on a digital computer. Hence, our result shows that the Fourier transform of certain signals cannot be computed on digital hardware of any kind, including CPUs, FPGAs, and DSPs. This also implies that there is no symmetry between the time and frequency domain with respect to computability. Therefore, numerical approaches which employ the frequency domain representation of a signal (like calculating the convolution by performing a multiplication in the frequency domain) can be problematic. Interestingly, an idealized analog machine can compute the Fourier transform. However, it is unclear whether and how this theoretical superiority of the analog machine can be translated into practice. Further, we show that it is not possible to find an algorithm that can always decide for a given signal whether the Fourier transform is computable or not.

AB - The Fourier transform is an important operation in signal processing. However, its exact computation on digital computers can be problematic. In this paper we consider the computability of the Fourier transform and the discrete-time Fourier transform (DTFT). We construct a computable bandlimited absolutely integrable signal that has a continuous Fourier transform, which is, however, not Turing computable. Further, we also construct a computable sequence such that the DTFT is not Turing computable. Turing computability models what is theoretically implementable on a digital computer. Hence, our result shows that the Fourier transform of certain signals cannot be computed on digital hardware of any kind, including CPUs, FPGAs, and DSPs. This also implies that there is no symmetry between the time and frequency domain with respect to computability. Therefore, numerical approaches which employ the frequency domain representation of a signal (like calculating the convolution by performing a multiplication in the frequency domain) can be problematic. Interestingly, an idealized analog machine can compute the Fourier transform. However, it is unclear whether and how this theoretical superiority of the analog machine can be translated into practice. Further, we show that it is not possible to find an algorithm that can always decide for a given signal whether the Fourier transform is computable or not.

KW - Fourier transform

KW - Turing computability

KW - algorithmic decision

KW - discrete-time Fourier transform

KW - frequency domain

UR - http://www.scopus.com/inward/record.url?scp=85079783831&partnerID=8YFLogxK

U2 - 10.1109/TSP.2020.2964204

DO - 10.1109/TSP.2020.2964204

M3 - Article

AN - SCOPUS:85079783831

SN - 1053-587X

VL - 68

SP - 532

EP - 547

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

M1 - 8950369

ER -