Notch Fourier transform

Abstract

This paper presents a new computational algorithm for the Fourier transform at arbitrary frequencies. This algorithm, if an input signal has r frequencies, can be implemented by one FIR filter composed of r second-order digital notch filters (2-DNF's; <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_{p}(z) = 1 - 2 \cos (2\pif_{p}/f_{s}) z^{-1} + z^{-2}</tex> ), and r second-order digital resonators (2- DR's; H' <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> (z)= 1/H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> (z)) where frequencies f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> (p = 1, ..., r) are arbitrary and f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</inf> is a sampling frequency. This algorithm is referred to as the notch Fourier transform (NFT). This NFT algorithm has the following main advantages, 1) It can be applied to the input signal composed of arbitrary frequencies. 2) In advance of the end of one period of the signal, Fourier coefficients can be computed. 3) Although we can analyze the input signal composed of r arbitrary frequencies by solving simultaneous equations, the amount of processing in the NFT is smaller by the order of magnitude N where N = 2r is the number of sample data. At every sampling time after one block of data, the NFT can compute all Fourier coefficients with the number of multiplications proportional to N. But for the exact computation of Fourier coefficients, we must know the signal frequencies in advance.