Estimating the length of a compact function from a noisy measurement of the modulus of its Fourier transform

Abstract

Consider f, zero outside an interval of length L, and its Fourier transform F. It is known that, if arg F is unknown, L can be found from (a) the Fourier transform of |F|2 or (b) the Fourier transform of |F| exp(jϕ), where ϕ = Hilbert transfrom of In|F|. It appears that method (b) may be superior if |F| is measured in the presence of noise.