Correcting for the Alias Effect When Measuring the Power Spectrum Using a Fast Fourier Transform

Abstract

Because of mass assignment onto grid points in the measurement of the power spectrum using a fast Fourier transform (FFT), the raw power spectrum [|delta(f)(k)|(2)] estimated with the FFT is not the same as the true power spectrum P( k). In this paper we derive a formula that relates h|delta(f)(k)|(2)] to P( k). For a sample of N discrete objects, the formula reads [\delta(f) ( k)\(2)] = Sigma(n) [\W( k + 2k(N)n)\(2) P( k + 2(k)Nn) +1\N\W(k+2k(N)n)\(2)], where W(k) is the Fourier transform of the mass assignment function W( r), k(N) is the Nyquist wavenumber, and n is an integer vector. The formula is different from that in some previous works in which the summation over n is neglected. For the nearest grid point, cloud-in-cell, and triangular-shaped cloud assignment functions, we show that the shot-noise term Sigma(n) (1/N)\W(k + 2k(N)n)\(2) can be expressed by simple analytical functions. To reconstruct P( k) from the alias sum Sigma(n)\W( k + 2k(N)n) \(2) P( k + 2k(N) n), we propose an iterative method. We test the method by applying it to an N-body simulation sample and show that the method can successfully recover P( k). The discussion is further generalized to samples with observational selection effects.