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

Y. Jing

Published 2004 in The Astrophysical Journal

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 ⟨|δf(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 ⟨|δf(k)|2⟩ to P(k). For a sample of N discrete objects, the formula reads ⟨|δf(k)|2⟩ = [|W( + 2kN)|2P( + 2kN) + 1/N|W( + 2kN)|2], where W(k) is the Fourier transform of the mass assignment function W(r), kN 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 |W( + 2kN)|2 can be expressed by simple analytical functions. To reconstruct P(k) from the alias sum |W( + 2kN)|2P( + 2kN), 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.

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