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 <δ^{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 <δ^{f}(k)^{2}> to P(k). For a sample of N discrete objects, the formula reads <δ^{f}(k)^{2}>=Σn[W(k+2k_{N}n)^{2}P(k+2k_{N}n)+1/NW(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, cloudincell, and triangularshaped cloud assignment functions, we show that the shotnoise term Σn(1/N)W(k+2k_{N}n)^{2} can be expressed by simple analytical functions. To reconstruct P(k) from the alias sum ΣnW(k+2k_{N}n)^{2}P(k+2k_{N}n), we propose an iterative method. We test the method by applying it to an Nbody simulation sample and show that the method can successfully recover P(k). The discussion is further generalized to samples with observational selection effects.
 Publication:

The Astrophysical Journal
 Pub Date:
 February 2005
 DOI:
 10.1086/427087
 arXiv:
 arXiv:astroph/0409240
 Bibcode:
 2005ApJ...620..559J
 Keywords:

 Galaxies: Clusters: General;
 Cosmology: LargeScale Structure of Universe;
 Methods: Data Analysis;
 Methods: Statistical;
 Astrophysics
 EPrint:
 12 pages, 2 figures