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

• Published in: The Astrophysical journal Vol. 620; no. 2; pp. 559 - 563
• Main Author: Jing, Y. P
• Format: Journal Article
• Language: English
• Published: Chicago, IL IOP Publishing 20-02-2005; University of Chicago Press
• Subjects: Galaxy clusters; Statistical data; Statistical method; Data analysis; galaxies: clusters: general; Statistical analysis; Fast Fourier transforms; Power spectra; methods: data analysis -methods: statistical; Large-scale structure; large-scale structure of universe
• ISSN: 0004-637X; 1538-4357
• DOI: 10.1086/427087

Because of mass assignment onto grid points in the measurement of the power spectrum using a fast Fourier transform (FFT), the raw power spectrum <"d super(f)(k)" super(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 <"d super(f)(k)" super(2)> to P(k). For a sample of N discrete objects, the formula reads <"d super(f)(k)" super(2)> = n ["W( super(k) + 2k super(n) sub(N))" super(2)P( super(k) + 2k super(n) sub(N)) + 1/N"W( super(k) + 2k super(n) sub(N))" super(2)], where W(k) is the Fourier transform of the mass assignment function W(r), k sub(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 n (1/N) "W( super(k) + 2k super(n) sub(N))" super(2) can be expressed by simple analytical functions. To reconstruct P(k) from the alias sum n "W( super(k) + 2k super(n) sub(N))" super(2)P( super(k) + 2k super(n) sub(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.