Correcting for the Alias Effect When Measuring the Power Spectrum Using a Fast Fourier Transform
Year: 2005
Type: article
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 ⟩ = [| W ( + 2 k N )| 2 P ( + 2 k N ) + 1/ N | W ( + 2 k 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 | W ( + 2 k N )| 2 can be expressed by simple analytical functions. To reconstruct P ( k ) from the alias sum | W ( + 2 k N )| 2 P ( + 2 k 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.
Source: The Astrophysical Journal
DOI: 10.1086/427087
Author: Y. P. Jing
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1993956748