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

Because of mass assignment onto grid points in the measurement of the power spectrum using the Fast Fourier Transform (FFT), the raw power spectrum |ᶠ (k) |² estimated with FFT is not the same as the true power spectrum P (k). In this paper, we derive the formula which relates |ᶠ (k) |² to P (k). For a sample of N discrete objects, the formula reads: |ᶠ (k) |²= ₍ |W () |²P () +1/N|W () |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 of previous works where the summation over n is neglected. For the NGP, CIC and TSC assignment functions, we show that the shot noise term ₍ 1/N|W () |²] can be expressed by simple analytical functions. To reconstruct P (k) from the alias sum ₍|W () |² P (), 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.