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 the Fast Fourier Transform (FFT), the raw power spectrum $\la |\delta^f(k)|^2\ra$ estimated with FFT is not the same as the true power spectrum $P(k)$. In this paper, we derive the formula which relates $\la |\delta^f(k)|^2\ra$ to $P(k)$. For a sample of $N$ discrete objects, the formula reads: $\la |\delta^f(k)|^2\ra=\sum_{\vec n} [|W(\kalias)|^2P(\kalias)+1/N|W(\kalias)|^2]$, where $W(\vec k)$ is the Fourier transform of the mass assignment function $W(\vec r)$, $k_N$ is the Nyquist wavenumber, and $\vec n$ is an integer vector. The formula is different from that in some of previous works where the summation over $\vec n$ is neglected. For the NGP, CIC and TSC assignment functions, we show that the shot noise term $\sum_{\vec n} 1/N|W(\kalias)|^2]$ can be expressed by simple analytical functions. To reconstruct $P(k)$ from the alias sum $\sum_{\vec n}|W(\kalias)|^2 P(\kalias)$, 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.