Correcting for the alias effect when measuring the power spectrum using FFT

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$P(k)$\mathrm{.}Forasampleof$N$ 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 ${\sum }_{\vec{n}}1\mathrm{/}N\mathrm{\mid }W(\text{kalias}){\mathrm{\mid }}^2]$ 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.