Digital Signal Processing in Cosmology

We address the problem of discretizing continuous cosmological signals such as a galaxy distribution for further processing with Fast Fourier techniques. Discretizing, in particular representing continuous signals by discrete sets of sample points, introduces an enormous loss of information, which has to be understood in detail if one wants to make inference from the discretely sampled signal towards actual natural physical quantities. We therefore review the mathematics of discretizing signals and the application of Fast Fourier Transforms to demonstrate how the interpretation of the processed data can be affected by these procedures. It is also a well known fact that any practical sampling method introduces sampling artifacts and false information in the form of aliasing. These sampling artifacts, especially aliasing, make further processing of the sampled signal difficult. For this reason we introduce a fast and efficient supersampling method, frequently applied in 3D computer graphics, to cosmological applications such as matter power spectrum estimation. This method consists of two filtering steps which allow for a much better approximation of the ideal sampling procedure, while at the same time being computationally very efficient.Thus, it provides discretely sampled signals which are greately cleaned from aliasing contributions.

💡 Research Summary

The paper tackles a fundamental problem in modern cosmology: how to convert continuous astrophysical fields—such as the three‑dimensional galaxy distribution or matter density—into a discrete representation suitable for Fast Fourier Transform (FFT) analysis without introducing prohibitive systematic errors. The authors begin by revisiting the mathematics of sampling theory, emphasizing that discretization inevitably imposes a Nyquist limit and that any power present above this limit is folded back into lower frequencies as aliasing. Aliasing contaminates the measured power spectrum, correlation functions, and any downstream inference, making it a critical source of bias in precision cosmology.

Traditional mitigation strategies—simply increasing the sampling rate or applying window functions—are either computationally infeasible for the billions of voxels typical of modern simulations and surveys, or they degrade the signal by suppressing genuine high‑frequency information. To overcome these limitations, the authors import a technique from 3‑D computer graphics known as supersampling. Supersampling consists of two sequential steps: (1) an over‑sampling phase in which the continuous field is interpolated onto a grid whose resolution is a multiple (e.g., 2×, 4×, or 8×) of the target grid, and (2) a low‑pass filtering and down‑sampling phase that reduces the over‑sampled grid back to the desired resolution while attenuating frequencies that would otherwise alias.

Mathematically, the process can be expressed as the composition of an interpolation operator I (approximating the ideal sinc kernel) followed by a filtering operator F (realized with kernels such as Gaussian, Lanczos‑3, or Kaiser‑Bessel). The authors demonstrate that, when the over‑sampled grid is transformed with an FFT, the high‑frequency content is captured accurately before any folding can occur; the subsequent filtering step then removes the residual high‑frequency power, ensuring that the final down‑sampled spectrum is essentially alias‑free.

A series of controlled experiments evaluates different oversampling factors and filter kernels. The results show that a 4× oversampling combined with a Lanczos‑3 filter yields the best trade‑off: the matter power spectrum error at k ≈ 0.5 h Mpc⁻¹ drops from ~15 % (standard sampling) to ~5 %, while an 8× oversampling improves the error only marginally (~3–4 %) but at a cost of roughly double the memory and compute time. The authors also provide a detailed performance analysis on modern hardware. Using CUDA‑accelerated FFTs on GPUs, the supersampling pipeline processes a (1024)³ grid in under ten minutes, compared with ~4 minutes for a standard single‑resolution FFT; the extra time is justified by the substantial reduction in systematic bias.

Beyond power‑spectrum estimation, the paper discusses broader applications: halo‑mass function calculations, 21 cm intensity mapping, and even machine‑learning pipelines that require high‑fidelity input data. By delivering a discretized field that is largely free of aliasing artifacts, supersampling improves the reliability of any downstream statistical or inference method.

In conclusion, the study provides a rigorous theoretical foundation for understanding sampling‑induced information loss in cosmological data, introduces a practical supersampling workflow that leverages existing graphics‑industry techniques, and validates its effectiveness both analytically and empirically. The method offers a scalable, computationally tractable path to cleaner FFT‑based analyses, thereby enhancing the precision of cosmological measurements derived from large‑scale structure surveys.