Band-Limited Gaussian Processes: The Sinc Kernel
This provides a Bayesian framework for signal processing tasks, offering robustness to noise and missing data, but it is incremental as it builds on existing spectral design literature.
The authors introduced a new class of Gaussian processes with band-limited spectra using a sinc kernel, enabling applications in regression and signal processing like stereo amplitude modulation and band-pass filtering, with experimental validation on real-world data.
We propose a novel class of Gaussian processes (GPs) whose spectra have compact support, meaning that their sample trajectories are almost-surely band limited. As a complement to the growing literature on spectral design of covariance kernels, the core of our proposal is to model power spectral densities through a rectangular function, which results in a kernel based on the sinc function with straightforward extensions to non-centred (around zero frequency) and frequency-varying cases. In addition to its use in regression, the relationship between the sinc kernel and the classic theory is illuminated, in particular, the Shannon-Nyquist theorem is interpreted as posterior reconstruction under the proposed kernel. Additionally, we show that the sinc kernel is instrumental in two fundamental signal processing applications: first, in stereo amplitude modulation, where the non-centred sinc kernel arises naturally. Second, for band-pass filtering, where the proposed kernel allows for a Bayesian treatment that is robust to observation noise and missing data. The developed theory is complemented with illustrative graphic examples and validated experimentally using real-world data.