Low-pass filtering as Bayesian inference

This is because the concentration of energy at a specific range of frequencies might be indicative of mechanical faults [1], cardiac anomalies [2], astronomical discoveries [3, 4], and whale calls from submarine audio recordings [5] to name a few. The standard practice to isolate components within a specific frequency range from a time-series observation, referred to as filtering, isto convolve the observations with an object called linear filter. This convolution removes all frequencies that do not correspond to the desired frequency range, thus, filtering out unimportant frequencies. Thetheoretical rationale behind this approach is supported by the application of the Convolution Theorem [6] to power spectral densities (PSD): the PSD of a filtered time series corresponds to the PSD of the linear filter (user-designed) multiplied by the PSD of the observed time series (not controllable). This result allows for designing thelinear filter so as to remove unwanted frequency components to then perform the numerical convolution.