Abstract

In this talk I will present a theoretical framework that links a set of widely used methods from signal processing to statistical inference procedures. This result will then be used as a conceptual springboard to improve upon the classical methods.

I will begin by describing a family of related inference problems that have optimal solutions corresponding to the short-time Fourier transform (STFT), spectrogram, filter bank, and wavelet representations of signals. The framework allows us to use modern techniques from statistical inference to improve upon these classical signal processing methods.

I will show two examples where such an approach has borne fruit. In the first example we use an inferential extension of the Hilbert method to produce high-quality approaches to joint amplitude and frequency modulation of signals. The new approach is uncertainty-aware and therefore noise robust which results in fewer artifacts. In the second example we extend the STFT into an adaptive time-frequency analysis using a hierarchical probabilistic model. The parameters of the new representation, including the channel centre-frequencies and bandwidths, can be learned directly from the signal. The adaptive representation can be used to remove noise from signals and to impute missing data. Surprisingly, the method is an excellent model for naturally occurring audio textures such as howling wind, falling rain, and running water.

I will wrap up by discussing how we might bring the fields of signal processing and statistical inference closer together and the benefits and challenges of such a research effort.