# Signal Processing Research

Signal processing seeks to answer questions like: What sinusoids are present in the signal at the current time? What is the power of the signal in a given frequency band at the current time? How are the frequencies of the signal changing at the moment? The solutions provided by signal processing to these three questions are called; time-frequency analysis, amplitude demodulation, and frequency demodulation. These methods have wide applicability including tasks like, speech recognition, removal of noise from audio, audio search, and voice manipulation. Typically these solutions are fast and efficient, but they are sensitive to noise and missing data. Normally, they are also fixed analyses which do not adapt to the signal.

 In my research I take a different view of signal processing which leads to adaptive methods that handle uncertainty automatically, and which are therefore robust to noise and missing data. The new perspective is to view signal processing as solving inference problems. For example, from this perspective time-frequency analysis is an inference about which sinusoids are present in the signal. The solution to this inference problem combines what is known before we see the signal (any prior information we have about the likely sinusoids) and what the data tells us (which sinusoids are consistent with the observed data). Often these problems involve estimating more quantities than there are data-points, and they are consequently ill-posed. For this reason, there isn't one solution to the problem, but a range of plausible estimates that are consistent with both prior knowledge and the observed data. Once the analysis are framed as inference problems, the machinery of probabilistic inference automatically provides methods for handling missing and noisy data and for adapting the parameters of the representations to match the statistics of the signal. For time-frequency analysis this means unevenly sampled data is automatically handled and the properties of the filters or wavelets can be adapted to the signal. The new view of signal processing feels quite different from classical perspective. However, many simple inference problems end up recovering standard signal processing algorithms. That is, signal processing and probabilistic inference are two sides of the same coin. Although inference problems are theoretically the most accurate way of solving estimation tasks, they are also often computationally intensive. However, the connection to traditional signal processing approaches allow us to borrow efficient methods and apply them to these inference problems.