Mathematical Foundations of Nonlinear, Non-Gaussian, and Time-Varying Digital Speech Signal Processing

Classical digital speech signal processing assumes linearity, timeinvariance, and Gaussian random variables (LTI-Gaussian theory). In this article, we address the suitability of these mathematical assumptions for realistic speech signals with respect to the biophysics of voice production, finding that the LTI-Gaussian approach has some important accuracy and computational efficiency shortcomings in both theory and practice. Next, we explore the consequences of relaxing the assumptions of time-invariance and Gaussianity, which admits certain potentially useful techniques, including wavelet and sparse representations in computational harmonic analysis, but rules out Fourier analysis and convolution, which could be a disadvantage. Then, we focus on methods that retain time-invariance alone, which admits techniques from nonlinear time series analysis and Markov chains, both of which have shown promise in biomedical applications. We highlight recent examples of non-LTI-Gaussian digital speech signal processing in the literature, and draw conclusions for future prospects in this area.