The aim of the book is to give an accessible introduction of mathematical models and signal processing methods in speech and hearing sciences for senior undergraduate and beginning graduate students with basic knowledge of linear algebra, differential equations, numerical analysis, and probability. Speech and hearing sciences are fundamental to numerous technological advances of the digital world in the past decade, from music compression in MP3 to digital hearing aids, from network based voice enabled services to speech interaction with mobile phones. Mathematics and computation are intimately related to these leaps and bounds. On the other hand, speech and hearing are strongly interdisciplinary areas where dissimilar scientific and engineering publications and approaches often coexist and make it difficult for newcomers to enter.

A model based sound amplification method is proposed and studied to enhance the ability of the hearing impaired. The model consists of mechanical equations on basilar membrane and outer hair cell (OHC). The OHC is described by a nonlinear nonlocal feedforward model. In addition, a perceptive correction is defined to account for the lumped effect of higher level auditory processing, motivated by the intelligibility function of the hearing impaired. The gain functions are computed by matching the impaired model output to the perceptively weighted normal output, and qualitative agreement is achieved with NAL-NL1 prescription on clean signals. For noisy signals, an adaptive gain strategy is proposed based on the signal to noise ratios (SNR) computed by the model. The adaptive gain functions provide less gain as SNRs decrease so that the intelligibility can be higher with the adaptivity.

We study reactive front speeds in randomly perturbed cellular flows using a variational representation for the front speed. We develop this representation into a computational tool for computing the front speeds without resorting to closure approximations. We demonstrate that the front speeds depend on flow statistics and topologies in a complex and dramatic manner.

A nonlocally weighted soft-constrained natural gradient iterative method is introduced for robust blind separation in reverberant environment. The nonlocal weighting of the iterations promotes stability and convergence of the algorithm for long demixing filters. The scaling degree of freedom is controlled by soft-constraints built into the auxiliary difference equations. The small divisor problem of iterations in silence durations of speech is resolved. Computations on synthetic speech mixtures based on measured binaural room impulse responses show that the algorithm achieves higher signal-to-inteference ratio improvement than existing method (natural gradient time domain algorithm) in an office size room with reverberation time over 0.5 second.

We study a class of nonlinear nonlocal cochlear models of the transmission line type, describing the motion of basilar membrane (BM) in the cochlea. They are damped dispersive partial differential equations (PDEs) driven by time dependent boundary forcing due to the input sounds. The global well-posedness in time follows from energy estimates. Uniform bounds of solutions hold in the case of bounded nonlinear damping. When the input sounds are multi-frequency tones, and the nonlinearity in the PDEs is cubic, we construct smooth quasi-periodic solutions (multi-tone solutions) in the weakly nonlinear regime, where new frequencies are generated due to nonlinear interaction. When the input consists of two tones at frequencies f 1, f 2 (f 1< f 2), and high enough intensities, numerical results illustrate the formation of combination tones at 2f 1 f 2 and 2f 2 f 1, in agreement with hearing experiments. We visualize