The objective of this section is to relate advances in time-frequency analysis to auditory-based algorithms so as to understand the robustness of these algorithms. Our proposed research focuses on three specific aspects of time-frequency analysis and their implications to auditory processing: Quadratic detectors, positive time-frequency distributions, variational frequency.
Quadratic detectors. Signal processing algorithms based on a linear transform, such as a spectrogram, suffer performance limitations arising from the trade-offs between time and frequency resolution associated to the Fourier transform. To effectively decouple frequency selectivity from temporal response, there now exist filters based on quadratic signal processing (originally due to Cohen). It has been shown in [. Loughlin Atlas Bernard Pitton .] that, suitable quadratic detectors based on quadratic time-frequency representations are very effective in applications to speaker verification, manufacturing sensing , and communications decoding. We propose to refine the theory and design of such detectors, investigate the striking formal resemblance between such detectors and proposed binaural stereausis networks [.Shamma 1989 Gopalaswamy.], and adapt such detectors to a variety of monaural tasks such as pitch estimation.
Positive time-frequency distributions. Three classically desirable properties of a time-frequency representation are that both time and frequency marginals are satisfied and that the representation is non-negative-definite. We intentionally call these representations ``distributions,'' and have developed a theory for their implementation which is based upon minimum cross-entropy iterations. Starting from a conventionally blurred representation, these distributions show the best possible resolution in time and frequency with no significant cross-term distortion [. Atlas Pitton 1992.]. An interesting possibility that we propose to investigate is the applicability of the iterative sharpening of time and frequency resolution to provide a new modeling paradigm for auditory tuning. These distributions have already been applied to problems in manufacturing sensing, where the increased resolution has been essential to automatic machining monitoring systems. Clearly, these distributions will also play a significant role in acoustic transient classification problems that we plan to address ( Thrust area V).
Variational frequency. Coming straight from the theory of time-frequency representations is the notion of one- or two-dimensional Fourier transforms of the time-frequency plane. A one-dimensional transform along the time axis of a time-frequency plane results in a frequency/frequency plot where one axis is conventional Fourier frequency while the other axis is a form of frequency of non-stationarity. We call this latter frequency ``variational frequency.'' We shall investigate in detail this new and underutilized construct which concisely represents a tremendous amount of useful information about a signal. For example, acoustic-phonetic and prosodic information in speech occupy distinct and consistent regions along the variational frequency axis: Phonetic information changes at a rate of about 15 Hz and prosodic information changes at a rate closer to 4 Hz. For monitoring of rotating machinery, short-term diagnostic information appears around the rotational frequency (and its harmonics) yet trends indicating deterioration appear at several orders of magnitude lower in log variational frequency. The notion of variational frequency closely resembles the cortical multiscale representation, especially along the temporal dimension discussed earlier ( Thrust area II(A)). This is because the variational frequency representation is essentially a double-Fourier transform (with respect to the time-axis), which offers an alternative way to quantify and encode spectral change. These distributions will be applied to signals where the contextual environment is important, such as in machine monitoring, where short-term events and long-term trends need to be integrated ( Thrust area V(B)).