Time-frequency representations (TFRs) extend traditional Fourier analysis by mapping a time-domain process into a function of time and frequency, which allows us to clearly see how the spectral characteristics of non-stationary signals vary with time. TFRs have been useful in many applications such as communications, radar signal processing, and speech signal analysis. In the presence of additive noise processes that are characterized as impulsive, however, TFRs rapidly degrade and are no longer able to resolve any time-frequency characteristics of the underlying signals. To date, no work has been done to characterize the effect of additive impulses on TFRs, nor has there been any real effort to develop methods for making TFRs more robust even though additive impulsive processes can be found arising from many diverse sources, such as lightning effects, car ignitions, and fluorescent lighting fixtures. In this dissertation, we address both of these issues, by first developing a comprehensive characterization of the effects of impulses on a variety of TFR types using simple models that yield a tractable analysis. We then propose two approaches to increasing the robustness of TFRs, both based on existing impulse mitigation methods. One approach uses phase-preserving fractional lower order moments to modify the signal prior to applying a standard TFR operation. The other approach modifies the short-time Fourier transform, which can be thought of as a sequential combination of modulation and filtering operations, by replacing the linear filter operation with a robust, nonlinear matched myriad filter. Both of these approaches produce TFRs that can be tuned to compensate for a wide variety of additive noise types, from Gaussian noise to very heavy-tailed noise distributions. As a result, they include the standard TFRs as special cases. In addition, the TFRs produced by both of these approaches retain many of the desirable properties of standard TFRs, such as time-frequency invariance. We demonstrate the utility of both of these approaches by comparing their performance to that of standard TFRs for the case of a measured signal corrupted by measured impulsive atmospheric noise.
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