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Bayes’ Rule Using Imprecise Probabilities

Bayes’ rule, as one of the fundamental concepts of statistical signal processing, provides a way to update our belief about an event based on the arrival of new pieces of evidence. Uncertainty is traditionally modeled by a probability distribution. Prior belief is thus expressed by a prior probability distribution, while the update involves the likelihood function, a probabilistic expression of how likely it is to observe the evidence.

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Discriminative and Generative Learning for the Linear Estimation of Random Signals

Inference tasks in signal processing are often characterized by the availability of reliable statistical modeling with some missing instance-specific parameters. One conventional approach uses data to estimate these missing parameters and then infers based on the estimated model. Alternatively, data can also be leveraged to directly learn the inference mapping end to end. These approaches for combining partially known statistical models and data in inference are related to the notions of generative and discriminative models used in the machine learning literature [1] , [2] , typically considered in the context of classifiers.

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Proper Definitions of Dirichlet Conditions and Convergence of Fourier Representations

Fourier theory is the backbone of signal processing (SP) and communication engineering. It has been widely used in almost all branches of science and engineering in numerous applications since its inception. However, Fourier representations such as Fourier series (FS) and Fourier transform (FT) may not exist for some signals that fail to fulfill a predefined set of Dirichlet conditions (DCs). 

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Synthesis of Fast-Decaying Window Functions

A window function is a mathematical function that is zero valued outside some chosen interval [1] , [2] . For applications like filtering, detection, and estimation, the window functions take the form of limited time functions, which are in general real and even functions [3] , [4] , while for applications like beamforming and image processing, they are limited spatial functions. A spatial window can be a complex function for optimizing the beams in magnitude as well as in phase, as in the case of certain antenna arrays, where the phasor currents in the array are complex numbers [5].

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Sliding Discrete Fourier Transform with Kernel Windowing

The sliding discrete Fourier transform (SDFT) is an efficient method for computing the N-point DFT of a given signal starting at a given sample from the N-point DFT of the same signal starting at the previous sample [1]. However, the SDFT does not allow the use of a window function, generally incorporated in the computation of the DFT to reduce spectral leakage, as it would break its sliding property.

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