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Generalization of the Weyl symbol and the spreading function via time -frequency warpings: Theory and application

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  • Engineering
  • Electronics And Electrical

Abstract

We propose a new generalized formulation of the narrowband Weyl correspondence to obtain the generalized Weyl symbol (WS) and the generalized spreading function (SF) of a linear operator. The generalized WS and SF are obtained from the conventional WS and SF via warping operators; they are useful tools for analyzing linear time-varying (LTV) systems and random processes characterized by dispersive frequency shifts. Using the generalized SF, we interpret a system output as the weighted superposition of generalized frequency shifted versions of the input signal, where the weight is the generalized SF. We provide various special cases of the generalized WS and SF matched to specific time-frequency (TF) characteristics, e.g. hyperbolic, power, exponential WS. We demonstrate the advantages of the generalized WS and SF by providing analysis and detection examples. ^ We extend the Shenoy and Parks' wideband version of the Weyl correspondence by providing a formulation for the wideband Weyl symbol (P0WS) based on the unitary Bertrand P0-distribution. We also propose a new generalized formulation of the wideband Weyl correspondence. We define the generalized P0WS and generalized wideband spreading function (WSF), which are useful tools for analyzing dispersive time shifts in LTV systems and random processes. We derive properties of the generalized P0WS, and provide special cases of the generalized P0WS and WSF. Analysis and detection application examples demonstrate the importance of the new formulations over the conventional narrowband WS. ^ We extend the idea of the narrowband WS and the P0WS using smoothing kernels. We propose classes of time-frequency symbols that satisfy common covariance properties. We define these new symbols as TF smoothed versions of the narrowband WS (TF shift covariant symbols) or as affine smoothed versions of the narrowband WS or of the P0WS (affine TF symbols). The smoothing depends on a kernel function that uniquely characterizes each TF symbol in a class. We also define the unitarity property of TF shift covariant symbols and affine TF symbols and derive the corresponding kernel constraints. We provide examples of unitary TF shift covariant symbols and unitary affine TF symbols. We generalize the class of TF shift covariant symbols and the class of affine TF symbols. ^

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