Quadratic Time-Frequency Representations with Scale Covariance and Generalized Time-Shift Covariance: A Unified Framework for the Affine, Hyperbolic, and Power Classes

We propose the generalized class of quadratic time-frequency representations (QTFRs) that satisfy the scale covariance property, which is important in multiresolution analysis, and the generalized time-shift covariance property, which is important in the analysis of signals propagating through systems with specific dispersive characteristics. We discuss a formulation of the generalized class QTFRs in terms of two-dimensional kernel functions, a generalized signal expansion related to the generalized class time-frequency geometry, an important member of the generalized class, a set of desirable QTFR properties and their corresponding kernel constraints, and a "localized-kernel" generalized subclass that is characterized by one-dimensional kernels. Special cases of the generalized QTFR class include the affine class and the new hyperbolic class and power classes. All these QTFR classes satisfy the scale covariance property. In addition, the affine QTFRs are covariant to constant time shifts, the hyperbolic QTFRs are covariant to hyperbolic time shifts, and the power QTFRs are covariant to power time shifts. We present a detailed study of these classes that includes their definition and formulation, an associated generalized signal expansion, important class members, desirable QTFR properties and corresponding kernel constraints, and localized-kernel subclasses. Also, we investigate the subclasses formed by the intersection between the affine and hyperbolic classes, the affine and power classes, and the hyperbolic and power classes. These subclasses are important since their members satisfy additional desirable properties. We show that the hyperbolic class is obtained from Cohen's QTFR class using a "hyperbolic time-frequency warping" and that the power classes are obtained similarly by applying a "power time-frequency warping" to the affine class. The affine class is a special case of the power classes. Furthermore, we generalize the time-frequency warping so that when applied either to Cohen's class or to the affine class, it yields QTFRs that are always generalized time-shift covariant but not necessarily scale covariant.