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 represent ations (QTFRs) that satisfy the scale covariance property, which is im portant in multiresolution analysis, and the generalized time-shift co variance property, which is important in the analysis of signals propa gating through systems with specific dispersive characteristics. We di scuss a formulation of the generalized class QTFRs in terms of two-dim ensional kernel functions, a generalized signal expansion related to t he generalized class time-frequency geometry, an important member of t he generalized class, a set of desirable QTFR properties and their cor responding kernel constraints, and a 'localized-kernel'' generalized s ubclass that is characterized by one-dimensional kernels. Special case s of the generalized QTFR class include the affine class and the new h yperbolic 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 hyperb olic time shifts, and the power QTFRs are covariant to power time shif ts. We present a detailed study of these classes that includes their d efinition and formulation, an associated generalized signal expansion, important class members, desirable QTFR properties and corresponding kernel constraints, and localized-kernel subclasses. Also, we investig ate the subclasses formed by the intersection between the affine and h yperbolic classes, the affine and power classes, and the hyperbolic an d power classes. These subclasses are important since their members sa tisfy additional desirable properties. We show that the hyperbolic cla ss is obtained from Cohen's QTFR class using a ''hyperbolic time-frequ ency warping'' and that the power classes are obtained similarly by ap plying a ''power time-frequency warping'' to the affine class. The aff ine class is a special case of the power classes. Furthermore, we gene ralize the time-frequency warping so that when applied either to Cohen 's class or to the affine class, it yields QTFRs that are always gener alized time-shift covariant but not necessarily scale covariant. (C) 1 998 Academic Press.