New class of affine higher order time-frequency representations

We propose a new class of affine higher order time-frequency representations (HO-TFRs) unifying HO-TFRs which satisfy the desirable properties of scale covariance and time-shift covariance. This new class extends to higher order (N\u3e2) the affine class of quadratic (N = 2) time-frequency representations. In this paper, we provide five alternative formulations of the class in terms of multi-dimensional smoothing kernels. We discuss important class members, including the new higher order scalogram that is related to the wavelet transform. We also list additional desirable properties and derive the associated kernel constraints. Finally, we consider a subclass of affine HO-TFRs that intersects with a Cohen\u27s class of time and frequency shift covariant HO-TFRs. A formulation for HO-TFRs satisfying three covariances in this higher order affine-Cohen intersection is derived