| We extend the theory of multirate linear digital signal processing to Volterra filters. We show that they have the same multirate properties as linear systems and we propose some inherent applications. In the first part of this dissertation, we construct the theoretical tools we need to study multirate Volterra systems. We extend to the discrete time domain the notion of basic second order Volterra system with one multiplier and its Z-transform is computed in terms of its first order components. pth order kernels are then inferred in terms of lower order components. Then, the equivalent of a tandem connection of Finite Memory Span (FMS) Volterra kernels is computed. Finally, we establish the constraints a p th order Volterra kernel must satisfy in order for it to annihilate a polynomial of degree "d".; In the second part of this dissertation, we build on the theoretical foundation of the first part to study the multirate properties of Volterra systems. We show that Volterra systems satisfy the Noble Identities. After, We extend the notion of half-band filters to homogeneous Volterra systems. We prove that polyphase decomposition of Volterra kernels is possible. Interestingly, as a corollary, we propose a new class of Volterra filters that annihilates any upsampled signal. Then, we construct half-band second order kernels that transform an interpolated polynomial of degree "d" into a polynomial of degree "2.d". We also construct second order homogeneous Volterra systems that, not only conserve polynomials, but also conserves the degree of the polynomials. We show that those filters are very smooth interpolators of an upsampled signal or of an upsampled image. Furthermore, we establish the Perfect Reconstruction equations of second order Volterra filter banks and we show that those equations have at least a solution. Finally, using the corollary of the polyphase decomposition of Volterra kernels, we introduce a new class of second order Volterra systems that can be equalized quasi exactly. |
