Complete parametrization of synthesis in multidimensional perfect reconstruction FIR systems

Many problems in FIR filter banks, via the method of polyphase representation, can be characterized by their transfer matrices. In the case of multidimensional (MD) FIR filter banks, the resulting transfer matrices have Laurent polynomial entries in several variables. For a given analysis filter bank, this method produces a multi-input multi-output (MIMO) system (e.g. an oversampled filter bank). Since such a system does not have a unique perfect reconstruction (PR) pair in general, there exists a certain degree of freedom in designing a synthesis system to make the overall system a PR system. In this paper, for a given analysis system, the author gives a complete parametrization of all the synthesis systems that ensure perfect reconstruction of inputs, which gives us a space to search for an optimal synthesis system. This involves an adaptation of Schreyer´s algorithm to carry out syzygy computations with Laurent polynomials