Weyl-Heisenberg Systems and the Finite Zak Transform

Abstract. Previously, a theoretical foundation for designing algorithms for computing Weyl-Heisenberg (W-H) coefficients at critical sampling was established by applying the finite Zak transform. This theory established clear and easily computable conditions for the existence of W-H expansion and for stability of computations. The main computational task in the resulting algorithm was a 2-D finite Fourier transform. In this work we extend the applicability of the approach to rationally over-sampled W-H systems by developing a deeper understanding of the relationship established by the finite Zak trans-form between linear algebra properties of W-H systems and function theory in Zak space. This relationship will impact on questions of existence, parameterization, and computation of W-H expansions. Implementation results on single RISC processor of i860 and the PARAGON parallel multiprocessor system are given. The algorithms described in this paper possess highly parallel structure and are especially suited in a distributed memory, parallel-processing environment. Timing results show that real-time computation of W-H expansions is realizable.