Theory, realization and design of one-dimensional and two-dimensional filters using delta operator

In many digital signal processing and wide-band communication applications, there is a need for sampling at rates much higher than the relevant underlying information and systems bandwidths. In such situations, the delta operator formulated discrete-time systems offer several advantages over the traditional shift-operator based discrete-time systems. The advantages arising out of this include better system accuracy and unification with the underlying continuous time system when the sampling period tends to zero. The γ complex frequency variable is the transform domain representation for the delta operator. The goal of this dissertation is to present new theories and designs for one and two-dimensional (1-D and 2-D) filters in this new γ-complex plane associated with the delta operator.; In Part I of the dissertation, a new theory and design procedure in γ-complex plane is developed for 1-D filters using lossless discrete time integrators (LDI). This procedure is a natural way of designing sampled data filters with integrators. In the development of the design process, this approach yields better numerical accuracy compared to the existing methods using shift-operator. The theory used is based on signal flow graphs with sensitivity properties similar to those which are derived from resistively terminated lossless two-ports. It is shown that these filters exhibit very low passband sensitivity to perturbation in coefficient and parameter values. In order to develop the design technique, a new form of continued fraction expansion for polynomials relating to delta operator is developed. Further, the dissertation presents lossless two-port theory in γ-complex variable.; In Part II of the dissertation, the design of 2-D discrete-time filters formulated in (γ₁,γ₂) complex variable is presented. The standard symmetries such as quadrantal, diagonal, four-fold rotational, and octagonal are defined for real and complex polynomials in (γ₁,γ₂). Further, the conditions that a given polynomial has to satisfy to possess the above symmetries are established. The results presented in the dissertation unify the symmetry properties of the discrete-time systems with the underlying continuous-time 2-D systems. So when one applies a limiting condition of the sampling period T tending to zero, the symmetry conditions of both systems coincide. This is not possible with the existing symmetry properties of shift-operator 2-D systems. Using the symmetry conditions, 2-D filters are designed and it is shown that when these filters are of narrow band in nature, they possess superior sensitivity properties compared to the designs based on 2-D shift-operator.