Unified Pascal Matrix for First-Order $s{\hbox {\textendash }}z$ Domain Transformations

The so-called generalized Pascal matrix is used for transforming a continuous-time (CT) linear system (filter) into a discrete-time (DT) one. This paper derives an explicit expression for a new generalized Pascal matrix called unified Pascal matrix from a unified first-order S-to-Z transformation model and rigorously proves the inverses for various first-order s-to-z transformations. After deriving a recurrence formula for recursively generating the inner elements of the unified Pascal matrix from its boundary elements, we also show that the recurrence formula leads to computationally unstable solutions for high-order systems due to the so-called catastrophic cancellation in numerical computation, but the unstable problem can be solved through partitioning the whole unified Pascal matrix into several small matrices (submatrices) and then using the recurrence formula to compute the submatrices individually from their boundary elements. This operation almost retains the same computational complexity while guarantees numerically stable solutions. Moreover, an interesting property of the unified Pascal matrix is proved.