The z-Transform for Initial Value Problems

Abstract

In the previous two chapters we investigated various boundary value problems associated with difference equations. In the process, we developed discrete transforms which are compatible with each kind of boundary value problem. This included the discrete Fourier transform (DFT) of Chapter 4 for periodic boundary conditions, the discrete sine transform (DST) and the discrete cosine transform (DCT) of Chapter 5, which are compatible, respectively, with the Dirichlet condition, i.e., when the value of the function is given at the two end points; and the Neumann condition where the slope (or the central difference) is given at the two end points. We now turn our attention to a different sort of problem in which difference equation may appear. This is the initial value problem which is often associated with time-dependent or evolutionary system.