New Graphical Tools for Visualizing the Laplace and z-Domains

A new computer-based plotting method for the Laplace and z-transforms and network transfer functions is presented. Pixels in a bit-mapped plotting image are colored to represent the phase of the transform with a mapping relation. This graphical approach lets designers visualize more fully the mathematical properties of the entire complex frequency domain and exhibits several important numerical advantages over previous techniques. Closely separated singularities may be readily resolved with arbitrary precision, and the complex frequencies of poles and zeroes may be found graphically without any of the convergence issues commonly encountered with the iterative pole-zero analysis of simulation tools. Design of circuits and feedback systems with complicated high order loci of roots in the Laplace or z-domain can be aided with simple computational tools.