A simplified approach to Bode´s theorem for continuous-time and discrete-time systems

A simplified approach to W.H. Bode´s (1945) theorem for both continuous-time and discrete-time systems, along with some generalization, are presented. For continuous-time systems, the constraints of open-loop stability and roll-off at s=∝ are removed. A counterexample shows that when the excess poles/zeros vanishes, the Bode integral drops from infinite to finite value when the open-loop gain crosses a critical value. A revised result is also developed. The salient feature of this approach is that at no stage are either Cauchy´s theorem or the Poisson integral invoked; the simplified proof relies only on elementary analysis. This approach carries over to the discrete-time cases in a straightforward manner