Some potential pitfalls with s to z-plane mappings

Design of digital infinite impulse response (IIR) filters is a compulsory topic in most signal processing courses. Most often, it is taught by using the bilinear transform to map an analogue counterpart into the corresponding digital filter. The usual approach is to define a mapping between the complex variables s and z, and hence, by substitution, derive a mapping between ω, analogue frequency, and θ, sampled frequency. This is rather elliptical, since the real aim is to establish the correspondence between the frequency response of a prototype analogue system H(jω), and H(e^jθ), the response of the sampled system. Here we provide a rigorous analysis for the mutual invertibility between the analogue frequency ω, and the digital frequency θ for this case. Based upon the definition of the tan and arctan functions, conditions of existence, uniqueness and continuity of such a mutually inverse mapping are derived. Based upon these results, simple proofs for the mutually inverse mappings ω→θ and θ→ω are given. This is supported by appropriate diagrams. This problem arose as a student question while teaching DSP