z-Transforms

Abstract

The Laplace transform plays an important role in the analysis of analogue signals or systems, since it uses a generalised complex frequency variable s = ± σ ± jω, with σ describing amplitude growth and decay of the sinusoidal signal having a radian frequency of ω,. However, complications arise in using the s-plane representation to analyse a sampled signal or sampled-data system due to their characteristic infinite number of complementary frequency spectra. Let us consider a sinusoidal signal cosω _b t which, using Euler’s identity (e^±jθ = cosθ ± jsinθ), can be expressed as

$$\cos \,\;{\omega _b}t = \frac{1}{2}\left( {{e^{j{\omega _b}t}} + {e^{ - j{\omega _b}t}}} \right)$$