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Relating Continuous-Time and Discrete-Time in the Classroom

Abstract It never ceases to be difficult to communicate to students the relationships between continuous- time and discrete-time systems, and their models. Those relationships span both the frequency domain and the time-domain, and are complicated by several fundamental differences, such as the stability boundaries in the frequency planes, and the nonlinearity and non-uniqueness of the mapping z = esT.

This paper presents a Matlab®-based method that helps instructors to graphically demonstrate important relationships between continuous-time and discrete-time systems. The author has been using this method successfully for several years in a variety of systems-related courses to help students understand these relationships. The software is also freely available to students, so they may reconstruct classroom demonstrations, and do much more.

In the demonstrations, frequency-response plots, pole/zero maps, and time-domain response plots in both domains are easily created and modified, and are programmatically linked. The plots are also annotated in a variety of ways to help emphasize relationships among them. All of the plots have interactive capability, and display much more information than simple line graphs. Making changes in any of the plots automatically causes appropriate changes to be made in all the other plots. This removes from the instructor the burden of creating, managing, and updating interrelated plots.

This method can be used, for example, to enhance discussions of (1) zero-order-hold and bilinear transformations between continuous-time and discrete-time domains, (2) graphical (vector) analysis of the Fourier transforms, (3) aliasing in the frequency and time domains, and (4) the nonlinearity and non-uniqueness of the mapping z = esT.

Complete details of the method will be presented, and the software is freely available to everyone, for educational purposes, in the form of a Matlab® toolbox.

1. Introduction The ability to use computers at the lectern, enabled by relatively cheap projection equipment, has tremendous potential for computer-aided teaching. This is not a reference to being able to present PowerPoint slides, using computers in essentially the same way we would use overhead projectors to show static transparencies. Rather, we now have the ability to create accurate mathematical plots on-the-fly, and dynamically manipulate graphical content to emphasize points of discussion.

This paper is about realizing that potential for the purpose of teaching the relationships between continuous-time (C-T) and discrete-time (D-T) systems. This is an area that requires a teacher to present several different types of plots – time-domain response plots, frequency-response (e.g., Bode) plots, and pole/zero maps in the s-plane and z-plane – and to discuss their inter- relatedness.

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Hopkins, M. (2008, June), Relating Continuous Time And Discrete Time In The Classroom Paper presented at 2008 Annual Conference & Exposition, Pittsburgh, Pennsylvania. 10.18260/1-2--4417