Explaining Convolution Using MATLAB

Students often have a difficult time understanding what convolution is. Students can often evaluate the convolution integral (continuous time case), convolution sum (discrete-time case), or perform graphical convolution but may not have a good grasp of what is happening. In other words, students can solve the formula but often do not understand the result or why they get that result. Most engineering texts explain convolution by giving the convolution integral (and/or convolution sum) and doing some mathematical and graphical examples. They often do not attempt to explain how convolution corresponds with what is happening between the system and the input to give the output response. In this paper, a more intuitive explanation of convolution is given and MATLAB and SIMULINK simulations of physical systems are used to give a more intuitive approach to understanding convolution from a systems perspective. Introduction Students are often introduced to convolution before they see the use for it. In most Electrical Engineering curriculums, convolution is introduced in sophomore or junior level signals and systems courses. Convolution is often performed numerically and students have a tendency to blindly accept the results their calculator or computer provides. Thus it is important for students to understand the use, along with the theory of convolution, so they can better evaluate the results they get from convolution. If convolution is explained from a systems perspective with good examples, they will see the use, and hopefully understand the theory and be able to determine if their results make sense. I have found several interactive web lectures/tutorials on convolution [1][2][3]. These web pages give a good definition of convolution and a good explanation of how to compute the convolution of two signals or functions. This paper proposes explaining convolution from a systems perspective using simulations of familiar systems. It is hoped that this will give students insight into what is happening with convolution. The system simulation examples are performed using MATLAB and SIMULINK. MATLAB and SIMULINK are used extensively in Electrical Engineering programs for signals and systems courses, control systems courses, and signal processing courses to name a few. MATLAB and SIMULINK work well for simulating most electrical, mechanical, and chemical systems. Basically any system that can be modeled in transfer function form, by differential equations, or state variable equations can be simulated using MATLAB and SIMULINK. The paper is organized as follows: the definition of convolution along with some system theory is provided, methods of computing convolution are discussed, a method of explaining convolution using familiar systems is proposed, and suggestions are provided for how to incorporate this in the classroom. 1 Assistant Professor of Computer Science, Georgia Southern University, P.O. Box 8093, Statesboro, GA, 30460 8093.