Transform theory and applications for engineering & technology students

This article presents a streamlined overview of Fourier, Laplace, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script">Z</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -transforms for a sophomore- or junior-level signals and systems course in electrical/computer engineering and engineering technology programs. Transform methods are introduced after foundational circuit analysis and serve as prerequisites to studies in communications, controls, power, and signal processing. Conventional textbooks define transforms formally and rely on tables of properties and transform pairs, often assuming prior mathematical preparation. However, this approach is challenging for engineering technology students, who may have completed only Calculus I and II yet are expected to meet ABET-driven learning outcomes requiring advanced mathematical competencies. To bridge the gap between formal theory and engineering applications, this article suggests an alternative instructional strategy that begins with an exposure to seven interconnected topics over one or two lectures, progressing from familiar to more complex material. These topics are revisited during the semester to reinforce understanding and contextualize applications. The sequence comprises: (i) logarithms and phasors; (ii) vector basis representation; (iii) polynomial functions; (iv) periodic functions and Fourier series; (v) Fourier transform for non-periodic functions; (vi) Laplace transform for functions not suited to Fourier analysis; and (vii) sampling and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script">Z</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -transform. By incrementally increasing conceptual complexity and leveraging visualization tools such as MATLAB, this approach demystifies transform theory and enhances students’ appreciation for the application of transforms in engineering contexts. The article offers a valuable resource for students, instructors, and textbook authors seeking a more incremental approach to transform theory instruction.