Abstract
A modern challenge in electrical engineering education is to keep the math at a sufficient level, with a goal to find an optimal balance between calculus competence and operative skills needed for real-life technical applications. It is not uncommon that some gaps emerge during this quest, which makes it difficult for undergraduates to entirely understand topics related on prior knowledge. This paper aims to draw attention to several important moments concerning total response of Continuous Linear Time Invariant systems, which are superficially or incorrectly explained in many textbooks, and to offer logical arrangement which can be easily understood and accepted by students. The base of discussion relies on classical calculus background, particularly on Picard’s theorem on existence and uniqueness. This theorem is rarely mentioned in signals and systems textbooks. However, mathematical models of many types of signals don’t satisfy the condition for continuity, which can easily produce difficulties in the learning process. It is shown that some reported disagreements and issues related to initial conditions, can be easily cleared out by using the smooth transition from classical calculus to mathematics used in system theory. It is also shown that the classical method is always the primary tool, even for determining the impulse response, while the impulse response is unnecessary or sufficient to determine the total system response, regardless of whether the convolution integral is used.
Highlights
After an introductory discussion about general properties of signals and systems, typical undergraduate signals and systems course accentuate the importance of continuous Linear Time Invariant (LTI) systems for which the input and output are related through the constant-coefficient linear differential equations (DE) [1]–[7]
It is shown that both methods, with some augmentation as presented in the paper, offer the same outcome, and that both methods possess the same possibilities in determination of total system response
Even when convolution integral is used for total response determination, the classical method is indispensable: it is used for determination of h(t)
Summary
After an introductory discussion about general properties of signals and systems, typical undergraduate signals and systems course accentuate the importance of continuous Linear Time Invariant (LTI) systems for which the input (excitation) and output (response) are related through the constant-coefficient linear differential equations (DE) [1]–[7]. In order to facilitate acceptance of GC with application to time-domain analysis of signals and systems, two main approaches are used: application of classical method which utilizes undergraduate calculus tools with the slight impact of GC, and the convolution method - which is strongly based on impulse response concept followed by a convolution integral. If both of them are taught carelessly in the same course, or in a group of related courses, they can raise more questions than they answer. Discussion in the paper is performed with application to SISO continual systems, while a conclusion for discrete case and MIMO systems can be done by analogy
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