Improving Conceptual Understanding of Signals and Systems in Undergraduate Engineering Students Using Collaborative In-Class Laboratory Exercises

Three MATLAB-based in-class collaborative laboratory exercises were introduced in conjunction with the traditional lecturing and problem-solving techniques in the signals and systems course at Vanderbilt University. These labs were developed to enhance students’ conceptual understanding and retention using MATLAB simulations of audio synthesis and processing, using guitar notes as signals and processing these signals through linear timeinvariant (LTI) systems to produce sound effects. The impact of the new curriculum on students’ conceptual understanding was evaluated through three techniques – quantitative assessment using the Signals and Systems Concept Inventory (SSCI), and qualitative assessment using a voluntary end-of-semester lab survey and a small group analysis. Analysis of SSCI scores from the first batch of students in the new curriculum indicated a course average normalized gain of 0.54 in the discrete time SSCI and 0.61 in the continuous time SSCI student performance. Student agreement on the labs (reinforcing the concepts of signal transforms and visualization, convolution and filtering) correlated well to their actual SSCI scores on questions based on these concepts. Analysis of subtest topics suggested persistence of common misconceptions, thereby motivating suitable changes to the lab exercises to be implemented in future semesters. Student performance and responses indicated that the collaborative laboratory exercises improved student learning and also suggested areas for improvement in the lab exercises for future semesters. Introduction The undergraduate electrical engineering program at Vanderbilt University offers an introductory signals and systems course (EECE 214: Signals and Systems) focusing on continuous and discrete time signals and systems representations and analyses. Sophomore and junior level electrical, computer and biomedical engineering students with the required prerequisites of Electric Circuits, Calculus and a basic programming course (MATLAB/C++/JAVA) take EECE 214 to acquire a strong foundation for advanced courses such as digital signal and image processing, biomedical signal processing and control systems. Traditionally, classroom lecturing and problem-solving sessions with MATLAB-based demos of practical applications have been the preferred teaching technique in this course, with a goal of emphasizing the interdependence of mathematical representations and tangible physical interpretations of concepts. Targeting this level of conceptual understanding in students has been a persisting challenge to instructors 1, 2 . Towards this end, MATLAB/LabVIEW based software simulations 3, 4, 5 and analog circuits based hardware lab exercises 6, 7 have been developed and implemented successfully in several universities with significant improvement in students’ conceptual understanding of signals and systems. Based on these existing lab curricula, three in-class MATLAB-based collaborative lab exercises were developed and implemented in the EECE 214 course curriculum in fall 2013. This paper outlines the significance and features of these lab exercises, their impact on students’ conceptual understanding as assessed by the SSCI and student feedback, and persisting student misconceptions that may be effectively addressed by modifying the lab exercises. EECE 214 introduces time and frequency domain representations and analyses of continuous and discrete time signals and LTI systems. New concepts such as convolution, LTI system theory, P ge 24715.2 sampling, Fourier analysis and, Laplace and Z transforms are presented through lectures and problem-solving sessions. Students can exhibit inabilities to apply the following learning skills required in this course (a) integration of their prior knowledge of calculus and complex numbers to develop a strong mathematical foundation of these concepts with a thorough understanding of the computational procedures involved, (b) graphical interpretation of the mathematical basis of these concepts to understand their physical meaning and hierarchical relevance in the course curriculum and, (c) successful application of these concepts with appropriate mathematical formulations to solve practical problems on signal filtering, modulation and optimal system design. Previous research 1 indicates many students find skills (a) and (b) challenging, specifically with concepts of convolution and Fourier transforms, thereby resulting in the inability to solve practical problems and learn advanced topics. Based on the success of MATLAB-based lab exercises as part of the signals and systems curriculum in many universities 8, 9, 10, 11 , the instructors at Vanderbilt University developed three in-class collaborative MATLAB-based lab exercises that were included in the existing EECE 214 curriculum. Software simulation tools such as MATLAB can facilitate step-by-step visualization of computational procedures and their results while solving signals and systems problems. A broad spectrum of problems ranging from simple mathematical computations in convolution or frequency domain transforms to application-based system design of filters and feedback systems can be effectively simulated using MATLAB. MATLAB-based homework problem sets in EECE 214 target the mathematical problem-solving component and the in-class labs focus on reinforcing concepts through application-based practical problem solving. The new curriculum with the in-class labs and problem sets was implemented in the fall 2013 semester with an enrollment of 19 students (18 electrical and computer engineering sophomores and juniors and one biomedical engineering junior). Since MATLAB programming was not an explicit prerequisite for this course, an initial class survey indicated over 50% of the students were inexperienced MATLAB programmers. These students were encouraged to utilize the multiple MATLAB tutorial sessions conducted by the teaching assistant. Since the primary goal of the course was to utilize MATLAB as a tool to reinforce concepts and provide a platform for hands-on simulation and modeling of signals and systems, the MATLAB code for the lab exercises was provided to the students. In addition to interpreting the results of the lab exercises in terms of relevant concepts, the technical lab report focused on code interpretation and execution, and user input parameter variation, thereby providing moderate training in MATLAB programming. The three in-class labs replaced three lecture sessions suitably to ensure continuity in the lecture content. In addition to a brief introduction of the labs in the prior lecture session, students were provided with pre-lab exercises that were to be reviewed before the in-class lab session. The prelab module introduced key concepts reinforced in the lab and suggested additional resources (online review articles and tutorials) relevant to the application of these concepts. The in-class lab module comprised of short MATLAB exercises implementing and testing one or more concepts. The in-class lab sessions foster collaborative learning among the students by having them work in randomly assigned small groups. The lab sessions began with the instructor giving an overview of the exercises and the corresponding MATLAB code, and providing relevant block diagrams of systems, expected graphical outputs, and suggestions for user inputs to P ge 24715.3 simulate various system behaviors. This introduction was followed by group work with instructor assistance as needed, thereby stimulating peer and instructor interaction. Each MATLAB exercise consisted of guided questions that helped students associate the MATLAB implementations with the underlying concepts and develop a qualitative and quantitative interpretation of the observed outputs. The final component of the in-class labs was the collaborative technical lab report with students’ answers to the pre-lab and in-lab questions and brief interpretation of the MATLAB code in terms of the concepts implemented and challenges encountered with its execution while simulating various system behaviors. Laboratory Exercises Audio signal synthesis and processing is a standard application that has been utilized successfully in several MATLAB-based signals and systems lab curricula 4, 8, 9 . The three in-class labs in EECE 214 are also based on this application with the goal of improving students’ conceptual understanding of signals and systems. Table 1 describes the applications and concepts targeted in the three in-class lab exercises. Week in Semester Applications Tasks Concepts Lab 1 – Week 6 (Modeling guitar music)  Frequency of music notes  Guitar harmonics on a vibrating string  Chords and tunes  Guitar strumming effect  Play music notes (4s sine wave) at fundamental frequency  Model open and fretted string guitar notes on a vibrating string using Karplus Strong algorithm  Model and play B minor chord and compare to actual guitar notes  Model and play 'Hot cross buns'  Create strumming effect on B minor with delay of 50ms between notes of the chord  Modeling guitar notes as LTI system (Karplus Strong algorithm low pass filter and a delay line)  Convolution of unit impulse function with impulse response of LTI system.  Time and frequency domain representations of exponentially decaying sinusoids to observe interaction of fundamental frequency and harmonics using Bode plots and spectrograms Table 1: Laboratory Exercises