Wow! Linear Systems and Signal Processing is fun!

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1 Wow! Linear Systems and Signal Processing is fun! Maurice F. Aburdene and Kundan Nepal Electrical Engineering Department, Bucknell University, Lewisburg, PA Abstract - We describe a recent offering of a linear systems and signal processing course for third-year electrical and computer engineering students. This course is a pre-requisite for our first digital signal processing course. Students have traditionally viewed linear systems courses as mathematical and extremely difficult. Without compromising the rigor of the required concepts, we strived to make the course fun, with application-based hands-on laboratory projects. These projects can be modified easily to meet specific instructors preferences. Index Terms- Linear Systems, Signal Processing, Signal Processing Education, Student Learning Counterintuitive, when moving from continuoustime domain to discrete-time domain Educators have tried to address the above concerns. For example, Ferre, Giremus and Grivel have tried to make the course more appealing by focusing on small-group based learning projects [5]. Wright, Morrow, Allie and Welch suggested ways to enhance engineering education and outreach [6], and Turner and Hoffbeck suggested ways to put theory into practice using software [7]. Warren suggested using computational projects to optimize student learning and retention of fundamental concepts [3]. Buck and Wage proposed active and cooperative learning (ACL) in signal processing courses [4]. INTRODUCTION Most undergraduate programs in electrical and computer engineering require a course in linear systems, signals and systems, signal processing, or a similarly named course in the third year. There are many excellent texts in these areas (too many to list here). The texts and courses focus on continuous-time signal representation, discrete-time signal representation, sinusoidal signal analysis and design methods, linear system definitions and methods, differential equations, difference equations, modeling of systems, spectral analysis, Laplace transforms, analog filters, sampling, Z-transforms, digital filters, continuoustime Fourier series, convolution, and continuous-time and discrete Fourier transforms. In addition, many texts and/or linear systems classes incorporate software, such as MATLAB, Spice, Mathematica, Mathcad or locally developed software. We have tried to teach all of these topics in our classes and as instructors, we think they are great and try to impart our excitement to our students. Pendergrass states that students have difficulty in such courses since they view them as only math and theory [1]. Greenberg, Smith and Newman found that many students had difficulty mastering the fundamentals of spectral analysis, appearing to be overwhelmed by the interaction of multiple variables [2]. In addition, if you teach one of the linear systems courses, then you might have observed that such a course is perceived by students as: Theoretical and mathematical[1],[2] Notation intensive[2] Difficult [1]-[4] Fast paced and time-intensive In the summer of 2009, we decided to tackle the issue and completely redesign the course, ELEC 320 (Linear Systems and Signal Processing), trying to make it as much a fun course for the students as it is for the instructors. The basic idea was to make the course application-based and to have hands-on laboratory projects. Our goal was to have students view the course as an opportunity to learn most of the above topics (very ambitious goal) and have fun. We decided not to choose a new text, but rather, use the texts that students had used earlier, along with web based-resources, and industrial application notes [8]-[11]. We taught the course in the fall of 2009 using seven projects (applications) in addition to an introduction to MATLAB and a final project. Each application and the major concepts involved were discussed first in class. Students were asked to perform experiments in lab to improve their understanding of the major concepts and their applications. Students were then required to write project reports (learn by writing). The course also required the students to propose a final project that covered one or more areas learned in the classroom. The students then devote the final three weeks of the semester designing and implementing their ideas to produce a functional product. Two of the projects are described in detail in [12]. PROJECTS AND APPLICATIONS I. Introduction to MATLAB Using Circuit Examples Throughout the semester, students used MATLAB to analyze linear systems and signals applications and problems. Our students have had prior programming experience using JAVA and we used two introductory labs to focus on helping students familiarize themselves with the F1G-1

2 MATLAB interface and learn its rules and syntax. The labs focused on the representation of signals as arrays and matrices; processing of these arrays and matrices; plotting of signals and conditional statements and loops. In order to show how MATLAB might be used to analyze signals, two examples were chosen. In the first example, students were asked to analyze an inverting op-amp circuit; create both a DC and AC input signal and analyze the output response in each case. For the second example, students were asked to create a signal by combining three sinusoidal waveforms to approximate a square wave signal with 1 khz frequency. Students were then tasked with using this signal as an input to a series RC circuit with bandwidth of 1.5 khz and to derive and plot the output voltage across the capacitor. These examples were an indirect introduction to the Fourier series and analog filters incorporated in some of the course projects. communication link with a given bandwidth. In particular, we were interested in having the students determine the harmonic content and the time-response of both the transmitted signal and received signal. Students programmed the Agilent 33220A 20 MHz Function Generator to create the ASCII character and created a filter to view and analyze the output response. Students also used MATLAB and Simulink to simulate the communication model. This project reinforced concepts of periodic signals, Fourier series, harmonics, phase shift, signal power, bit rate, link bandwidth, Nyquist's theorem and Shannon s theorem for data transmission on a noisy channel. Details of this project are presented in [12]. IV. Modeling and Control of a DC Motor II. Satellite Communications FIGURE 1 A SIMULINK MODEL OF A SATELLITE COMMUNICATIONS SYSTEM. Communication systems provided us with excellent applications of important concepts that students had studied in ELEC 120 (Foundations of Electrical Engineering) and ELEC (Circuit Theory I &II). This project introduced students to a satellite communication system where a signal is transmitted from a ground transmitter, received by the satellite after some delay and reflected back to the ground receiver [12]-[14]. Students analyzed the signal received by the ground station as the sum of the signal sent from the station and the signal reflected back from the satellite. This interesting project helped students review sinusoidal signals, phasors, power, and understand the effect of delays, echoes and fading. In addition, the students got an opportunity to learn more about MATLAB and Simulink while getting to implement fundamental concepts they had previously learned as shown in Figure 1. This project is fully described in [12]. III. Data Communications and Fourier Series In this project, we focused on Fourier analysis of periodic signals and applications to data communication [12]-[15]. Students were given the task of analyzing an 8-bit ASCII character transmitted at a particular bit rate over a FIGURE 2 TOP VIEW OF BOARD WITH DC MOTOR AND SENSORS. The purpose of this project was to build, test and characterize a speed control system [16]. The primary objective of this lab was to model systems using differential equations and Laplace transforms. Students were given a differential equation model of a DC motor and were asked to use Laplace transforms to obtain the open-loop and closed loop transfer functions. The experiment used a DC motor with an AC tachometer as shown in Figure 2. Using rectifiers, the students converted the tachometer signal to DC and using op-amps created a proportional feedback speed control system. Vref + - Rectifier Amplifier AC Tach M DC Motor FIGURE 3 MODEL OF THE SPEED CONTROL SYSTEM. Out For this project, students were asked to build the system shown in Figure 3 and answer the following questions: F1G-2

3 Let v a be the input voltage to the motor, τ=6.7 sec and K= 46 rad/sec-volt. The motor speed, the output, can be determined directly from the equation written as: What is the step response of the system? What is the transfer function H(s) = W(s)/V al (s) based on this equation? Draw a block diagram of the closed-loop (CL) control system. Obtain the transfer function of the CL system. Use a square wave as your input voltage and observe the speed of the motor and the rectifier output voltage using the oscilloscope. What is the time constant of the closed loop system? Can you determine K and τ? V. Touch-tone Decoding and Filters FIGURE 4 PROGRAMMABLE FILTER BOARD. In this project, we focused on analog filter design for the Touch-tone telephone dialing system [17], using the MF10 (universal monolithic dual switched capacitor filter) chip, including a discussion of the history of the development and design issues. To minimize the time devoted to building the detection filters, a circuit board with an MF10 filter (shown in Figure 4) was given to the students. The circuit board had the necessary clock generator, audio jacks, operational amplifiers and an area to wire circuit components. Students were tasked with using the MF10 datasheet to create the required filter to detect a particular touch-tone. Based on their calculations, students then programmed the board by proper selection of resistors. Here is a sampling of some questions from this project: In your own words describe the Touch-tone telephone dialing system. How many tones are generated per pushbutton? What are the frequency assignments for the pushbuttons? Suppose the transmitter was nonlinear, list the first three harmonics of each frequency. Compare these harmonics with the original frequencies. Develop a block diagram(s) for a detection system using analog filters. Design a lowpass/highpass filter with a 1 khz bandwidth. Test your design using a tone generator program (Dual-Tone Multi-Frequency, DTMF) and the sound card. Observe and record the output of the filter when you push the 1 button. Observe and record the output of the filter when you push the # button. Using the output of the filter as an input, design a bandpass filter to detect the 7 button. Compare the output of the filter with output when depressing other buttons. What is the frequency response of the filter? What would happen to your detector circuit if the transmitter frequencies varied by ±2%? VI. Analog-to-Digital and Digital-to-Analog Converters The objective of this project was to help reinforce the concepts of analog-to-digital converters (A/Ds), digital-toanalog converters (D/As), sampling and digital signal processing. Students used the Keithley KUSB-3100 data acquisition hardware, the sound card and MATLAB for signal acquisition and processing. Students created waveforms using a function generator, sampled the waveforms at various sampling rates using the KUSB hardware and then processed the samples using MATLAB to generate the magnitude and phase plots using fast Fourier transforms (FFT). Students also sampled and recorded sound at various sampling rates using the sound-card and processed it using MATLAB/Simulink. Here is a sampling of some questions from this project: Please study the Keithley KUSB 3100 data sheet and answer the following questions: How many bits does this A/D have? What is the maximum analog input voltage? Develop the digital output vs. analog input characteristic? What is the maximum, minimum and average conversion error? Assume that the conversion error introduces noise, what is the signal-to-noise ratio for a 12-bit A/D? In your own words, describe (a) Nyquist s sampling theorem and (b) Shannon s theorem for noisy channels. Using the MATLAB program provided, sample a 200 Hz sinusoidal 1V peak-to-peak signal. Now modify the program to sample (a) two periods of the input signal, (b) the input signal at 50 Hz, 200Hz, 500Hz, and 1 khz. Relate the above to the Nyquist sampling frequency. Use the MATLAB fft program to determine magnitude and phase spectrum of the input signal. Repeat this for a square wave signal. Use the program to sample a sinusoidal input signal using the A/D and to output the sampled signal using the D/A. Observe the input and output on the oscilloscope. Just like the Keithley board, the sound card in your computer can be used to acquire analog signals. Assume the sampling frequency is 8 khz and a signal is sampled for 10 seconds. How many samples are collected? What is the time difference between two samples? F1G-3

4 Set up your function generator to generate a sinusoidal signal of 2 khz. Connect a speaker to the function generator. Now set the sampling frequency of your program to 8 khz and acquire 2 seconds of data. Plot the data just acquired and play it using the sound command in MATLAB. Now change the function generator frequency to 6 khz. Acquire 2 seconds of data and view the data using the plot command. Does the data look the same as before? Play the sound using the sound command. Does it sound the same? Repeat this for a sampling frequency of 11,025Hz. Instead of some tone generated through the function generator, let us actually work on real speech signals. Using the MATLAB program, record a voice at a sampling frequency of 22,050 Hz. Play it using the sound command. Now change the sampling frequency to 11,025 Hz. Play it using the sound command again. Do you hear any difference? VII. Digital Filter Design This project used the previous project on A/D, D/A and sampling as a foundation to introduce students to Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) filters. Students used MATLAB/Simulink to describe the difference equations and interpret the transfer function of the filters and perform an analysis of the frequency response of the filters. In addition, students tested the different filters by using their favorite music as inputs to the filter. Here is a sampling of questions from this project: In your own words, describe FIR and IIR filters. What are the advantages/ disadvantages of both the IIR and FIR filters? Let x[n] = [n]. Find y[n] = x(n-n). Let N=4. Sketch y[n] vs. n. Find H(z) =Y(z)/ X(z). Sketch the amplitude of H(z) vs. ω and phase of H(z) vs. ω. Discuss your results. Let x[n] = [n]. Find y[n] = (x[n] + x[n-1] x[n- (N-1)/N]) with N=4. Sketch y[n] vs. n. Find H(z) =Y(z)/ X(z). Sketch the amplitude of H(z) vs. ω and phase of H(z) vs. ω. Discuss your results. Load your favorite music file into MATLAB, x[n]. Implement the filter and listen to the filtered output using the sound card. Experiment with various values of N. Discuss your results. Assume zero initial conditions. Let x[n]= [n]. Find y[n] = -αy[n-1] +x[n-1]. Sketch y[n] vs. n. Find H(z) = Y(z)/X(z). Sketch the amplitude of H(z) vs. ω and phase of H(z) vs. ω. Discuss your results. Implement this filter and listen to the filtered output of your music file. Experiment with various values of α. Discuss your results. Now implement the filter y[n] = x[n] - x[n-1]. Listen to the filtered output of your music and discuss your results. VIII. Convolution: Computations and Applications This project was designed to help students learn about the impulse response of continuous-time systems and see the practical application of convolution. Students were first asked to determine the impulse response of an RC circuit both analytically and experimentally. Then they were asked to use convolution to determine the response to various input signals. In addition, the class went to measure the impulse response of the Powers Theatre (on campus). As an application of convolution, students were asked to experiment with music by taking their favorite music and using convolution to determine how the music would sound in the theatre. The trip to the Powers Theatre also demonstrated some of the basic concepts discussed in the satellite communications project. Here is a sampling of some questions from this project: F1G-4 Consider a series RC circuit driven by a voltage source, with the output voltage measured across the capacitor. We will use R = 10 kω and C = 0.1 μf. Analyze this circuit and derive the expression for the impulse response, h(t). In your analysis, consider applying a rectangular pulse that gets briefer and briefer while maintaining unit area. Analyze the circuit and derive an expression for the output, y(t). Devise a procedure to experimentally measure the step response of the circuit. Obtain the impulse response of the circuit, h(t). Use convolution to predict the response of this circuit to a 5 volt step function. Apply a 5 volt square wave and compare the response with your prediction. Use convolution to predict the response of this circuit to a 2 volt sinusoidal signal, f(t) = 2 sin(ωt). Then apply a 2 volt sinusoidal signal and compare the response with your prediction. Using the Keithley USB board, sample the impulse response, h(n), of the RC circuit. Load a music file, x(n) and convolve it with h(n) using the conv function. What effect does convolution with h(n) have on the music, i.e. what is different about the music after the processing? Load the impulse response of the Powers Theatre. The objective is to use convolution to hear digitized music and then play the resulting music and to listen to it as if you are in the Powers Theater. Plot the impulse response, h(n), of the Powers Theater vs. sample number. Load a music file, x(n) and convolve it with h(n). Play x(n) and y(n). What effect does convolution with h(n) have on the music? IX. Final Projects Students were asked to propose their own project that covered one or more concepts learned in the course. The

5 final project, which spanned three weeks, involved the design, implementation, test and documentation of a complete system (either hardware or software). At the end of the semester, students were required to demonstrate a working system to the class and guests, and prepare a professional presentation. In addition, students were required to write a technical report that documented their design, a product specification datasheet for use by the marketing group, and a sales brochure. The open-ended nature of the project yielded a number of interesting projects. ASSESMENT Since the course underwent a major redesign, and we had tried a new project based approach to the course, it was important for us to find out how the students perceived the different projects. At the end of the semester (Fall 2009), students were asked to fill out an anonymous survey which included that they indicate what labs you found helpful so we can improve them. The result of the survey is presented in Figure 5 for each project. A total of 24 students responded to the survey and rated each project on a scale of 1 to 5. A rating of 1 indicates that the students did not find the project helpful while a rating of 5 indicates the students found the projects very helpful in enhancing their understanding of the concepts involved. Students overwhelmingly seemed to like the first project titled Introduction to MATLAB using Circuit Examples. The final project received high scores as well. This is not surprising since the students had themselves worked to create the project specification, planned it, and implemented the project. The median for each project was 4 out of 5, indicating general satisfaction with the projects in helping the students understand the concepts involved. FIGURE 5 RESULTS OF THE END-OF-SEMESTER STUDENT SURVEY FOR EACH PROJECT (TOTAL RESPONDENTS= 24 STUDENTS). F1G-5

6 TABLE I COMPARISON OF COURSE LEARNING GOALS FOR FALL 2008, 2009 AND Course Learning Goal Mean Median Std. Dev Mean Median Std. Dev Mean Median Std. Dev Identify and use signal models Develop models of engineering systems, physical systems, and social systems Analyze continuous-time system models by applying Fourier and frequency response methods Analyze discrete-time systems Develop computer models using available software packages for analysis and design Design a hardware or software system and formulate system specifications Table I shows a comparison of the course learning goals for the course offerings in 2008, 2009 and 2010 using the same scale as the survey. The classes had 10, 24, and 15 students respectively. The course offered in 2008 was a traditional text [14]/lecture/lab format and did not use the projects discussed in this paper. Students responses show that scores for the course learning goals improved from 2008 to 2009 as a result of the move to a project based course. The learning goals results for 2010 shows further improvements in all but analyze discrete-time systems. SUMMARY This paper presented an overview of our project based course on linear systems and signal processing. As instructors, our motivation was to dispel the notion that this type of course is too mathematical and difficult to be fun and exciting. Through application oriented projects, we tried to convey the enthusiasm we have for the subject to our students in the course. An end of semester survey (Fall 2009) showed that while students did in fact enjoy the project driven approach, we have room for improvement. Based on the survey and our own analysis, the course was modified from this offering in the fall of We dropped the modeling and control of a DC motor and have gone back to requiring a textbook [14] for the course. Textbooks from previous classes that we thought would have been kept by students as reference materials for their own private libraries were, in fact, sold back. ACKNOWLEDGEMENTS We thank Professors Rich Kozick, Heath Hansum, James Baish, and Mr. Tom Thul for their valuable help with the design of the projects. REFERENCES [2] J. E. Greenberg, N. T. Smith, and J. H. Newman, Instructional module in Fourier spectral analysis, based on principles of How People Learn, Journal of Engineering Education, vol. 92, pp , [3] S. Warren, Optimizing Student Learning and retention of time and frequency domain concepts through numerical computation projects, Proc. of the American Society for Engineering Education Annual Conference & Exposition, [4] J. R. Buck, and K.E. Wage, Active and Cooperative Learning in Signal Processing Courses, IEEE Signal Processing Magazine, vol. 22 no. 2, pp 76-81, [5] G. Ferre, A. Giremus, and E. Grivel, Small-group learning projects to make signal processing more appealing: From speech processing to OFDMA synchronization, IEEE International Conference on Acoustics, Speech and Signal Processing, pp , [6] C. H. G. Wright, M. G. Morrow, M. C. Allie, and T. B. Welch, Enhancing engineering education and outreach using real-time DSP, IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) pp , [7] J. Turner, and J.P. Hoffbeck, Putting Theory into Practice with Simulink, Proc. of the American Society for Engineering Education Annual Conference & Exposition, [8] J.W. Nilsson, and S.A. Riedel, Electric Circuits (8th Edition), Prentice-Hall, [9] L. S. Bobrow, Fundamentals of Electrical Engineering, Oxford University Press, New York, [10] A. S. Sedra, and K.C. Smith, Microelectronics Circuits (5th Edition), Oxford University Press, [11] Agilent Technologies, The Fundamentals of Signal Analysis, Application Note 243. [12] M. F. Aburdene, and K. Nepal, Satellite Communications, Data Communications, and Simulation, Proc. of the American Society for Engineering Education Annual Conference & Exposition, [13] D. R. Fannin, W. H. Tranter, and R. E. Ziemer, Signals and Systems: Continuous and Discrete: Fourth Edition, Prentice Hall, [14] M. J. Roberts, Fundamentals of Signals & Systems, McGraw Hill, New York, [15] A. S. Tanenbaum, Computer Networks (3 rd Edition), Prentice-Hall, 1996, (Chapter 2, pp ). [16] R. C. Dorf, and R. H. Bishop, Modern Control Systems (11th Edition), Prentice-Hall, [17] L. Schenker, "Pushbutton Calling with a Two-Group Voice Frequency Code," The Bell System Technical Journal, vol. 39, pp , January [1] N. A. Pendergrass, Using Computers, Simulators and Sound to give hands-on experience, Proc. of the American Society for Engineering Education Annual Conference & Exposition, F1G-6