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Signals and Systems I

by: Antonina Wuckert

Signals and Systems I ECE 220

Antonina Wuckert
GPA 3.94

Kathleen Wage

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Kathleen Wage
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This 7 page Class Notes was uploaded by Antonina Wuckert on Monday September 28, 2015. The Class Notes belongs to ECE 220 at George Mason University taught by Kathleen Wage in Fall. Since its upload, it has received 57 views. For similar materials see /class/215020/ece-220-george-mason-university in ELECTRICAL AND COMPUTER ENGINEERING at George Mason University.

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Date Created: 09/28/15
ECE 220 Problem 21 Old exam questions Consider the continuous time LTI system that has the impulse response ht shown in Figure 211 and the frequency response H jw shown in Figure 212 e 5 4 l EiD Q LQZQZ hm 2X nt3 3 2 ht Figure 211 Impulse response ht of the CT LTI system in Problem 3 IHG 12 2Hjm 3m 107c 1 7 Figure 212 Frequency response magnitude and phase plots of the CT LTI system in Problem 3 In parts a d below you are given 4 signals that are inputs to the LTI system de ned above Determine and sketch the output of the system corresponding to each input Make sure to label your sketches a Input to system wat 6 Determine the output yat Provide a sketch of yat and justi cation of your answer b Input to system wbt cos27rt Determine the output ybt Provide a sketch of ybt and justi cation of your answer c Input to system wct pt cos1007rt where pt and its Fourier transform Pjw are shown in Figure 213 Determine the output yea Provide a sketch of yct and justi cation of your answer d Input to system wdt shown in Figure 214 Determine the output ydt Provide a sketch of ydt and justi cation of your answer Comparing Student Understanding of Signals and Systems Using a Concept Inventory a Traditional Exam and Interviews John R Buck Kathleen E Wage Margret A Hjalmarson and Jill K Nelson October 2007 Proceedings of the 37th ASEEIEEE Frontiers in Education Conference pp SlGlSIG6 2007 IEEE Personal use of this material is permitted However permission to reprintrepublish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists or to reuse any copyrighted component of this work in other works must be obtained from the IEEE Session SlG Comparing Student Understanding of Signals and Systems Using a Concept Inventory a Traditional Exam and Interviews John R Buckl Kathleen E Wagez Margret A Hjalmarson3 and Jill K Nelson4 Abstract Concept inventories play a growing role in assessing student understanding in engineering curricula A common application of concept inventories is a prepost test assessment in a course For this reason it is important to confirm the validity of any new concept inventory ie to verify that the inventory measures what it is designed to assess The Signals and Systems Concept Inventory SSCI is a 25questi0n multiplechoice exam assessing core concepts in undergraduate signals and systems courses This paper presents two analyses supporting the validity of the SSCI The first analysis compares the responses of 40 students to final exam questions with their responses to related SSCI questions This analysis finds statisticallysignificant correlations between the SSCI and the final exam for questions on convolution and Fourier transform properties The second analysis examines the interview responses of 18 students to SSCI questions on frequencyselective ltering and convolution The interviews suggest students have a strong understanding of high and low frequency have some understanding of the relationship between time and frequency domains but struggle to interpret frequency responses The interviews also suggest that many students retain some conceptual understanding of convolution after their memory of the convolution integral has faded Index Terms 7 Assessment concept inventory interview signals and systems validity INTRODUCTION Students who can organize facts and ideas within a conceptual framework learn new information quickly and are more likely to recognize appropriate applications of these ideas in novel situations I Consequently many instructors make conceptual understanding one of the desired outcomes in undergraduate courses even though they are often unsure how to assess this outcome New accreditation criteria 2 and the increasingly international nature of engineering education is increasing the demand for outcome assessments that are portable across campus and national boundaries The Signals and Systems Concept Inventory SSCI is a 25 question multiplechoice test designed to evaluate students understanding of fundamental concepts in signals and systems Popularized by Hestenes et at developers of the Force Concept Inventory for physics 3 concept inventories provide an objective instrument for quantifying students conceptual grasp of particular topics Development of the SSCI began in late 2000 with support from the National Science Foundation N SFfunded Foundation Coalition and continues with ongoing funding under NSF s Assessment of Student Achievement program The SSCI has both continuoustime CT and discretetime CDT versions representing the two common courses in an undergraduate signals and systems 5amp5 sequence Each SSCI question includes four possible answers the correct answer plus three distractors that re ect common student misconceptions Over 1400 students have taken the SSCI and the resulting data has been used to analyze the effects of active and collaborative learning 4 projectbased courses 56 graphical vs textual programming interfaces for signal processing 7 and the impact of prerequisite courses on SampS conceptual understanding 8 A significant component of this paper is an extension of our analysis presented in 9 of student interviews based on the SSCI Relatively little other research has been published on SampS conceptual understanding Nasr et al 101 I conducted an extensive sequence of interviews probing student misconceptions in SampS This research was conducted in the context of an aeronautical engineering course rather than the electrical engineering curriculum and predominantly focused on electronic circuit implementations of CT linear time invariant systems In spite of the different emphasis between these disciplines this unrelated research identified similar misconceptions to those found in previous SSCI analyses 8 The following section describes an analysis correlating students performance on openended SampS exam problems with their performance on related CTSSCI questions Subsequent sections describe our protocol for student interviews based on a subset of CTSSCI questions and the results of analyzing those interviews The final section summarizes the evidence from these two related studies 1 John R Buck ECE Dept amp SMAST University of Massachusetts Dartmouth N Dartmouth MA johnbuckieeeorg 2 Kathleen E Wage ECE Dept George Mason University Fairfax VA kewageieeeorg 3 Margret A Hjalmarson College of Education and Human Development Fairfax VA mhjalmargmuedu 4 Jill K Nelson ECE Dept George Mason University Fairfax VA jnelsongmuedu 1424410843072500 2007 IEEE October 10 13 2007 Milwaukee WI 373911 ASEEIEEE Frontiers in Education Conference supporting the content and construct validity of the SSCI SSCI vs TRADITIONAL EXAM QUESTIONS How does the level of students conceptual understanding affect their ability to work problems Do students who perform well on concept inventory questions perform similarly on more traditional measures of student learning such as openended analytical questions These issues must be addressed in the process of establishing the content validity of the SSCI As a part of the SSCI validation study this section compares students performance on the conceptual questions with their performance on the openended analytical problems that made up part of a final exam in an SampS course George Mason University GM U Study At GMU Signals and Systems I ECE 220 is the first SampS course that students take This course focuses exclusively on CT linear systems and is often taken concurrently with courses in both circuits and differential equations ECE 220 follows an introductory course on signal analysis that provides students with basic knowledge about sinusoids complex exponentials and simple filtering methods in addition to teaching them Matlab programming skills Students take Signals and Systems 1 during the second semester of their sophomore year or first semester of their junior year The workload for ECE 220 consists of two class meetings one recitation session and one laboratory session per week In fall 2006 one of the authors KEW taught the course using active and collaborative learning methods 4 Class time was divided between short lecture segments on key concepts and inclass group exercises Students were expected to do the assigned reading prior to coming to class so that they would be adequately prepared to participate in the interactive problemsolving sessions Each class meeting began with a quiz on the assigned reading The textbook for the course was Linear Systems and Signals by Lathi 13 For more information on how ECE 220 was taught refer to the course webpage httpecegmuedukwageece220fall06 Fortyfour students took ECE 220 in fall 2006 The CT SSCI was a required component of the course but students had a choice of whether to participate in the research study Of the 44 students enrolled 40 signed informed consent forms agreeing to have their exams and other data analyzed The SSCI was administered as a pretest during the initial laboratory session Students had up to one hour to complete the pretest The SSCI was also administered as the first part of the final exam and was worth 25 of this exam The other 75 of the final exam grade was based on students answers to five traditional openended exam problems Students were given both parts of the final at the beginning of the exam period They were required to complete the multiplechoice SSCI questions within the first hour and then had the remainder of the 2 hour and 45 minute exam period to complete the openended problems While students worked on the SSCI they were not permitted to use calculators 1424410843072500 2007 IEEE Session SIG books or notes Once they turned in the SSCI they could use three 85 xl 1 sheets of notes but calculators and books were still not permitted Results of the GM U Study In fall 2006 the CTSSCI posttest average was 67 for the 40 students participating in the study 33 of these 40 students also took the SSCI pretest and their pretest average was 39 The average gain for the course as defined by Hake 14 is ltggt046 By Hake s definition this is a medium gain consistent with the gain achieved in other courses using interactive learning methods Figure 1 shows a scatter plot of scores on the SSCI versus scores on the openended problems for the 40 students taking ECE 220 in fall 2006 As the figure indicates the SSCI scores are significantly correlated plt0001 with scores on the remainder of the final exam For this population students who have greater conceptual understanding as measured by the SSCI perform better on openended analytical questions Correlation064 Significance0000 A25 i 3 L020 0 00 0 z I N o o o o 0 H615 o ooo o r 0 gm o r 6 57 7 D D O l l l l 0 10 20 30 4O 50 60 70 Final exam out of 75 pts FIGURE 1 COMPARISON OF SCORES ON THE CTSSCI WITH SCORES ON FINAL EXAM PROBLEMS FOR 40 STUDENTS WHO TOOK ECE 220 IN FALL 2006 Individual parts of two final exam problems are closely related to two SSCI questions Problem lc on convolution and Problem 3b on Bode plots relate to CTSSCI questions 8 and 22 respectively Two of the authors KEW and JKN reviewed the final exams independently and coded students responses to 1c and 3b using a 4level scheme 3correct 2minor errors lmajor errors 0completely wrong The independent coding differed for only 6 of 40 students These six cases were discussed and a consensus on the coding was reached Problem 1c Plc on the final exam asked students to compute the output of a linear time invariant LTI system with an impulse response equal to a square pulse when the input consists of the sum of two nonoverlapping rectangular pulses The correct answer is the convolution of the impulse response and the input which for this problem results in a triangular pulse followed by a trapezoidal pulse The related CTSSCI question Q8 asks students for the output of an LTI system when the impulse response is a square pulse and the input consists of a single rectangular pulse The answer to this question is a trapezoidal pulse Since the startingending points of the pulses in Plc and Q8 are not the same the October 10 13 2007 Milwaukee WI 37th ASEEIEEE Frontiers in Education Conference SIG2 trapezoidal pulses in the correct answers are not identical The distractors for Q8 probe whether students multiply instead of convolve add instead of convolve or convolve to obtain the right shape but make a mistake in determining the extent of the output Table 1 compares student responses to Plc and SSCI Q8 As the table shows all students who were able to correctly convolve the signals for the openended problem also chose the correct answer to the related SSCI question The majority of students who made minor errors in the analytical convolution were also able to choose the correct answer for the multiplechoice question Some students with major errors or completely wrong answers to the analytical convolution were still able to answer Q8 correctly but others chose the answer with the right shape but wrong extent It is notable that the only student who selected the SSCI distractor indicating he would add the input and impulse response to obtain the output of an LTl system was also completely unable to do the analytical convolution problem Assuming that students who made minor errors had a correct understanding of convolution whereas those who made major errors did not it is possible to reduce Table 1 to the 2x2 contingency table shown in Table 11 Applying Fisher s exact test 15 to this table indicates a statistically significant plt0004 positive correlation between students performance on Plc and CT SSCI Q8 TABLE I TABLE 11 sscr Q8 Problem 3b P3b on the ECE 220 final exam gave students a system function and asked them to sketch the frequency response magnitude of the system using a standard Bode plot The system had two poles and one zero and a DC offset Sketching the Bode plot should be straightforward for students assuming they understand how poles and zeros affect the magnitude response of a system P3b is closely related to SSCI Q22 which gives students the Bode magnitude plot for a system with frequency response H 03950 and asks them to identify the magnitude plot for a new system obtained by cascading H 0a with another system which has a pole located at 100 Q22 assesses whether students understand what adding a pole does to the magnitude response plot The distractors for this question probe whether students confuse poles and zeros cannot distinguish between a pole at 10 and a pole at 100 and whether they think a pole causes an additional 1424410843072500 2007 IEEE Session SIG DC offset Table 111 summarizes the comparison between students responses to the openended Bode plot question and the SSCI Bode plot question The majority of students who prepared the correct plot for their answer to P3b also chose the correct answer for CTSSCI Q22 Many students with minor and major errors in their answers to P3b were also able to select the correct answer to Q22 Overall the correlation between the SSCI question and the analytical problem is not as clear as it was for the convolution question Analysis via Fisher s exact test does not reveal a statistically significant correlation between P3b and Q22 TABLE III sscr Q22 Another problem on the ECE 220 final exam is closely related to a set of SSCI questions Final Exam Problem 4 P4 focuses on the Fourier transform and Fourier transform properties Rather than relating to a single SSCI question this problem relates to six SSCI questions that probe students understanding of various aspects of the Fourier transform These six questions 911 15 16 21 make up a Fourier transform subtest of the CTSSCI 8 The first part of P4 tested students ability to execute the mechanics of a routine inverse Fourier transform with a familiar transform pair rectangle to sinc function The second part of P4 required students to synthesize several Fourier transform properties to obtain the correct answer Figure 2 shows scatter plots of students scores on the CTSSCI Fourier subtest with their scores on P4a and P4b The maximum possible score on P4a was 7 points while that on P4b was 8 points As the plots indicate the Fourier subtest score is statistically significantly correlated with the responses to P4b but not with P4a This is consistent with P4a testing the students ability to execute a procedure amenable to rote memorization and thus not necessarily related to conceptual understanding P4b on the other hand requires conceptual insight in order to deconstruct a complex problem into its constituent subproblems and thus it is reasonable that performance on this problem correlates significantly with the conceptual subtest on Fourier transform properties October 10 13 2007 Milwaukee WI 373911 ASEEIEEE Frontiers in Education Conference Corr012 Sig0448 Corr062 Sig0000 e e 86 o oo 86 o 000 2 o o o oo 2 o o o o 00 40 000000 4ooooooooo o oo o g o o o o 22 o o oo 22 o o o 9 9 E E LEO LEO 2 4 6 8 O 2 4 6 Final Problem 43 Final Problem 4b FIGURE2 COMPARISON OF SCORES ON THE SSCI FOURIER SUBTEST WITH SCORES ON FINAL EXAM PROBLEM 4 THESE SCATTER PLOTS CONTAIN THE RESULTS FOR 40 STUDENTS SOME POINTS OVERLAP THE FOURIER SUBTEST CONSISTED OF 6 QUESTIONS MAX SCORE6 AND THE TOTAL POINTS POSSIBLE FOR PROBLEMS 4A AND 4B WERE 7 AND 8 POINTS RESPECTIVELY The analysis of the ECE 220 nal exam data with the linked CTSSCI data provides evidence supporting the content validity of the convolution and Fourier transform problems of the CTSSCI There is a statisticallysignificant correlation linking the performance on these concept inventory questions to traditional openended exam problems assessing the same topics At the same time this data suggests we may need to revisit CTSSCI Q22 if performance on this problem remains uncorrelated with related exam problems Our investigation of these correlations is ongoing and we plan to analyze additional data from the Spring 2007 offering of ECE 220 at GMU INTERVIEW DESIGN Students choices on multiplechoice concept inventories provide some information about their understanding and suggest possible confusions when they choose distractors Interviewing the students provides a more complete picture of their thought processes and may reveal lurking confusions even when they choose the correct answer These interviews are an important component of verifying both the content and construct validity of a concept inventory instrument Students responses provide insight about whether the questions are clear and whether they assess the intended concepts Interview Protocol The interviews focused on six conceptual questions about SampS including five questions taken from the CTSSCI The remaining question was written specifically for the interviews The interviews were semistructured in that the students were asked multiplechoice questions but the interviewer could probe students further based on their answers Students were provided with a paper copy of the questions and then asked to explain their reasoning aloud as they arrived at their answer to each question The interview sessions were recorded with a digital voice recorder After completing the conceptual questions students were asked some general questions about SampS and the SSCI instrument including the following 0 Do you feel you have a good grasp of the material covered in your signals and systems class 1424410843072500 2007 IEEE Session SIG 0 Did this exam cover the material in the course Were there topics on the exam that were not included in your course Or vice versa 0 Did you have enough time to complete the exam 0 Were the questions hard to read or easy to read At the conclusion of the interview students had the opportunity to ask questions about the exam or the research study Conceptual Questions Five of the six interview questions focused on concepts related to frequencyselective filtering which is one of the most important topics covered in SampS courses To master filtering students must also understand sinusoidal signals frequency responses and the relationship between the time and frequency domains The sixth interview question focused on convolution which is another fundamental topic covered in SampS courses The list below summarizes the conceptual questions used for the interview Reference 8 provides a complete description of the SSCI questions and discusses the development process for the SSCI 0 CTSSCI Question 1 Q1 This question probes whether students understand the meaning of frequency The students must choose which of four graphs of sinusoids has the highest frequency 0 CTSSCI Question 6 Q6 This question involves filtering a single frequency cosine with a lowpass filter The students must use graphs of the magnitude and phase responses of the filter to determine the output signal 0 CTSSCI Question 7 Q7 This question investigates students understanding of timefrequency relationships The students are shown a windowed sinusoid and its associated Fourier transform They are then shown a sinusoid at a higher frequency and asked to chose the correct plot for the Fourier transform of the second signal 0 CTSSCI Question 8 Q8 This question was described in the second section of the paper 0 CTSSCI Question 25 Q25 This question requires students to synthesize the concepts probed in Questions 1 6 and 25 They are given an input signal with two sequential windowed sinusoidal pulses at different frequencies and the Fourier transform of the input signal The students are also given the frequency response of a lowpass filter and must choose which time signal represents the output of the filter for the input given 0 Additional Question Qadd In addition to the CT SSCI questions listed above the interviews included a new question This question probes students understanding of the relationship between time and frequency domains in a manner similar to CTSSCI Question 7 but instead of using windowed sinusoids it uses two random signals These signals are obtained by filtering white noise with two lowpass filters with different bandwidths For most students this is a novel application of the concept and intuition developed in Question 7 See 9 for the exact text of this question October 10 13 2007 Milwaukee WI 37th ASEEIEEE Frontiers in Education Conference SIG4 Interview Pool We interviewed 18 5855 students from Ullass Dartmouth as a part of this study The first group of interviews took place in Feb 2006 and the second group in Feb 2007 All of the interviews were conducted by an instructor KEVD from George Mason University At the time of the interviews the students had completed the CT 5855 course and were enrolled in the DT 5855 course taught by one of the authors JRB Each student received 20 compensation for participating in the interviews and signed an informed consent form to authorize the anonymous use of their responses as well as the collection of academic and demographic data Three of the interview participants were female one was an under represented minority and one is registered as a disabled student with dyslexia The interview subjects represented a wide range of grades in CT SampS CD to A overall GPA 19 to 39 and CTSSCI posttest scores 44 to 84 INTERVIEW ANALYSES AND RESULTS Three of the authors JRB KEW MAH coded the transcripts independently and then discussed discrepancies until consensus could be reached on a code for each response in each category Four categories were used to code the transcripts as follows The first category was whether the students final answer was correct or not Table TV shows the number of students in each group who answered each question correctly Note that the convolution question Q8 was only asked in the second group of interviews Though the sample sizes are small the distribution and character of the responses across groups 1 and 2 is similar so the remainder of the discussion describes the interview sample as a whole TABLE IV The second category of coding reasoning indicated the quality of the students reasoning in answering each question Students were classified in one of four categories right partial muddled and wrong Partial reasoning indicated the students solution process was moving in the right direction but there may have been minor errors Muddled indicated that students had difficulty explaining their reasoning and justifying their response Table V summarizes the reasoning scores for the entire sample Note that the results for Q8 1424410843072500 2007 IEEE Session SIG include only answers from the second group of interviews The third category of coding process indicated the process students had used to generate their solution The coding specified whether students used a logical chain of reasoning or process leading to the answer the process of elimination or guessing The category blend indicates that students blended these types of processes to arrive at their answer Table VI summarizes the reasoning for questions 6 7 8 25 and the additional question Question 1 is not included in these results because it could be solved by inspection ie just looking at the graphs The final category was students confidence in their answer For the first group of interviews the coders assigned the confidence level based on how the student stated their answer For the second group the students were explicitly asked to identify whether they were very con ident somewhat confident not confident or guessing about their answer The analysis shows that students were overall either somewhat confident or very confident about their responses to a majority of questions For group 1 responses were coded as either very confident or somewhat confident for 36 out of 45 responses For group 2 students reported they were somewhat or very confident on 42 of the responses Part of our goal with the interviews was to establish the validity of the SSCI questions as capturing students conceptual understanding or the lack thereof The interviews have helped confirm that students do have conceptual understanding in some areas and lack the ability to articulate their reasoning in other areas In addition for the questions they were most likely to answer correctly eg Ql regarding the frequency they were also most likely to provide clear explanations As the conceptual complexity increases eg Q25 which requires synthesis of a few ideas the students have greater difficulty obtaining the right answer and provide less clear explanations TABLE V TABLE VI October 10 13 2007 Milwaukee WI 373911 ASEEIEEE Frontiers in Education Conference


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