ACF Matlab[1]d

IEEE TRANSACTIONS ON EDUCATION, VOL. 55, NO. 3, AUGUST 2012 349

MATLAB-Based Program for TeachingAutocorrelation Function and Noise Concepts

Gordana Jovanovic Dolecek, Senior Member, IEEE

Abstract—An attractive MATLAB-based tool for teaching thebasics of autocorrelation function and noise concepts is presentedin this paper. This tool enhances traditional in-classroom lecturing.The demonstrations of the tool described here highlight the de-scription of the autocorrelation function (ACF) in a general casefor wide-sense stationary (WSS) processes, and for several impor-tant special processes including Gaussian white noise (GWN), non-Gaussian white noise, and Gaussian non-white noise. The demon-strations were designed for and introduced in the graduate-levelcourse “Introduction to Communications,” but would also be ap-propriate at the undergraduate level in various courses on signalsor communications, or in a basic course on probability and randomvariables. The course description and program evaluations are alsoincluded.Index Terms—Autocorrelation function (ACF), Gaussian band-

limited noise, Gaussian white noise (GWN), non-Gaussian whitenoise, power spectral density (PSD).

I. INTRODUCTION

T HE COMPLEXITY of the systems encountered in elec-trical and computer engineering calls for engineers to

have an understanding of random variables and noise con-cepts [1]. Yet, it is generally considered that it takes more timeto acquire a good understanding in this than in other subjectareas [2]. An example of work focusing on teaching randomsignals and noise using an experimental approach can be foundin [3].Computer-aided learning has become both an important

educational tool and a research topic in various engineeringdisciplines since using educational software can be of great helpin improving the quality of teaching of any given subject [4].It is widely accepted that introduction of a visual illustrationreinforces the lessons learned in the classroom in differentdisciplines [5]–[11]. Learners can better understand the issueunder investigation through a graphic presentation ratherthan through mathematical equations [9]. “Through graphicsdisplay, abstract mathematical results can be demonstratedimpressively and effectively; hence, students’ attention shouldbe captured and sustained” [10].In particular, the use of demo programs in teaching random

signals and processes gives students a visual and intuitive rep-resentation of random signals and processes, which have tradi-

Manuscript received November 30, 2010; revised March 06, 2011 and July18, 2011; accepted October 31, 2011. Date of publication December 02, 2011;date of current version July 31, 2012. This work was supported in part by theConacyt.The author is with the National Institute INAOE, 72000 Puebla, Mexico

(e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TE.2011.2176736

tionally been stated in terms of abstract mathematical descrip-tions. To this end, several demo programs have been developedto enhance the traditional classroom teaching of random signalsand processes, [12]–[15]. An earlier initiative pursuing similargoals was proposed in [16]. This paper presents a demo programfor understanding the concept of the autocorrelation function forthe wide-sense stationary (WSS) process , which was notexplicitly discussed in [12]; it was designed for and introducedin the graduate-level course “Introduction to Communications,”given for Electrical Engineering majors at the Institute INAOE,Puebla, Mexico.The discussion here begins by recalling the theoretical

underpinnings of the cumulative distribution function (CDF),the probability density function (PDF), autocorrelation func-tion (ACF), and power spectral density (PSD) for the WSSprocess [1].The CDF of a random variable is defined as the probability

that the variable is less or equal to any value of(1)

where means the probability of the event .The PDF of , is defined as the derivative of

(2)The demo program to estimate the PDF using (2) is described

in [15].The PDF completely specifies the behavior of the random

variable. It is worth noting that the students were already fa-miliar with CDF and PDF and their estimation before taking theclass on ACF.The ACF for the WSS process is defined as [1]

(3)where indicates the expectation operator.However, these probabilistic concepts can sometimes be con-

fusing for students; they may not understand, for example, whyPDF is not enough to describe a random process. The goal ofthis demo is to provide a better understanding of the autocorre-lation function than can be achieved by its typical introductionthrough its mathematical form (3).The Fourier transform of the autocorrelation function is

called the power spectral density (PSD), [1]

(4)

where stands for the Fourier transform of .0018-9359/$26.00 © 2011 IEEE

350 IEEE TRANSACTIONS ON EDUCATION, VOL. 55, NO. 3, AUGUST 2012

In electrical engineering, it is customary to present the meansquared value of as the average power of , which isrelated to PSD as [1]

(5)

The term “white noise” usually refers to a randomprocess whose PSD is constant for all frequencies [1]

for all (6)The ACF of such a process is

(7)

where is the delta function.In the computer experiment, the sequences are observed in-

stead of continuous signals. If the process is the white sequencewith zero mean and variance , the autocorrelation of thisprocess [1] is

otherwise. (8)

The corresponding PSD is equal to the variance [1]for (9)

where is the frequency band of interest.This PSD obtained from (9) is commonly referred to as an

ideal PSD because it is derived and is typically not estimated.While there are various methods for estimating PSD, this topicis beyond the scope of a basic course on random signals andprocesses for which this demo is designed. For this reason, thisdemo considers ideal PSD instead of estimated PSD.White noise whose amplitude is described by a Gaussian

PDF is called Gaussian white noise: A demonstration ofGAUSSIAN WHITE NOISE that describes the ACF, PSD,and PDF of the process is also included in the tool. As thecharacteristic “white” is related not to the amplitudes, butto the flat spectrum, the white process need not necessarilybe Gaussian, and vice versa. Lopez-Martin in [3] states that“this validation often surprises the student since in most com-munication systems they study noise is modeled as additive,white, and Gaussian for mathematical convenience.” To allowstudents to verify this for themselves, the tool includes thedemonstration NON-GAUSSIANWHITE NOISE where, as anexample, a zero mean uniform white noise is chosen, showingthat the non-Gaussian process may also be white. The demoGAUSSIAN NON-WHITE NOISE is also included to demon-strate that the Gaussian process may not always be white.Section II provides a brief description of the program.

The demonstrations are described and discussed in detail inSections III–VI. Section VII describes the course the wayin which this demo is used in the teaching process and pro-vides an evaluation of the demo program. Section VIII drawsconclusions.

Fig. 1. Menu.

II. DESCRIPTION OF DEMO PROGRAMThe content of the program is given in the menu shown in

Fig. 1.The topics in the menu are chosen to give a better insight in

the information contained in an ACF and to provide a betterunderstanding of the different types of noise in the frameworkof the ACF. The programs are written in MATLAB since this isencountered by electrical engineering students in a wide varietyof classes. Nevertheless, a student need not be a priori proficientin MATLAB or any other programming language, and this willnot preclude their giving all their attention to the topic beingpresented.At each step, the program provides the student-user with all

necessary instructions, including what to do to reach the nextstep. For example, to continue

pause % Strike any key to continue!or to present the corresponding plot

pause % Strike any key for the plot!The student has to play an active role during interactive di-

alogs prompted through the program by choosing all the neces-sary parameters for the demo program. For example

N=input(’Give the value N, N = ’);Give the value N, N =

where she or he types the desired values of the random variable,or

VA=input(’Give the value of variance, VA = ’);Give the value of variance, VA =

where the student types the desired value of the variance.The demo programs can be executed repeatedly with different

parameters.III. SIGNIFICANCE OF THE AUTOCORRELATION FUNCTIONAssuming that the process is zero-mean, i.e.,

(which is the case generally met [1]), if the autocorrelationfunction drops off slowly, then the probability of a large changein in seconds is small. In other words, the larger theautocorrelation function is at time , the more nearly “depen-dent” two measurements taken seconds apart are, as shownin (3) [1]. Therefore, the autocorrelation function containsthe information about the change of the process with time.

JOVANOVIC DOLECEK: MATLAB-BASED PROGRAM FOR TEACHING AUTOCORRELATION FUNCTION AND NOISE CONCEPTS 351

Fig. 2. Signals.

Fig. 3. Estimated PDFs.

Consequently, for the same values of lag , a fast process willhave smaller values of autocorrelation function than will a slowprocess irrespective of its amplitude distribution described bythe PDF.The goal of this demonstration is to show visually that the

autocorrelation function contains information about the changesof the process with time. Consequently, two processes may havethe same PDF but different autocorrelation functions. To thisend, two signals and are generated, as shown in Fig. 2,for the first 250 values. The signal is obtained by filtering thesignal with a low-pass filter and subsequent scaling in orderto keep the same variance as the input signal. At this point inthe demo, there is purposely no parameter to choose because thestudents have not yet been exposed to the problem of filteringof random signals.It can easily be observed that the signal changes slowly

compared to signal , which changes very fast. In the fol-lowing, the PDFs for both signals are estimated. Students areusually surprised by the result, which shows that both signals arezero-mean Gaussian signals with the same variance, i.e., bothhave the same PDF, as shown in Fig. 3.

Fig. 4. Autocorrelation functions.

Fig. 5. Signals and the corresponding ACFs.

At this point, it becomes clear that an additional characteristicof the process is required, which must include informationabout the difference between these two processes. This is pro-vided by the ACF for both signals, shown in Fig. 4. Both ACFsare decaying over time because both signals are zero-mean sig-nals. Additionally, both functions are equal at the origin becausethe signals have the same variance (Fig. 3). The ACF of thesignal (solid line), however, is decaying faster than that ofthe signal (dotted line) because the signal is changingfaster than the signal . As a consequence, the similarity of theamplitudes, i.e., their correlation (3) for the same value of , isless than for the signal .Both signals and their ACFs are plotted again in Fig. 5 to

show the relationship between the shape of the ACF and thechanges in the corresponding signals.

IV. GAUSSIAN WHITE NOISEThe goal of this demonstration is to show the characteristics

of the Gaussian zero-mean white noise (GWN): PDF, ACF, andPSD. The student-user chooses the variance of the noise. Asan example, Fig. 6 shows the generated noise with mean value


Fig. 6. Gaussian white noise.

Fig. 7. Estimated PSD.

zero and the variance 4, along with the estimated PDF, ACF, andthe ideal PSD, which is equal to the variance. The PDF clearlyindicates that the amplitudes are described by the Gaussian PDF,and consequently the noise is Gaussian noise.The estimated ACF, however, indicates that the process is

white; see (5). Therefore, Fig. 6 also shows the ideal PSD,which is flat and equal to the variance for the given frequencyrange. For completeness, the estimated PSD, created using asimple method called PERIODOGRAM [1], is also shown inFig. 7 just to remind the students that the estimated PSD isan estimation from (9), where the quality of the estimationdepends on the chosen estimation method. The topic of PSDestimation is addressed in an advanced course on digital signalprocessing (DSP).

V. NON-GAUSSIAN WHITE NOISEThe goal of this demonstration is to show that the white

process need not necessarily be Gaussian (and vice versa),and that the characteristic “white” is related not to amplitudes,but to the flat spectrum. In this demonstration, the zero meanuniform white noise in the interval is chosen, where

Fig. 8. Uniform white noise.

Fig. 9. Magnitude characteristic of the filter.

the student-user chooses the value on her/his own. Theestimated PDF and ACF and the ideal and estimated PSD areshown. As an example, Fig. 8 shows the noise in the choseninterval [ 4, 4], with the estimated PDF, ACF, and the idealPSD.The variance of the uniform variable is equal to [1]

(10)Note that the estimated PDF shows that the process is notGaussian, but uniform, and the estimated ACF and the plot ofthe ideal PSD show that the process is white.

VI. GAUSSIAN NON-WHITE NOISEThe goal of this demo is to show that Gaussian noise may

also be non-white noise. To this end, the GWN is filtered bythe low-pass filter whose magnitude characteristic is shown inFig. 9. The parameters are the variance of GWN and the cutofffrequencies of the filter. The input and output signals of the filterare shown in Fig. 10.Additionally, the input and output estimated PDF are shown

in Fig. 11.


Fig. 10. Input and output signals.

Fig. 11. Estimated probability density functions.

Fig. 11 shows that both the input and the output signals areGaussian with zero mean and different variances.Observing the estimated input and output ACF in Fig. 12, it

can easily be seen that the input signal is white noise, and thatthe output signal is not white noise. The spectrum of the outputsignal is band-limited by the filter, and consequently the outputsignal is a band-limited signal. Therefore, the output signal isGaussian, but not white noise.In order to visualize this statement, the output signal, along

with the estimated PDF, ACF, and PSD are all shown togetherin Fig. 13.VII. COURSE CONTEXT, USE OF DEMO, AND EVALUATION

A. Course ContextThe demo program described here was designed for and

introduced in the graduate-level course “Introduction toCommunications,” given for Electrical Engineering majors atthe Institute INAOE, Mexico. The course has a total of assigned45 class hours, and the main topics covered include modelingof communication systems, the characteristics of signals andsystems in the time and frequency domains, and random signals

Fig. 12. Input and output autocorrelation functions.

Fig. 13. Gaussian non-white noise.

and processes. This last topic, random signals and processes,occupies 25 class hours and includes the descriptions of oneor more random variables, transformation of random variables,Gaussian, Rayleigh, Rice, exponential, lognormal variables,random processes, autocorrelation, power spectral density, andnoise.The demo would also be appropriate at the undergraduate

level in a course on signals and systems or in other upper-levelsignal processing courses such as digital signal processing orcommunication systems. It may also be valuable in a basiccourse on probability and random variables.B. Use of DemonstrationThe students are already familiar with the PDF and its estima-

tion, and this demo aids them in better understanding the ACFand the noise concepts. As already mentioned, this introductorycourse does not discuss how to estimate PSD, so the ideal ratherthan the estimated PSD (9) is used to visualize the concept ofwhite noise. Students are shown the estimated PSD, however,so they can see the difference between the mathematical (ideal)and the estimated PSD.


Learning science research indicates that students learn muchbetter through active participation in the teaching/learningprocess rather than passive learning in the traditional class-room [16], and that collaboration among learners has asignificant impact on learning outcomes [17]. Following thisgoal, the use of demonstrations has two objectives: to providea visualization of the concept and to transform passive learninginto more active learning:First, theoretical descriptions of the ACF and noise are pro-

vided in the classroom. A total of 3 h are used for the theoreticalexplanations and numerical examples, followed by an additional3 h for the presented demo. The demo program is used in twosessions, of 1- and 2-h duration, respectively.In the first session, the instructor presents the demo programs

using the projector. In order to encourage active participationof students during the presentation, the students are asked topropose the demo parameters.The next session is used for students to run the demo program

on their own laptops, choosing their own parameters. Studentsare also invited to copy the results of the demo into a Worddocument for the following discussion. After carrying out eachof the four demos from Fig. 1, there is a discussion in which thestudents compare the results they obtained and participate in adiscussion. Typical discussion topics are the following: 1) whydoes the ACF decay faster in a more rapidly changing process?2) how does a student’s choice of variance value affect the resultobtained for Gaussian white noise? 3) how does the choice ofthe range of the uniform interval affect the uniform white noise?4) how does the choice of the variance and the cutoff frequenciesof the filter affect input and output PDFs and ACFs?The students demonstrated very high levels of motivation and

actively participated in all steps of the demonstrations and dis-cussion. Students can also use the demo by themselves, after theclasses, as an additional tool in solving homework assignments.C. QuizIn order to verify the utility of the demo as a teaching tool,

a simple 5-min quiz is given before and after the demo presen-tation. Additionally, at the end of the course, the same quiz hasbeen incorporated in the overall quiz used to evaluate all demoprograms used in the course.Typical questions for the quiz are the following.Q1) Why is the ACF needed if the PDF of the process is

known?Q2) Please write the expression for the autocorrelation

function.Q3) Please briefly explain the meaning of this expression.Q4) Is Gaussian noise always white?Q5) Can the uniform noise also be white?This quiz was introduced as a part of the evaluation process

for first time in 2010. Table I gives the number of correct an-swers as a percentage of the total number of students (12).As expected, early in the course students typically had diffi-

culty giving correct answers to questions Q1 and Q3–Q5. Afterthe demo presentation, however, all students answered ques-tions Q4 and Q5 correctly. Also, the number of correct answersto questions Q1 and Q3 increased by 33.3% and 16.7%, respec-tively, after the demo presentation, compared to the number of

TABLE IQUIZ RESULTS

correct answers before the demo presentation. Moreover, at theend of the course, when the students are ready for the exam,the number of correct answers rose by an additional 8.34% and8.53%, respectively.

D. EvaluationTo evaluate the quality of the demo in the teaching–learning

process, a student survey was conducted at the end of the course,after students had ample time to evaluate the potential benefits.The topics are chosen on the base of the literature [17], [18],

are used in all the questionnaires used in this work [12]–[15],and are dynamically modified each year in order to obtain morecomplete results.The following evaluation topics were covered in the 2010

survey:1) usefulness of the demo in teaching random variables;2) program design aspects;3) user perceptions.All questions on evaluation forms 1 and 2 are rated from 1

(lowest) to 4 (highest).Evaluation forms:1) Usefulness of the Demo in Teaching Random Variables

1.1) How justified is the use of the demo pro-gram in teaching the autocorrelation function?( unjustified; absolutely justified)

1.2) Did this demo help you to better understand theconcept of the autocorrelation function? ( NO;

Absolutely YES)1.3) Did this demo help you to better under-

stand the Gaussian white noise? ( NO;Absolutely YES)

1.4) Did this demo help you to better understandthe non-Gaussian white noise? ( NO;

Absolutely YES)1.5) Did this demo help you to better understand

the Gaussian non-white noise? ( NO;Absolutely YES)

1.6) How clear were the explanations given? (confusing; absolutely clear)

2) Program Design Aspects2.1) How easy was it to operate the demo? (

complex; very easy)2.2) How flexible and repeatable was the demo? (

not at all; very)


2.3) Were special knowledge or programming skills re-quired? ( to a high extent; not at all)

2.4) What was the general quality of presentation (fig-ures, resolution, visibility, etc.)? ( poor;excellent)

3) User Perception (General questions without marks)This third part of the survey was added in 2010 to elicit thegeneral user’s impression of the demo program.3.1) What was themost important thing you learned from

the demo program?3.2) What are your comments on the best and worst as-

pects of the content characteristics of the program?3.3) What are your comments on the best and worst as-

pects of the methodological characteristics of theprogram?

3.4) What recommendations do you have for improve-ment of the demo?

3.5) What other comments do you have?The evaluation was completed by a total of 59 students from

in the graduate-level course “Introduction to Communications,”in which the demo program was used as a complementary tool.Fig. 14 shows the result of the evaluation for the first two topicsin terms of the average marks for all questions. This shows thatstudents gave the maximum mark to questions Q2–Q5, sug-gesting that they found this demo to be very useful in givingthem a better understanding of the autocorrelation function andstatistical properties of noise. They also highly rated the pro-gram design aspects of the demo.Results of the third element of the questionnaire all indicated

that the demo has fulfilled its education objective. Some typicalanswers are listed as follows.Q1) What was the most important thing you learned from the

demo program? Answers:— “The meaning of the ACF.”— “The processes can have the same PDF but differentACFs.”

— “Gaussian noise is not always white.”— “White noise is not always Gaussian.”

Q2) What are your comments on the best and worst aspectsof the content characteristics of the program? Answers:Best aspects:— “Explanations of ACF and PSD in the contest ofnoise.”

Worst aspects:— “Include more details in the Demo 1.”— “Include the properties of ACF.”

Q3) What are your comments on the best and worst aspectsof the methodological characteristics of the program?Answers:Best aspects:— “Visualization of theoretic classes.”— “Complement to theoretical classes.”— “The way demo is applied.”— “Active participation in learning process.”Worst aspects: None mentioned.

Q4) What recommendations do you have for improvementof the demo? Answers:— “Include the properties of ACF.”

Fig. 14. Evaluation. (a) Usefulness of the demo for teaching autocorrelationfunction. (b) Program design aspects.

— “Include the cross-correlation function.”— “Improve the graphical design.”

Q5) What other comments do you have? Answers:— “Include demo programs in all topics of the course.”— “Include demo programs in other courses.”— “Excellent way of teaching.”

E. HomeworkThe demo was also incorporated into homework, where the

students were asked to estimate the ACF of the given process,the Gaussian white and non-white noises, and the uniform whitenoise. They were also asked to draw conclusions on how dif-ferent parameters (mean value and variance in Demos 2–4, andthe filter parameters in Demo 4) affect the obtained results.

VIII. CONCLUSIONThis paper presents a new initiative to include the use of demo

programs in teaching the concepts of autocorrelation functionand noise. Students are led step by step through the demonstra-tions that visualize the main characteristics of the autocorrela-tion functions in the context of white and band-limited noise.


The software was used as a complement to theoretical classes,but can also be used alone as a self-study tool. A student evalu-ation survey indicated that the demo program was an excellentcomplement to the teaching process, providing a visualization ofthe topic and allowing active participation by the students in theteaching–learning process, and that it helped students to betterunderstand the concepts of the autocorrelation function. In con-trast to an experimental approach [3], the proposed demo doesnot need any laboratory equipment and consequently is cheaper,requiring only a laptop and projector. Additionally, students canrun the programs themselves during and after classes, varyingthe different parameters, and can use the demo as additionalhelp in solving homework. The experimental and the demo ap-proaches are complementary. The proposal made in [9] dealswith MATLAB, but does not use the demo programs, which aredeemed more convenient for classroom teaching.

REFERENCES[1] A. Leon-Garcia, Probability and Random Processes for Electrical En-

gineering, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2008.[2] R. Quere, M. Lalande, J. N. Boutin, and C. Valente, “An automatic

characterization of Gaussian noise source for undergraduate electronicslaboratory,” IEEE Trans. Educ., vol. 38, no. 2, pp. 126–130, Feb. 1995.

[3] A. Lopez-Martin, “Teaching random signals and noise: An experi-mental approach,” IEEE Trans. Educ., vol. 47, no. 2, pp. 174–179,May 2004.

[4] J. J. Castro-Sanchez, E. Castillo, J. Hortolano, and A. Rodriguez, “De-signing and using software tools for educational purposes: FLAT, a casestudy,” IEEE Trans. Educ., vol. 52, no. 1, pp. 66–74, Feb. 2009.

[5] C. Depcik and D. N. Assanis, “Graphical user interfaces in an engi-neering educational environment,” Comput. Appl. Eng. Educ., vol. 10,no. 13, pp. 48–59, Feb. 2005, 2005.

[6] J. Wilson and W. Jennings, “Studio courses: How information tech-nology is changing the way we teach, on campus and off,” Proc. IEEE,vol. 88, no. 1, pp. 72–80, Jan. 2000.

[7] N. I. Sarkar and T. M. Craig, “Teaching wireless communication andnetworking fundamentals using Wi-Fi projects,” IEEE Trans. Educ.,vol. 49, no. 1, pp. 98–104, Feb. 2006.

[8] P. Pandey and C. Zimitat, “Medical student’s learning of anatomy:Memorization, understanding and visualization,” Med. Educ., vol. 41,no. 1, pp. 7–14, Jan. 2007.

[9] T. C. Hung, S. K. Wang, S. W. Tai, and C. T. Hung, “An innova-tive improvement of engineering learning system using computationalfluid dynamics concept,”Comput. Educ., vol. 48, no. 1, pp. 44–58, Jan.2007.

[10] G. S. Ng, “Teaching effectively with visual effect in an image-pro-cessing class,” Comput. Appl. Eng. Educ., vol. 5, no. 2, pp. 111–114,1997.

[11] M. Duran, S. Gallardo, and S. Toral, “A learning methodology usingMATLAB/Simulink for undergraduate electrical engineering coursesattending to learner satisfaction,” Int. J. Technol. Design Educ., vol.17, no. 1, pp. 53–73, 1997.

[12] G. Jovanovic Dolecek, “RANDEMO—Educational software forrandom signal analysis,” Comput. Appl. Eng. Educ., vol. 5, no. 2, pp.93–97, 1997.

[13] G. Jovanovic Dolecek and F. Harris, “On MATLAB demonstrations ofnarrowband Gaussian noise,” Comput. Appl. Eng. Educ., vol. 19, no.3, pp. 598–603, 2011.

[14] G. Jovanovic Dolecek, “Interactive MATLAB-based demo programfor sum of independent random variables,” Comput. Appl. Eng. Educ.,2010, to be published.

[15] G. Jovanovic Dolecek and F. Harris, “Understanding histograms, prob-ability and probability density using MATLAB,” in Proc. ASEE PSWConf., San Diego, CA, Mar. 2009, pp. 332–345.

[16] A. Herrera, “Design of course of random signals using MATLAB,” inProc. 26th Annu. Frontiers Educ. Conf., Salt Lake City, UT, Nov. 1996,vol. 3, pp. 1219–1222.

[17] O. Iglesias, C. Paniagua, and R. Pessacq, “Evaluation of universityeducational software,” Comput. Appl. Eng. Educ., vol. 5, no. 3, pp.181–188, 1997.

[18] E. D. Lindsay and M. C. Good, “Effects of laboratory access modesupon learning outcomes,” IEEE Trans. Educ., vol. 48, no. 4, pp.619–631, Nov. 2005.

Gordana Jovanovic Dolecek (M’96–SM’04) received the B.S. degree from theUniversity of Sarajevo, Sarajevo, Bosnia and Herzegovina, in 1969, the M.Sc.degree from the University of Belgrade, Belgrade, Serbia, in 1975, and the Ph.D.degree from the University of Sarajevo in 1981, all in electrical engineering.She was a Professor with the Faculty of Electrical Engineering, University of

Sarajevo, until 1993, and from 1993 to 1995, she was with the Institute MihailoPupin, Belgrade, Serbia. In 1995, she joined the Department for Electronics, In-stitute INAOE, Puebla, Mexico, as a Professor. During 2001 to 2002 and 2006,she was with Department of Electrical and Computer Engineering, Universityof California, Santa Barbara, as a Visiting Scholar. She was also with San DiegoState University, San Diego, CA, as a Visiting Professor from 2007 to 2008. Sheis the author of three books, the editor of one book, the author/coauthor of 18book chapters, and the author/coauthor of more than 250 papers. Her researchinterests include digital signal processing and digital communications, and theapplication of new methods in teaching process.Prof. Jovanovic Dolecek is a member of the Mexican Academy of Sciences

and the National Researcher System (SNI) Mexico.

Documents

ACF Matlab[1]d