teaching random signals and noise: an experimental approach

174 IEEE TRANSACTIONS ON EDUCATION, VOL. 47, NO. 2, MAY 2004

Teaching Random Signals and Noise: AnExperimental Approach

Antonio J. López-Martín, Member, IEEE

Abstract—A practical approach for teaching random signalsand noise is described, where theoretical aspects are comple-mented by several laboratory experiments enriching the student’sunderstanding on basic topics, such as histograms and estimationof probability density function, autocorrelation function, andpower spectral density. The equipment required is minimum andinexpensive. In fact, the existing equipment of laboratory benchesemployed for an electronic instrumentation course has been used.No investment from our institution has been necessary due to thefull exploitation of the potentials of the existing instruments andtheir PC connectivity.

Index Terms—Communication systems, noise, random process,random signals, undergraduate laboratory.

I. INTRODUCTION

ASOUND foundation concerning random signals and noiseis essential for evaluating the performance of communi-

cation systems. In order to teach such topics, concepts of signalanalysis and probability are combined to construct mathemat-ical models (random processes) [1], [2]. The resulting mate-rial is usually rather theoretical and abstract for the student,and the lack of an adequate correlation with real-world phe-nomena and measurements often discourages him or her. Theauthor’s teaching experience confirms this generally extendedclaim, which also occurs in related engineering topics with thislevel of theoretical content [3]. In this context, the author hastried to solve this issue by redefining the way such conceptsare transmitted; theoretical aspects on such topics are translatedto real measurements via electronic instrumentation as soon asthey are introduced, by means of carefully designed laboratoryexperiments.

Former initiatives have been proposed pursuing similar goals.In [4], computer simulations are presented for enhancing thestudent’s understanding on electrical noise. A similar approachis followed in [5] for the related topic of quantization noise.Nevertheless, real measurements on real-world signals allow thestudent to address practical issues and phenomena often not ob-served in computer simulations. In this sense, a pseudorandomnoise generator circuit is proposed in [6] for use in undergrad-uate labs. Building dedicated hardware by the students, such asthis generator, can be a rewarding experience since it involvesknowledge of many engineering disciplines. However, the tightschedules and excessive segmentation of topics in contempo-rary engineering programs often make this alternative difficult.

Manuscript received July 31, 2002; revised May 6, 2003.The author is with the Department of Electrical and Electronic Engi-

neering, Public University of Navarra, E-31006 Pamplona, Spain (e-mail:[email protected]).

Digital Object Identifier 10.1109/TE.2004.824838

For this reason, the use of general-purpose instrumentation isusually preferred. This latter choice has been followed in thelaboratory experiments described here.

Such a novel approach has been successfully applied forthree years now by the author in the course Communica-tion Theory, established in the Spanish Public University ofNavarra, Pamplona, Spain, as mandatory for junior students inTelecommunication Engineering. The course covers the mainaspects of analog communications, including the fundamentalconcepts of random signals and noise. The laboratory exper-iments designed have been revised according to the students’feedback during this time; their final version is presented inthis paper. One byproduct objective of such experiments was,as mentioned previously, the use of general-purpose equip-ment, readily available in most departments of electrical andcomputer engineering. Two obvious benefits of this approachare the avoidance of dedicated investments and the familiarityof the students with the laboratory bench acquired in formerlaboratory sessions.

II. COURSE CONTEXT

The course on Communication Theory at the Public Univer-sity of Navarra, in which the aforementioned laboratory exper-iments are introduced, has a total of 60 contact hours assigned.The main topics treated are linear modulation and frequencymultiplexing, angular modulation, random signals and noise,pulse modulation, time multiplexing, and information theory.As can be readily noticed, although many topics are studied,the main core of the course is analog modulations and noise incommunications systems, which are studied at length. When thestudent completes the course, he or she should have a sound un-derstanding on which modulation method best fits a certain ap-plication, how it can be generated and detected, and how noiseinfluences signal transmission.

III. LABORATORY BENCH

The laboratory bench employed for the experiments is thesame used in another course on electronic instrumentation inthis electrical and electronic engineering department, which wasdescribed in [7]. Hence, no additional investment was neces-sary by the institution. The equipment and components requiredare inexpensive and readily available in standard undergraduate,electrical, and computer engineering laboratories. This noveluse of existing technology instead of the employment of ded-icated (expensive and complex) instruments is a theme consid-ered important in undergraduate laboratory programs [8]–[10].

0018-9359/04$20.00 © 2004 IEEE

LÓPEZ-MARTÍN: TEACHING RANDOM SIGNALS AND NOISE: AN EXPERIMENTAL APPROACH 175

The complete laboratory bench is shown in Fig. 1. Eachstation in the undergraduate laboratory is outfitted with thefollowing items:

• PC Pentium II, 32 Mb RAM, 1 Gb HD;• IEEE-488 interface (PCIIA card);• HP 33 120A arbitrary signal generator;• PM 3335 analog/digital oscilloscope;• dc power supply; and• breadboard and some discrete components (resistors and

capacitors).

All the instruments include an IEEE-488 interface. The signalgenerator provides several built-in signals ranging from peri-odic sine or square waves to amplitude- or frequency-modulatedwaves or Gaussian noise. The oscilloscope has a dual analog anddigital nature, providing the didactical advantage of a simplecomparison between analog and digital operating modes. Bothinstruments, connected to the PC through the IEEE-488 bus,lead to a great deal of flexibility in configuring the laboratorybench. In particular, they notably augment the measurement ca-pability of the benches by emulating a spectrum analyzer and anetwork analyzer, which are not physically available in the lab-oratory bench. Dedicated software has been developed that, inconjunction with the measurement and connectivity hardware,configures the experimental bench. Such software was devel-oped using National Instruments LabView. However, the finalversion was implemented as stand-alone executable programs,so that the LabView software is not required in the bench-topPCs.

IV. LABORATORY EXPERIMENTS

The proposed set of laboratory experiments provides applica-tion of theoretical concepts, showing the student how to analyzeand characterize random signals and noise in practice. In partic-ular, they introduce basic tools and measurements for estimatingthe behavior of a random signal, assuming that it comes from anergodic random process [1], [2]. They are divided in two maingroups.

1) “Shape-independent” measurements. They are intendedto characterize the expected value of the random signalsampled at any instant. They include the most importantstatistical averages (mean value, variance, standard devi-ation, etc.) and the estimation of the distribution functionand probability density function (pdf) [1], [2] by meansof the evaluation of the amplitude histogram [1] over acertain time interval.

2) “Shape-dependent” measurements. They are aimed tocharacterize the expected signal variation over time (itsshape) and, therefore, also its expected spectral content.This information is provided by the autocorrelationfunction [1], [2] in the time domain and by the powerspectral density (PSD) function [1], [2], also known aspower spectrum, in the frequency domain. Both func-tions can be obtained in practice by the evaluation of theperiodogram [11] (square of amplitude spectrum scaled).

Since each of these sets of measurements provides only a par-tial description of the random signal, both are required to fullycharacterize it. Three experiments were designed to introduce

Fig. 1. Undergraduate laboratory bench employed.

the student to these issues and to provide insight about how tocarry out this task. They employ the experimental environmentdescribed in Section III, running a different program on the PCfor each experiment.

The first two experiments just require the PC since they arebased on simulated signals. They introduce important mea-surements required for characterizing a random signal. Fig. 2shows the main window of the first experiment. This programtrains the student to evaluate the shape-independent measure-ments mentioned previously in 1). The student is asked toselect sequentially among simulated Gaussian noise, uniformnoise, and sinusoidal waves with random initial phase. Theprogram generates such signals continuously and estimatesthe main statistical averages (mean, variance, etc.) for every

sample, being an integer value chosen by the student.It also calculates the histogram corresponding to these sam-ples. To do so, it just divides the amplitude range intointervals (bins) selected by the student and counts the numberof samples falling into each bin. The resulting bar diagramcorresponds to the histogram, as shown in Fig. 3. When theamplitude range cannot be limited (e.g., Gaussian noise) arange of is selected, where is the standard deviationcalculated previously. The successive histograms obtained arecontinuously averaged. The resulting average histogram is in-tegrated and normalized to a maximum value of 1, leading toa good estimation of the distribution function. Once derived,one gets the estimated pdf, as shown in Fig. 2. Mathematically

pdf with

(1)

with being the histogram obtained for the th set ofsamples. Variable corresponds to the signal amplitude, andis the number of acquisitions (and therefore histograms). Notethat since histograms are discrete-time functions, integrationand derivation in (1) actually correspond to accumulation anddifferentiation operators, respectively, in the practical imple-mentation. As a result, the estimated pdf is also a discrete-time


Fig. 2. Front panel of program for the first experiment.

function. The student can select a linear interpolation method toobtain a continuous-time estimation if necessary.

The second experiment shows the student how to obtain thesecond group of measurements, mentioned previously as shape-dependent measurements in 2). Fig. 4 shows the main screen ofthe program employed. As before, the student is asked to chooseamong different random signals. Then, the program generatessuch signals continuously and estimates the periodogram every

samples, where is an integer value chosen by the student.The successive periodograms are averaged, leading to a fair es-timation of the PSD function [11]. Mathematically

PSD (2)

with being the th periodogram of the set ofsamples, and the number of acquisitions (and thereforeperiodograms). In addition, by applying the inverse FastFourier transform to the estimated PSD, an estimation of theautocorrelation function is obtained [11]. Both estimations arediscrete-time functions, and the student can convert them tocontinuous-time by linear interpolation.

By means of this program, the student can readily relate thetime variation of the different types of noise implemented totheir autocorrelation function and power spectrum. Thus, thestudent can validate, for instance, that the Gaussian or uniformnature of noise is not directly related to its PSD, and in partic-ular, that Gaussian noise need not necessarily be white, and viceversa. This validation often surprises the student, since in mostcommunication systems they study noise is modeled as additive,white, and Gaussian for mathematical convenience.

Finally, the student is asked to generate periodic signals (tri-angular, square, ramp, etc.) with or without random initial phaseand to contrast again their shape with their autocorrelation func-tion and power spectrum. This exercise broadens the student’sperspective concerning these important measurements, showingthat they are not exclusive for noise, or even for random signals.

Once the students are introduced to the basic measurements inthe two former experiments, the third one applies them to real-

Fig. 3. Calculation of the histogram.

Fig. 4. Front panel of program for the second experiment.

world signal acquisitions with real instrumentation. It is basedon a new program that remotely controls the instruments andcalculates all the required measurements at the same time. Theprogram uploads the signal acquired by the oscilloscope andmakes the complete set of measurements described previouslyin order to characterize it statistically. Fig. 5 shows a screenshotof the program, where the estimated statistical averages, pdf,autocorrelation, PSD, etc., can be observed.

The first task of this laboratory experiment is to characterizethe noise generated by the signal generator using this program.The student connects the generator output to the input of theoscilloscope, which performs the analog-to-digital (A/D) con-version and uploads the noise samples to the PC. By observingthe program windows, the student can identify the relationshipbetween the PSD and the autocorrelation function (which is pre-sented simultaneously) and evaluate them in practice for a realnoise signal. He or she should identify the Gaussian nature ofsuch noise from the estimated pdf and relate the flatness of itsPSD to its lack of correlation. The student is then encouragedto measure the noise power using several methods: calculatingvariance and adding the squared dc value, measuring the auto-correlation function at the origin, or estimating the area underthe PSD. Finally, the student is asked to modify different noiseparameters in the signal generator (root-mean-square value andoffset) and to identify their effect on the different measured pa-rameters and functions.

In the second part of the experiment, the student builds asimple resistance–capacitance (RC) low-pass filter and appliesthe generator noise at its input, now characterizing the outputnoise. The reduction of noise power and increase in correlation


Fig. 5. Front panel of program for analysis of real signals.

as a result of the filtering can thus be observed by comparing thefilter input and output power and by observing the widening inthe autocorrelation function for the output noise, respectively.To assess this increase in correlation quantitatively, the studentis asked to measure decorrelation time, equal to about 4.61 RCfor this filter [11]. This procedure can be accomplished by mea-suring when the estimated autocorrelation function falls to 1%of its maximum value. The zoom function implemented in thegraphs facilitates this measurement.

The student is subsequently asked to evaluate how the PSD ofthe filter output is a good estimate of the (squared) filter ampli-tude response if the input noise has uniform PSD, an importantconcept in network analysis. The 3-dB cutoff frequency of thefilter is measured from the output PSD and compared with itstheoretical value, equal to . Finally, the equivalentnoise bandwidth of the filter can be readily estimated and com-pared with the theoretical expectation, given by 1/(4RC) [11].The student does so by applying [1]

Noise Equiv. BW (3)

with and being the measured power and the peak powerdensity (obtained from the PSD graph) of the filtered noise,respectively.

The last part of the experiment is employed to characterizeperiodic signals whose randomness is in their initial phase. Todo so, the student selects the desired periodic signal in a control

menu of the program, which sends the signal generator the re-quired bus command to generate it. The signal is then acquiredin the oscilloscope, with the trigger disabled (which producesrandomness in the initial phase of the successive acquisitions).First, sinusoidal waves are analyzed. The student usually ex-pects, from basic theory, a sinusoidal autocorrelation functionand the corresponding impulse function in the PSD. Surpriseoften arises when he or she observes a sinusoid with amplitudedecreasing linearly from the origin in the autocorrelation func-tion and a shape in the PSD. The laboratory teacher atthis point can justify this result by the truncation inherent to theacquisition of a periodic wave.

After this, periodic square waves are employed, and the de-pendence of the statistical measurements with their duty cycleis analyzed. Finally, triangular and ramp waves are studied. Theresulting estimated pdf and statistical parameters (mean, vari-ance, etc.) are identical to those of a noise with uniform pdf.This exercise helps the student to understand that these mea-surements are not enough for the statistical characterization ofthe signal and that PSD and autocorrelation functions are im-portant additional measurements.

The three laboratory experiments described are covered intwo laboratory sessions of 3 h each. The first one corresponds tothe first and second experiments, whereas the second one coversthe third experiment. The laboratory work of the course is com-plemented by two previous sessions, also of 3 h each, focused onlinear and angular modulations, respectively. In these sessions,


students generate amplitude- and frequency-modulated signals,analyze and relate their time and frequency representations, andstudy important effects (variation on modulation index, captureeffect in frequency modulation, etc.). Use of the same labora-tory bench, for these initial sessions, provides familiarity withthe laboratory equipment when the student faces the more de-manding experiments on random signals and noise.

In this way, 12 laboratory hours cover most of the coursetopics cited in Section II. Laboratory sessions are scheduled ina coordinated fashion with theoretical lectures, avoiding exces-sive delays between the acquisition of a novel concept and itsexperimental application.

V. IMPACT OF THE LABORATORY EXPERIMENTS ON STUDENT

LEARNING AND COMPREHENSION

As mentioned in Section I, the proposed laboratory experi-ments have been used for the last three years, and the experienceof the author is very positive. Students show more motivationonce they apply the theoretical concepts to real-world signals,a fact also confirmed by student feedback. In contrast to theexperience of former approaches (carried out prior to 2000in the Public University of Navarra), where only occasionalcomputer simulations complemented the theoretical material,a deeper grasp on how noise is characterized and how it affectssignal transmission has been witnessed in the students whocompleted the laboratory experiments. Particular topics wherethe students’ comprehension has shown clear enhancement are,for instance, the following:

• basic measurements for characterization of random sig-nals (How are they related? How are they performed usingconventional laboratory instrumentation?);

• practical issues affecting such measurements and theirsolution (e.g., signal truncation inherent to acquisition,solved by ensuring a sufficient acquisition time);

• effect of filtering on random signals;• application of noise signals in network analysis.

In order to quantitatively estimate this improvement, an anal-ysis of the student marks in the written final exams for thelast two years has been conducted. Since the described labo-ratory experiments are optional for the students to complete thecourse, only a subset of the students completed the laboratoryexperiments. This situation allows assessment of the impact ofsuch experiments among students with the same background,exposed to the same theoretical material, with the only distinc-tive variable being the laboratory work.

Table I shows results of such an analysis. The first row cor-responds to the students who completed the laboratory work,whereas the second row corresponds to those who just attendedclass lectures. The table includes the number of students whocompleted the laboratory experiments, their percentage over thetotal number of students enrolled, and their average results forthe last two spring semesters. To obtain the “results” coefficient,the numerical value of the part of the exam corresponding to thetopics covered by the laboratory experiments was normalized to10 and averaged among the students of each group.

By inspecting Table I, two main points are noticed: 1) there isan increase in attendance at the laboratory sessions in 2002 (be-

TABLE IIMPACT OF THE LABORATORY EXPERIMENTS IN THE LAST TWO

SPRING SEMESTERS

cause of the favorable references of students enrolled in 2001),which confirms that the experiments presented motivate stu-dents, and 2) there is an enhancement in comprehension of thetopics of random signals and noise gained by the students at-tending the laboratory, as reflected by their results in the finalexam. One could argue whether the results of a written finalexam precisely reflect the laboratory work or that students vol-untarily enrolling in laboratory sessions are a priori more mo-tivated and potentially receive better marks, even without thelaboratory. Considering this last argument, the “results” coef-ficient was calculated for the spring semester 1999, leading toa value of 3.79. This semester was the last without the labora-tory initiative. Comparing this value with the data of the last twosemesters in Table I, and with the results averaged over all thestudents (4.64 and 5.03 for 2001 and 2002, respectively), onecan make the following observations.

1) Better results are obtained on the average in 2001 and2002, leading to an overall improvement in terms of aca-demic success.

2) This increase is a result of the students attending the lab-oratory, since those not enrolled in the laboratory do notsignificantly deviate from the data of 1999.

All things considered, the proposed initiative has a positiveeffect on student motivation, learning, and academic successwith regard to most theoretical approaches employed in the past.

VI. CONCLUSION

Theoretical aspects of statistical signal analysis and char-acterization can be efficiently complemented with properlydesigned laboratory work so that students can have a deepercomprehension on the fundamental principles involved. Thelaboratory experiments described in this paper offer studentsthe opportunity to experience the fundamental concepts ofrandom signals and noise in a hands-on environment. Therequired laboratory benches are the same as those employedfor a course on electronic instrumentation in the department.They consist of inexpensive and widely available equipment,thus avoiding additional investment. Such experiments havebeen given for three years, providing reward to the author interms of academic achievement. The author would be glad toprovide the laboratory notes and programs to anyone interestedin them.

ACKNOWLEDGMENT

The author would like to thank the reviewers, whose construc-tive comments have remarkably improved the overall qualityof the paper. The author also wishes to thank M. Holmes andJ. A. Conner, who enhanced the manuscript’s grammar.


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Antonio J. López-Martín (M’04) was born in Pamplona, Spain, in 1972. Hereceived the M.Sc. and Ph.D. degrees (with honors) from the Public Universityof Navarra, Pamplona, Spain, in 1995 and 1999, respectively.

He has been at the New Mexico State University, Las Cruces, and the SwissFederal Institute of Technology, Zurich, as a Visiting Professor and Invited Re-searcher, respectively. Currently, he is Associate Professor with the Public Uni-versity of Navarra. He also holds the position of Adjunct Professor with the NewMexico State University. His research interests include very-large-scale-integra-tion (VLSI) microelectronics, analog and digital signal processing, and commu-nication systems. He has published more than 100 papers on these topics in in-ternational journals and conferences, holds two international patents in the field,and leads various research projects for local companies.

teaching random signals and noise: an experimental approach

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