IEEE TRANSACTIONS ON EDUCATION, VOL. E-25, NO. 1, FEBRUARY 1982

[2] B. D. Carroll, "A computer engineering nucleus of an electricalengineering curriculum," IEEE Trans. Educ., vol. E-22, pp. 76-80,May 1979.

[31 M. C. Mulder, "Computer science and engineering education:Introduction and overview," Computer, vol. 10, pp. 72-75, Dec.1977.

[41 R. H. Austing, B. H. Barnes, D. T. Bonnette, G. L. Engel, andG. Stokes, "Curriculum '78: Recommendations for the under-graduate program in computer science," Commun. Ass. Comput.Mach., vol. 22, pp. 147-165, Mar. 1979.

[51 G. L. Engel, "A comparison of the ACM/C3S and the IEEE/CSEmodel curriculum subcommittee recommendations," Computer,vol. 10, pp. 121-123, Dec. 1977.

[61 Guidelines for IEEE Ad Hoc Visitors on EDPD AccreditationTeams-Computer Engineering Programs. Engineers Council forProfessional Development, 1979.

W*iiam J. Bamett (S'61-M'67) received theB.S. and Ph.D. degrees from Clemson Univer-sity, Clemson, SC, in 1963 and 1972, respectiv-ely, and the MS. degree from Rutgers Univer-sity,New Brunswick, NJ, in 1965, all in electricalengineering.He has worked for Bell Laboratories, Holmdel,

NJ and Greensboro, NC and E.I. DuPont deNemours in Aiken, SC. When the above paperwas written, he was an Associate Professor atClemson University, working in the areas of

software engineering, microprocessor systems, and digital signal analysis.He has recently moved back into industry as Manager of the AutomationDepartment in the Engineering Group ofDanielConstruction Company,Greenville, SC.

Short Notes

A Note on the Fourier Series

SHLOMO KARNI

Abstract-A convenient pedagogical tood is recalled, in an effort toalleviate one of the difficulties in the presentation of the Fourier series.

The tutorial presentation of the Fourier series to undergrad-uate students is, admittedly, not easy. Without adequatepreparation in such topics as convergence conditions andorthogonal functions, a typical student is told, in so manywords, "Here it is. Take my word for it, it works!"While this arbitrariness may be unavoidable for the initial

presentation of the sine and cosine series

f(t) = aO + , (ak cos kwo t + bk sin kwo t), (I)k=1

the conscientious instructor should try hard to make all subse-quent steps and derivations more "palatable." One such stepis in the presentation of the equivalent cosine (or sine) series

f (t) = ao + E Ck cos (kwo t + 0k)- (2)k=1

A quick survey among some of the texts which treat Fourierseries [ I] -[4] reveals here 1) the use of yet another trigono-metric identity, or 2) the presentation of (2) as an ad hoc (andtherefore unsatisfactory) step, or 3) an apparently unmoti-vated multiplication and division by the factor (a2 + b2)1I2at this stage.Such a situation can (and should) be remedied in a rather

Manuscript received December 31, 1980; revised April 7, 1981.The author is with the Department of Electrical and Computer En-

gineering, University of New Mexico, Albuquerque, NM 87131.

,a,,

bi'Fig. 1.

effortless and acceptable way to the student by the use ofphasors at this point; after all, that student has been usingphasors recently.Since sin x = cos (x - 900) (most students will accept this!),

we can write a typical term in (1) as follows.

ak cos kcoot + bk sin kwot = ak cos kwot+ bkcos (kwo t - 900) (3)

and use phasor notation to represent the first term as ak andthe second term as bk. (See Fig. 1). Then phasor additionimmediately yields the resulting phasor

(ak + bk2)12/tan 1 (-bklak)

whose corresponding time expression is given in (2).

REFERENCES[1] D. E. Johnson et al., Basic Electric Circuit Analysis. Englewood

Cliffs, NJ: Prentice-Hall, 1978.[2] O. Wing, Circuit Theory with ComputerMethods. New York:

McGraw-Hill, 1978.[31 J. R. O'Malley, CircuitAnalysis. Englewood Cliffs, NJ: Prentice-

Hall, 1980.[4] L. S. Bobrow, Elementary Linear CircuitAnalysis. New York:

Holt, Rinehart and Winston, 1981.

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