64 PROCEEDINGS OF THE IEEE JANUARY

I t is our feeling that (4) and ( 5 ) will form a theoretical basis for improved approximation techniques in the frequency domain, and at present we are beginning to investigate approximation and synthesis techniques based on these results.

J. W. CRAIG L. M. GOODMAN

Lincoln Laboratory’ Mass. Inst. of Tech.

Lexington, Mass.

REFERENCES [ l ] W. Cauer. Synthesis of Linear Communicalion Ndruwks. New York: McGraw-

Hill, 1958. ch. 8. [2] L. Weinberg. N c t w k Analysis and Synthesis. New York: McGraw-Hill. 1962.

ch. 11. [3] M. Dishal. ‘Gaussian-response filter design,” Elccf. Commun.. vol. 36, pp. 1-26,

January 1959. 141 E. Ulbrich and H. Piloty. “Uber den Entwurf von Allpawn. Tiefpassen und

Gruppenlaufzeit” Arch. Elekt. Ubertragung. vol. 14. PP. 451467, October 1960. Bandpassen mit einer im Tschebyscheffschen Sinne approximierten konstanten

[SI T. A. Abele. ‘UbertragungsLaktoren mit Tqchebyscheffscher Approximation konstanter Gruppenlaufzelt. Arch. E l e k f . Obnlragung. vol. 16, Pp. 9-18,

[6] N. Wiener and Y. W. Lee, ‘Electrical network system,” U. S. Patents 2 024 900, January 1962.

Y. W. Lee, “Synthesls of electnc networks by means of the Fourier transforms 1935; 2 128 257. 1938; 2 124 599, 1938.

[7] S. Darlington, “NetwArk synthesis using Tchebyscheff polynomial series.” of Laguerre’sfunctions ” J . Moth. Phys., vol. 1 1 . pp. 83-113, 1932.

Bell Sys. Tech. J vol 31 pp. 613-665. July 1952. [SI H. Watanabe, “Apprbximation theory for filter networks.” I R E Trans. on

Circuit Theory. vol. CT-8, pp. 341-356. September 1961. [9] G. A. Mathews et al.. .Approximating transfer funchons by linear program-

[lo] R. M. Lerner, ‘Band-pass filters with linear phase.” Proc. I E E E . vol. 5 2 . ming.” Proc. Firsf Allerlon Conf. on Circuit Theory. pp. 191-204, 1963.

[ l l ] R. Unbehauen. ‘Die rationale Approximation von Frequenzcharakteristiken.” pp. 249-268. March 1964.

[12] T. Fultsaw.a. Theory and procedure for optimization of low-pass attenuation Arch. Elekt. U!crlragung. vol. 18. pp. 607-616. October 1964.

characteristics, I R E Trans. on Circuit Theory. vol. CT-11. pp. 449-456.

[13] C. Pottle and S. J. Thorp, ”On the optimum least-squares approximation to December 1964.

the ideal low pass.” Proc. Sccond Annual Allerton Conf. on Circuit and Sysfcm Theory, pp. 201-219. 1964.

[141 J. Vlach. ‘.4pproximation graphisch gegebener Amplituden-, Phasen-. und Gruppenlaufzeitkurven.” Arch. Elekf . Ubcrtragung, vol. 19, pp. 5&58. January

151 A. Papoulis. The Fourier Integral and i f s Applications. New York: McGraw-Hill. 1965.

1965, pp. 198-199.

1 Operated with support from the U. S. Air Force.

On Determination of Cutoff Frequencies of Waveguides with Arbitrary Cross Section

Several approximate methods have been adopted by authors to determine the cutoff frequencies of waveguide with complicated cross section [1]-[7]. Each method has its own advantages for certain types of waveguides and its own disadvantages for other typesof waveguides. For example, the transverse resonance analysis has been applied to shallow ridge guides [l ] and shallow groove guides [8] and has given solutions which are simple and quite accurate. How- ever, this method gives solutions for trapezoidal waveguides [9], which are not as accurate as Yashkin’s approximate technique [2].

In a recent correspondence, Laura [lo] stated that the point- matching method is not a satisfactory technique for solving eigen- value problems with complicated configurations. He implied also that conformal mapping, along with various approximate techniques, excels the previous method. Although the point-matching method is not applicable for some types of problems, as analyzed in a recent paper [7], it will handle other types of problems more easily than other methods and will give quite accurate solutions. Consider the cutoff frequencies of a waveguide with cross section of a pentagon or hexagon, as an example. Using the point-matching method, the nor- malized cutoff wave numbers can be calculated easily by a computer with accuracy of three to four places. However, solving this problem by conformal mapping requires finding the mapping function, and

it may not be possible to obtain its exact solution. Using approximate techniques to solve an approximate partial difference equation (which is the result of the approximate mapping function) yields a double approximation for the solution. Furthermore, it is not easy to find the approximate mapping function and, if one is not found, the graphical method must be used.

In fact, conformal mapping, along with other approximate tech- niques, and the point-matching method are (almost) dual methods for considering two dimensional eigenvalue problems. The solu- tions of an eigenvalue problem must satisfy the partial differential equation and the boundary conditions. In general, the solutions of the former method satisfy the boundary conditions exactly but sat- isfy the partial differential equation approximately, while the solu- tions of the point-matching method satisfy the partial differential equation exactly but satisfy the boundary conditions approximately. It is obvious that problems with complicated boundary conditions are easier to handle by conformal mapping, along with various ap- proximate techniques, while problems with simple boundary condi- tions and simple configuration are easier to handle by the point- matching method.

H. Y. YEE Research Institute

University of Alabama Huntsville, Ala.

REFERENCES S. B. Cohn. “Properties of ridge waveguide.” Proc. I R E , vol. 35, PP. 783-789 August 1947. A. Ya. Yashkin. ‘A method of approxiyate calculation for waveguides of tri- angular and trapezoidal cross-section. Radio Engrg.. vol. 13, pp. 1-9. October 1958.

guides of very general cross-sections. Proc. IEEE. vol. 51. pp. 1436-1443. H. H. Meinke. K. P. Lange, and J. F; Ruger. ‘TE- and TM-waves in wave-

November 1963. F. J. Tischer and H. Y. Yee. ‘Waveguides witharbitrarycross-section consid- ered by coafdpmal mapping.” UARI research rept. 12. January 1964”; also

versity d Alabama Research Institute. Huntsville, Ala., UARI research rept. ‘Parabolic and elliptic waveguides considered by conformal mapping, Uni-

16. May 1964. M. Chi and P. A. Laura, .Approximate method of determining the cutoff

mmc T h w y and Techniques (Correspondence). vol. MTT-12, pp. 248-249, frequencies of waveguides of arbitrary c r o s section,” I E E E Trans. on Mino-

March 1964.

cross sections.” Proc. I E E E (Cmcspondcncc) , yol. 53. pp. 6?7-638. June 1965. H. Y. Y e and N. F. Audeh, ‘Cutoff frequencies of waveguides with arbitrary

considered by the point-matching method,’ I E E E Trans. on Minowarre Thcory H. Y. Yee and N. F. Audeh. ‘Uniform waveguldes with arbrtrary cross-sxtlon

and Techniques. vol. MTT;l3. pp. 847-85;. November 1965. J. W. E. Griemsmann. Grove guide, presented a t the 1964 Symp. on Quasi-Optics. Polytechnic Institute of Brooklyn. N. Y. S. T. Uptain and N. F. Audeh, ‘Transverse resonance Flution of uniform trapezoidal waveguides.’ to appear in I E E E Trans. on Mmowasc Tkeory and

P. A. Laltra. ‘Detection of cutoff pequencies of waveguides with arbitrary Techniques. March 1966.

cross sections by point-matching, Proc. I E E E (Correspondence), vol. 53. pp. 1660-1661, October 1965.

System Identification via Discrete Differential Approximation

Recently, several methods [l], [2] have been suggested for the identification of dynamic systems. For continuous time dynamic sys- tems, it is shown [3], [4] that the differential approximation method offers many advantages, the major one being the possibility of on-line application. In this letter, the discrete differential approximation method is offered as a useful method for identifying the parameters of noisy, nonlinear, and nonstationary discrete time systems.

Suppose a dynamic system is modeled by a difference equation

x @ + 1) = f(x(k), u ( k ) , ~ ) 0 I k I N (1)

where x(k) is the nX 1 state vector a t the k t h sampling instant, u(k) is the mX 1 known input vector, a n d p is the r X 1 unknown constant parameter vector. The actual value of p is the one for which

Manuscript received November 22.1%5. ported in part by a Nattonal Saence Foundatton G n n t GK-303. Manuscript received, November IS , 1965. The research reported here w a s SUP

1966 PROCEEDINGS LETTERS 65

x(k + 1) - , f (x (k ) , u ( k ) , p ) E 0 0 I k I N . (2)

However, a reasonable approximation to p would be a value pl

such that a suitable function of

x(k + 1) - f ( x ( k ) , u ( k ) , p d 0 I k I N is close to zero in some acceptable sense. For example, pl is obtained as a solution of

where (,) is the Euclidean inner product. The minimization operation implied by (3) results in 7 simultane-

ous algebraic equations that can be solved for the parameter vector. These equations are

where af/ap is an r X n matrix with elements

In case pi is nonstationary, one could represent it by the poly- nomial

aiki M

i-0

and now the minimization is performed over ai, i = O , 1, * . , M. Numerical experiments on the identification of several systems

will appear elsewhere. One important unresolved question is the fol- lowing. The previous scheme assumes that all the components of the state vector can be measured. Can this restriction be overcome? In the continuous time case, the answer is yes [4].

K. S. P. KUMAR Dept. of Elm. Engrg.

University of Minnesota Minneapolis, Minn.

REFERENCES [l] S. G. S. Shiva, .Parameter estimation of a dass of linear fixed differential sys-

tems.= Proc. I E E E (Conespondcnce). vol. 53, pp. 632-634, June 196.5. 121 J. E. Diamessis. 'On the determination of the parameters of certain nonlinear

systems," Proc. I E E E (Conesfiondencr). vol. 53. PP. 319-320. March 1965. 131 R. Bellman. R. Kalaba, and R. Sridhar, "Adaptive control via quasilinearization

and differential approximation. The RAND Corporation. Santa Monica, Calif.,

141 P. N. Hamilton and K. S. P. Kumar. 'System Identification via Differential RM-3928-PR. November 1%3.

Approximation' (to be puhhshed).

Fermat's La.st Theorem and the Smith Chart Pierre Fermat (1601-1665) wrote in the margin of a book which

he was studying that he had a proof for the theorem: It i s impos- sible to find integers x , y , z , and n which satisfy the equation

3? + y" = 6

if n > 2 . However, "the margin is too narrow to contain it." For three centuries, mathematicians have tried to find a proof without being successful. Proofs have been developed for many values of n. Since the theorem has been proved for n=4, it is necessary to consider only the case where n is an odd prime integer p.

I t is interesting to note that an equivalent theorem can be stated for a section of transmission line. I t follows from the well-known

Manuscript reodved Novembu 15,1965.

properties of the cydotmnic equation that

xp + YP = fil k-O ( x + Y L W , where e = Z r / p . If one assumes that y < x and lets r = y / x , the factors x ( l + r Lke) may represent voltages on a Smith chart.

Plot the point

Z+Y 2 0 = -

2 - Y

on a Smith chart, as shown in Fig. 1. Draw a circle with its center a t Z= 1 through the point 20. Divide the circle into p equal arcs be- ginning a t 20, and designate the end points as t . Draw lines from the point Z=O to each point b . Let Lk denote Ek/(radius of chart). Finally, it follows from Fermat's Last Theorem that

PdLoLILt ' * * L p - I

must be an irrational number.

-. ".

Fig. 1. Construction on a Smith chart.

The construction on the Smith chart can represent a section of transmission line, as shown in Fig. 2. The termination RL and the characteristic impedance Ro may be any rational numbers such that RL #Ro and RL >Ro. The length of the section is one-half wave- length, and it is divided into p equal parts. If V,= VO is any rational number, then

PdV"VlV2 * * * V P l must be an irrational number. If one wishes to become famous, all he has to do is to prove the last statement or find a contradiction.

H. F. MATHIS Dept. of Elec. Engrg.

R. F. MATHIS Dept. of Mathematics

The Ohio State University Columbus, Ohio