NETWORK SYNTHESIS GG5 



network singularities. This makes it possible to apply power series ap- 

 proximation methods, in terms of z, to obtain approximations based on 

 TchebychelT poljaiomial series, in terms of frequency. 



"Maximally flat" approximations in terms of z may be used to match 

 tlie first m terms in the Tchebycheff polynomial series representing net- 

 work gain or phase to the corresponding terms in the series representing 

 assigned gain or phase. In this way, a Tchebycheff polynomial type of 

 least squares approximation to the network function is made identical 

 to the corresponding least stjuares approximation to the ideal function. 

 The overall error, network function minus ideal function, is then the 

 difference between the two least squares errors. 



The s'-plane analysis may also be manipulated, in a quite different 

 way, to approach an equal ripple type of approximation (which usually 

 represents approximation in the Tchebycheff sense). The complications 

 are such that applications have been limited to problems of certain 

 quite special types. On the other hand, analysis of this sort has been 

 found useful in clarifying various other ways of seeking equal ripple 

 approximations . 



REFERENCES 



1. S. Darlington, "The Potential Analogue Method of Network Synthesis," Bell 



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2. G. L, Matthaei, "A General Method for Synthesis of Filter Transfer Func- 



tions as Applied to L-C and R-C Filter Examples," Stanford University 

 Electronics Laboratory Technical Report No. 39, Aug. 31, 1951 (for Office of 

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3. T. R. Bashkow, "A Contribution to Network Synthesis by Potential Anal- 



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 June 30, 1950 (for Office of Naval Research, NR-078-360). 



4. E. S. Kuh, "A Study of the Network-Synthesis Approximation Problem for 



Arbitrarj' Loss Functions," Stanford University Electronics Laboratory Tech- 

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5. R. Courant and D. Hilbert, Methoden der Mathematischen Physik, Vol. I, 



Chap. 2, Julius Springer, Berlin, 1931. 



6. C. Lanczos, "Trigonometric Interpolation of Empirical and Analytic Func- 



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7. H. A. Wheeler, "Potential Analog for Frequency Selectors with Oscillating 



Peaks," Wheeler Monograph No. 15, Wheeler Laboratories, Great Neck, 

 N. Y., 1951. 



8. S. Darlington, "Synthesis of Reactance 4-Poles," J . of Math, and Phys., 18, 



pp. 257-353, Sept., 1939. 



9. T. C. Fry, "Use of Continued Fractions in Design of Electrical Networks," 



American Math. Soc. Bulletin, 35, pp. 463-498, July-Aug., 1929. 



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11. H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Co., 



New York, 1948. 



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