NETWORK SYNTHESIS ()17 



is especially appropriate for general network applications, because the 

 odd ordered polynomials contribute to the imaginary parts of complex 

 network functions — such as ijS in a -\- ib.'\ 



It is apparent from (3) that the Tchebycheff polynomials become 

 simply Fourier harmonics, if they are plotted against a distorted fre- 

 quency scale — that is, against 0. This means that they must be ortho- 

 gonal, over that particular range of frequencies which corresponds to 

 real values of 4>. From the relation between 4> and co, it is clear that real 

 \'alues of (/) cover the frequency interval between — Wc and -\-Uc , which 

 is our useful interval. In other words, the interval of orthogonality coin- 

 cides witii the useful frec}uency inter\'al. The corresponding interval of 

 p is of course p = —iojc to -\-iuc . 



If a given fiuiction is plotted against <^, instead of w, it may be ex- 

 panded in a Fourier series. Each term in the series may be replaced by 

 a Tchebycheff polynomial, to obtain an expansion of a given function 

 in terms of polynomials, for the specific useful interval co = — Wc to 

 -\-cx}c . Established techniques are available for expanding experimental, 

 or other numerical data, in Fourier series, as well as actual analytic 

 fiuictions. 



In Fig. 2, some of the Tchebycheff polynomials are plotted against co. 

 The frequencies — co^ and +coc are also indicated. Frequencies between 

 these limits correspond to real values of the angle variable (p. If this 

 part of the frequency' scale is stretched, in the proper non-uniform way, 

 the various "loops" not only have the same maximum values, but also 

 the same shapes. In other words, they become periodic. More specifically, 

 a stretch which changes the frequency scale into a <A scale changes the 

 plots into sin /:0 or cos k4>. 



4. TRANSFORMATION OF VARIABLE 



An alternate to (3) may be obtained by relating a new variable, z, to 

 4> l^y 



z = e'* (4) 



Substituting z in the exponential equi\'alent of sin </>, in the first equa- 

 tion of (3), gives an alternative definition of z directly in terms of p, 

 namely: 



t A small change in the definition of would bring the definitions closer to 

 convention, by repkicing both sines l)y cosines (without altering Tk as a function 

 of p). This however, would complicate our later analj'sis. 



