736 BELL SYSTEM TECHNICAL JOURNAL 



The final expression for i^(.v-) may now be written as 



i^(.x:2) = fix') + eWlix) = (/!„ + AyX' + • • • + Anx'") + 



(A[x~+ ••• -i-Anx'"). (20) 



In terms of eq. (20), the corresponding expression for the special transfer 

 function for equalization purposes becomes 



Zn(jx) 2 



K 



Ro 



1 



(21) 



Ao + (^1 + A[)x' + (^2 + A2)x' + ... + (^„ + A'Jx' 



When all the Ay and A;, are known in a particular design, the coefiticients 

 Bi • • ■ Bn of eq. (7) may be readily determined. Hence, the elements of the 

 network may be found by using the appropriate equations of Section 2. 



4. Approximation Method 



This section will consider the second of the two main tasks in the formu- 

 lation of the design method. Broadly speaking, the special transfer function 

 derived in the previous section, eq. (13), provides the approximating func- 

 tions to be used in this problem while this section develops the systematic 

 method of determining the coefficients of these functions for a finite number 

 of network elements. The function of most interest in the approximation 

 problem is the in-band approximating function f{x-). Thus, the develop- 

 ment of the approximation method for reactive equalizers is concerned 

 specifically with the determination, consistent with the previous limitations 

 and requirements, of the coeflScients, Ao • • • ^„ , of the polynomial /(x-) . 



The Fourier method of polynomial approximation, first introduced by 

 Wiener, ^1 is characterized by a transformation of the independent variable 

 to make the approximating function in the new frequency domain a periodic 

 function. Thus, the well-known method of Fourier analysis is available as a 

 general polynomial approximation method. This method has not been ap- 

 plied extensively in practical applications. However, the uniform nature of 

 Bix"^) over the useful frequency range makes its application to the design 

 of reactive equalizers of the type described here seem feasible. 



By the transformation x = tan <p/2 the frequency domain, < x < <» , 

 is transformed to a corresponding (p domain, < ^ < tt. Since the range of 

 interest is to tt in the <p domain, all functions may be assumed to be either 

 even or odd with a period 2x. Thus, any amplitude approximating function 



2' Ref. 4. 



