746 



BELL SYSTEM TECHNICAL JOURNAL 



In this illustration a simplified f{x-) = .4o + AiX + A2X + A3X of 

 order (n — 2) may be chosen such that the transfer characteristic is matched 

 within the specified tolerance over the useful band.^^ The specification a{<p) 

 is determined from B{x^) by (1) calculating the B((p) which corresponds to 



B{x~) in the <p domain, and (2) multiplying B((p) by cos " ;^ to obtain a{(p) = 



B{(p) cos'" - . The results of these calculations in the <p domain are indicated 



by the fifth and sbcth columns of Table IL 

 The Fourier cosine coefficients, oo • • • an , are found by solving the set of 



IT 



linear equations expressed by eq. (25) for n = 3 and s = - . The Ck which 



depend on the approximated function a(<^) are computed from eq. (26). 

 After the indicated graphical integration is carried out, these constants have 

 the following values in this illustration: 



The existence of a solution of eq. (28) depends on || P,vt || 9^ 0. In this case 

 this determinant becomes 



II Pjk II ^ 0.00009. 



Thus, for all practical purposes, the linear equations for n = 3 represent a 

 dependent set. However, when Poo = t is substituted for -J above," the 



^ For the value of the tolerance specified in this illustration, an f(x^) of order 3 turns 

 out to be satisfactory. In the general case, where a higher degree of precision is desired, 

 it is, of course, expedient to choose an /(x^) of order n. 



'^ See discussion on p. 742. 



