DESIGN OF REACTIVE EQUALIZERS 734 



It is to be recof^mized in the following derivation and procedure thatfidp) 



represents the actual response of the network while Bdp) cos^" - , the pre- 



distorted specification for B{x-) in the ip domain, represents the desired 



response. For convenience, B(ip) cos^" | may be called the amplitude function 



a(<p). In addition, it is important to note that a(ip) is specified only over the 



range < (p < -, and the restrictions on the behavior of the approximating 



function /i(<^) outside this range are related to the restrictions on f(x-) in 

 the out-band region of the x domain. The general problem is thus one of 

 approximating the amplitude function a{(p) by a Fourier cosine series 



/ . ai; COS kip. 



k=0 



The first step towards a systematic method of obtaining the Fourier 

 cosine coelficients, ao ■ ■ ■ a^ , is the specification of the manner in which the 

 tolerance of match is to be minimized. In this case, the approximation is 

 always specified in the mean-square sense, i.e., the optimum coefficients are 

 obtained by solving the set of linear equations which are determmed when 

 the integral of the error squared, 



/ = / a((p) — 22 dk cos kip dip, 

 •> L k=0 J 



(23) 



is minimized. 



The set of linear equations which relates the a^ of the approximating 

 function /ly to the approximated function a(ip) is derived for a range to 5 

 in the ip domain with s < t hy minimizing eq. (23).- The minimum con- 

 dition is specified when the derivative with respect to each coefficient aj is 

 zero. Thus, 



da-^ i ^ I ^(^) - IJ ak cos kip [-cos 7V] dip = (24) 



is the analytical expression for this condition. Collecting terms, 



— = -2 / [a(ip) cos jip] dip + 2 / X) a.f, cos kip [cos />] dip 

 "";■ ''0 ^0 L'^^^o J -"-■' T- 



= —2 [a(ip) cos jip] dip + 2a j / cos 7V cos kip dip = 



•'0 JQ ' 



cos jip cos kip dip and C^ = / [a(ip) cos Mdip, the set of 

 '^ This derivation is similar to one given by R. M. RedhelTer in Ref. 6, pp. 8-10. 



