DESIGN OF REACTIVI': EQl' AL// Eh'S 



solution for I he a-j according to eci. (29) is 



147 



As jireviously stated, these coefficients represent the practical minimum 



IT 



of the average error in the mean-square sense over the range to - in the (p 



domain. However, they do not represent the best match over the useful band 

 for this illustration. The adjustment of these coefficients to produce a more 

 satisfactory match at high frequencies in the useful band begins by changing 



the value of ao to make/i f ^ j = ao — ao = 0.125. This condition is satisfied 



when the general level of response is lowered so that co = —0.025. The only 

 further adjustment that is necessary in order to compensate for the de- 



3 



creased tolerance oifi((p) = ^oy cos 7^ at high frequencies in the useful band 



3=0 

 is a change in the value of as . When as is adjusted to 03 = 0.623 a suitable 

 approximating function for a((p) in this illustration is 



n 



/i(v) = ^ o-j cos 7V = —0.025 + 2.527 cosc^ 

 3=0 



— 0.150 cos 2(p + 0.623 cos 3</?. 



Hence, the approximating function for B((p) is 



fi(<p) _ -0.025 + 2.527 cos <p - 0.150 cos 2cp + 0.623 cos 3^ 



M = 



M^) 



COS^ ~ 



These functions are tabulated in the last two columns of Table H. 



The coefficients .4o •■• A3 oi f(x-) are easily calculated from {he J\((p) 

 a.nd f(<p) above by the relation of the Ak- to the Oj expressed in Table I. Thus, 

 fix'-) = 2.975 - 6.143.1-2 ^ 7.493.v^ _ 3.325.1;^ 



The tinal operation in the solution of the approximation problem is the 

 choice of the squared Tchebycheff polynomial, e-I^^i'.v), which satisties a 

 resistance efficiency of 65 per cent. The Tchebychefif polynomial for ;/ = 5 is 



Vb{x) = 5x — 20.1; + 16x . 



