DESIGN OF REACTIVE EQUALIZERS 733 



IHjrtiou of the useful band an increased precision of equalization.'^ This 

 adjustment represents an increased transmission at low frequencies. Thus. 

 it is sometimes necessary to employ an ecjualizer of the constant resistance 

 t_\-pe when additional equalization is desired at low frequencies. Figures 16 

 and 1 7 have been drawn to reflect this condition on Aq . 



After an/(.v-) which conforms with the requirements outlined above has 

 been found, it is necessary to find a 



eW\{x) = A'J + .I2V + • • • + A'J" (15) 



which, when added to/(.v'-), produces the desired B{x~). This procedure is 

 greatly facilitated by the known properties of Tchebycheff polynomials: 

 A Tchebycheff polynomial of order n is defined by 



Vn{x) = cos (« cos~%). (16) 



This function oscillates between plus one and minus one for | .v | < 1 and 

 approaches ± » for | .t | > 1 . Tabulated below are the expanded analytical 

 expressions for the polynomials for n = 1 through ;; = 8. 



Fi(.v) = X F6(-v) 



F2(.v) - 2.V- - 1 V,{x) = 32/ 



Vz{x) = 4x - 3x V7{x) 



\\{x) = 8.V* - Sx-' + 1 T^8(.v) = 128x' - 256.v' + 160.v' - 32x' + 1 



With the help of the recursion formula, 



xVn(x) = ^[Vn+l{x) + Vn-l(x)], (17) 



the corresponding expressions for w > 8 may be systematically calculated. 

 Figure 18 shows a plot of the Tchebycheff polynomial for n = 5. 



In the case of low-pass filters'^ and impedance matching networks,'^ 

 Tchebycheff polynomials are often used for the solution of the ai)proxima- 

 tion problem. The function | Zi2(jx) |- in these cases has an oscillatory be- 

 havior which approximates unity in the useful band, and has all its zeros 

 at infinity so that the network consists of n elements of an unbalanced 

 ladder structure of alternating series inductances and shunt capacitances. 

 The appropriate function for | Zi2{jx) |- is 



I ^'='^'^' I' = 1 + } V'M ' ''«^ 



" There is a practical limit to the reduction of A below e^"o. Referring to Figs. 13 and 



14, it is apparent that A' = — . Thus, .4o is a direct measure of the impedance level over 



the useful band, and must not be made too small if the highest practical level of response 

 is to be attained. 



'8 Ref. 2, pp. 53-79. 



" Ref. 3, pp. 26-34. 



