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BELL SYSTEM TECHNICAL JOURNAL 



where e is an arbitrary constant. Figure 19 shows the plot of the squared 

 Tchebycheff polynomial, eWn{x), for the values of » = 5, and e = 0.5 

 and 6 = 0.1, while Fig. 20 shows a j)lot of the transfer function expressed 

 in eq. (18). 



It is to be noted that the oscillatory behavior with equal maxima and 

 minima of squared Tchebycheff polynomials for values oi x < 1 and the 

 rapid approach to + °c for values of x > 1 make their use particularly 

 suitable as the solution of the approximation problem for low-pass filters 

 and impedance matching networks. It is now apparent that these same 



Fig. 18 — Tchebycheff polynomial, F„(.t), for w = 5. 



properties validate their use as the out-band approximating function for 

 reactive equalizers.-" 



Another useful property of squared Tchebycheff polynomials as ap- 

 proximating functions for low-pass filters and impedance matching net- 

 works is the inclusion of the specification of the tolerance as a factor in the 

 transfer function. The allowable db deviation over the useful band is related 

 to e by 



€' = e 



- 1, 



where ap is the maximum pass-band loss in nepers. Thus, the appropriate 

 choice of e always realizes the specified tolerance over the useful band. 



2" When better tolerances are required and when the network configuration is not 

 rigidly specified, Jacobian elliptic functions, rather than TchebychefT polynomials, might 

 be employed. 



