A METHOD OF IMPEDANCE CORRECTION 



815 



to the "constant-^" type image impedance, has also been added for 

 comparison. It will be seen from the curves that these values of 

 the coefficients Ai . . . A„ give very good approximations for small 

 values of x, but inferior ones for values near unity. It is preferable in 

 most designs to sacrifice something at the lower end of the characteristic 

 in order to secure better performance in the higher range. 



N 



UJ 



O 



z 



^ 



o 



o 

 a 

 z 

 o 

 o 



tr 

 o 



1.2 



1.0 



0.8 



0.6 



O 



N 



UJ 



o 



z 

 < 

 I- 



(0 



in 

 111 

 a 



0.4 



0.2 



0.1 



0.2 



0.3 



0.4 



as 0.6 



0.7 



0.8 



0.9 



Fig. 



6 — Resistance and conductance characteristics secured from the binomial 



expansion. 



The advantage of an approximation distributed over the band is 

 gained by an expansion in terms of Legendrian harmonics. These 

 functions are discussed in standard reference books, such as Byerly 

 "Fourier Series and Spherical Harmonics" or Whittaker and Wat- 

 son "Modern Analysis." It is important to mention here, how- 

 ever, that they are simply polynomials. Any polynomial such as 



|5(1 +^ix2 + 



AnX^") can be broken up into a linear combina- 



tion of even ordered harmonics, and, conversely, any linear com- 

 bination of even ordered harmonics can be reduced to the form 



^ (1 + Aix^ -j- . . . Anx"^"). It is therefore easy to convert an ex- 



pansion in terms of even harmonics into a power series of the sort 

 with which we are directly concerned. The property of these functions 

 of most interest here is the fact that, for an expansion of any given 

 degree, they give the best "least squares" approximation to the 

 desired function. In the range between x = and x = 1, therefore, 

 the approximation they furnish is much better for most purposes than 

 that given by the binomial theorem. The expansion of \1 — x'^ in 



