IDEAL FILTERS 



231 



^ 10 = 



H 



a. 

 < 



10" 



103 



10' 



10 



0.1 02 0.3 0.4 0.5 06 0.7 0.8 0.9 10 



FREQUENCY 



Fig. 4. — Chart for phase computations. 



design parameters, may therefore be correspondingly important.-^ 

 Since slight adjustments in the design parameters will normally occur, 

 these charts are chiefly of value in making preliminary estimates. 



It is apparent that the approximation error at a given frequency 

 can be diminished either by increasing m or reducing a. Element for 

 element, an increase in m is much the more powerful method. Since 

 the total number of elements in the network is nearly proportional to 



" A simple example is furnished by the choice of the numerical constant multiply- 

 ing tanh Bll as a whole. It will be remembered that the constant was left undeter- 

 mined in the solution for the c's. In the original equation (14) it was chosen to give 

 the best characteristics in the neighborhood of / = 0. In preparing Figs. 3 and 4, 

 on the other hand, it was chosen with reference to the characteristics near/ = 00, 

 since the error expression used in these figures vanishes at that point. The two condi- 

 tions are very nearly equivalent; as we can see, for the half-spaced cut-off solution at 

 least, by means of Wallis' theorem. Since they are not identical, however, a change 

 from one to the other may produce a relatively large, though practically unimportant, 

 effect at extreme frequencies. 



