IDEAL FILTERS 239 



represents a sketch of tanh djl corresponding to the m = 2 curve of 

 Fig. 6. It will be seen that the curve rises monotonically toward the 

 line unity, at which = co . What we should evidently like to obtain 

 by slight alterations in the design parameters is a curve which rises 

 more rapidly, or perhaps one which ripples about unity. It is also 

 evident that the curve approximates unity so closely that even slight 

 adjustments may produce a radical effect. To take the simplest 

 possibility, if the constant multiplier of tanh 6/2 is slightly increased, 

 so that the curve crosses unity at a finite frequency, the appearance of 

 the resulting attenuation characteristic will be greatly altered. The 

 net gain in the general level of attenuation secured, however, will be 

 not more than 6 db. Similar remarks might be made with respect to 

 the phase characteristic. 



The relation between the phase and attenuation characteristics 

 where such adjustments are made can be illustrated most easily by 

 reference to the elementary half-spaced cut-off solution for the transi- 

 tion factors. It will be recalled that this solution was obtained by 

 equating the coefficients of the first powers of 1/(1 — /) and 1/(1 +/) 

 in (17) and (20). The approximation error thus depends chiefly upon 

 the succeeding term involving 1/(1 — /) and 1/(1 +/) to the second 

 power. A study of the expression shows that the error makes Q too 

 small in both the transmitting and attenuating ranges. If the phase 

 characteristic is the more important this error can be partly com- 

 pensated by slightly increasing the normal half-space between the 

 cut-off and the preceding critical frequency. On the other hand, the 

 attenuation will be improved if the interval between the cut-off and 

 the preceding critical frequency is decreased. To a more limited 

 extent, both characteristics can be improved by increasing the constant 

 factor which multiplies tanh 6/2 as a whole. 



A similar study might be made of the other groups of transition 

 factors, although the discussion would naturally become more com- 

 plicated. In general it appears, as with the half-spaced cut-off 

 solution, that the attenuation characteristic will be improved by a 

 slight decrease in the spacings of the transition factors, while the phase 

 characteristic will be improved if they are slightly increased. It 

 should be remarked, however, that as the network becomes more 

 complicated, either by a reduction in a or an increase in m, the de- 

 sirable modifications in the theoretical spacings are reduced. This 

 becomes evident if it is recalled that the transition factor spacings are 

 proportional to a while the error is roughly proportional to 0:"'+^ 

 It is therefore to be expected that the appropriate modifications in the 

 spacings between transition frequencies will be of the order of magni- 

 tude of a^ times their original values. 



