IDEAL FILTERS 



241 



obtained if the progressive reduction in the spacing between critical 

 frequencies extends over the complete transmission band, so that the 

 phase characteristic should resemble that of familiar ladder type filter 

 structures by becoming continually steeper as the cut-off is approached. 

 The exact arrangement will, however, depend upon the desired type 

 of best approximation to perfect suppression. If the approximation is 

 to be best at frequencies most remote from the cut-off, the critical 

 frequencies must be evenly spaced along an ordinary arc sine curve. 

 In the Tschebychefifian type of approximations studied by Cauer, on 

 the other hand, the spacing must be uniform along the arc of a certain 

 sn function. 



Design of a Band-Pass Filter 



To illustrate the manner in which small modifications of the theo- 

 retical frequency spacings may be employed to control the relative 

 emphasis placed on the phase and attenuation characteristics, we may 

 consider the design of a practical band-pass filter. Suppose that the 

 practical transmitting band is the 2,250-cycle interval between 11,375 

 and 13,625 c.p.s., in which the approximation of the phase charac- 

 teristic to linearity is specified by the requirement that dBjdco, the so- 

 called "delay," deviate from its average value by less than 0.1 milli- 

 second. The transition intervals are 500 c.p.s. each, beyond which 

 the loss is to be not less than 50 db. 



The comparatively liberal tolerances suggest that the approximation 

 furnished by m = 1 will be adequate. We notice that we can fit 10 

 uniform intervals of 250 c.p.s. between 11,250 and 13,750 c.p.s., which 

 locates the half-spaced cut-offs at 11,125 and 13,875 c.p.s. respectively. 

 When the characteristics corresponding to this design are checked, it is 

 found that the phase characteristic is rather better than required, 

 while the loss characteristic is weak. 



We then turn to the solution with m = 2, making a compensating 

 reduction in the number of uniform intervals. The critical frequency 

 allocation for this case is shown in Table IV. This arrangement meets 



TABLE IV 

 Critical Frequency Allocation for Linear Phase Shift Band-Pass Filter 



