IDEAL FILTERS 225 



where the A's are constants. In (20) this series is convergent. In 

 (17) it is merely asymptotic. It is known, however, that the error 

 due to ending (17) at any term is numerically less than the first 

 omitted term. Since we are at present interested in small values of a, 

 therefore, we can estimate the error in the approximation from the 

 first term alone. 



Inspection of this term shows that the error is greatest in the vicinity 

 of the transition interval, where the factor 1/(1 — /) is large. It de- 

 pends upon all three of the quantities /, a and m ; but by choosing 

 them in the proper order there is no difficulty in showing that an 

 indefinitely close approximation can be obtained. 



The transition interval must first be selected on the absolute fre- 

 quency scale. It may be as small as we choose. Next, a value of m 

 must be chosen. What value is used is immaterial for our present 

 purposes, although it is important for later applications. Finally, 

 a must be taken small enough so that all transition factors lie in the 

 prescribed transition interval. Otherwise it may be varied at will. 

 But by choosing it small enough, the error of approximation (22) can 

 obviously be reduced without limit for any value of / outside the 

 interval (/a, /b). We may thus conclude that only considerations of 

 expense and of manufacturing precision restrict the accuracy of 

 approach to the ideal filter. 



For purposes of future reference approximate formulae for the 

 attenuation and phase in the limiting condition are given below: 



A -u _ 



e ^ = 



2 ^-+1'^""^' [ (1 - /)™+i + (1 + /)-+! J ' ^^^^ 



""^'[71 — "^-K-T-. + .1 i^x ^. 1 sin-^- (24) 



a 



It will be seen that the attenuation rises monotonically as we recede 

 from the transmission band while the phase curve ripples about the 

 ideal straight line in a sinusoid of varying envelope. The ripples, of 

 course, increase in frequency as a is diminished but since the exponent 

 m -f 1 is always at least 2 they flatten out so rapidly that dB/doo 

 approaches constancy nevertheless. We may also observe that, 

 although the absolute time of delay increases indefinitely as a de- 

 creases, it varies only as 1/a, whereas the precision of approximation 

 can be made indefinitely great by choosing m large. 



Filters of Other Types 

 While the preceding analysis has been restricted formally to low- 

 pass filters, its application to filters of other transmission types is a 



