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BELL SYSTEM TECHNICAL JOURNAL 



attenuation is prescribed at all frequencies. In the same way (4) 

 shows how the complete attenuation and phase characteristics can be 

 determined if we begin by prescribing the attenuation below /o and the 

 phase shift above /o. Since /o can be chosen arbitrarily large or small 

 this is evidently a more general formula than either (1) or (2), while 

 it can itself be generalized, by the introduction of additional irrational 

 factors, to provide for more elaborate patterns of bands in which A 

 and B are specified alternately. 



As an example of this formula, let it be assumed that A = K for 

 f < fo and that B = Ut/I for / > /o. These are shown by the solid 

 lines in Fig.. 7. Substitution in (4) gives the A and B characteristics 

 in the rest of the spectrum as 



B = k sin-i {■ , 

 Jo 



A = K + k\og 



J 1 + =^ 



2 ^ ^ r 



f<fo 



(5) 



U ^ ' /oj' 

 These are indicated by broken lines in Fig. 7. In this particularly 



Fig. 7 — Construction of complete characteristics from an attenuation character- 

 istic specified below a certain frequency and a phase characteristic above it. The solid 

 lines represent the specified attenuation and phase characteristics, and the broken 

 lines their computed extensions to the rest of the spectrum. 



simple case all four fragments can be combined into the single analytic 

 formula 



A -\riB = K + k\og 



f 



, + i 



(6) 



