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BELL SYSTEM TECHNICAL JOURNAL 



theorem. It reads as follows: 



B 





dA 

 du 



log coth -x-du, 



(2) 



where B(fc) represents the phase shift at any arbitrarily chosen fre- 

 quency fc and u = logflfc. This equation, like (1), holds only for 

 the minimum phase shift case. 



Although equation (2) is somewhat more complicated than its 

 predecessor, it lends itself to an equally simple physical interpretation. 

 It is clear, to begin with, that the equation implies broadly that the 



03 ^ 



O 



_l 



II 



0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 



3 4 5 6 8 10 



Fig. 2 — Weighting function in loss-phase formula. 



phase shift at any frequency is proportional to the derivative of the 

 attenuation on a logarithmic frequency scale. For example, if dA/du 

 is doubled B will also be doubled. The phase shift at any particular 

 frequency, however, does not depend upon the derivative of attenua- 

 tion at that frequency alone, but upon the derivative at all frequencies, 

 since it involves a summing up, or integration, of contributions from 

 the complete frequency spectrum. Finally, we notice that the contri- 

 butions to the total phase shift from the various portions of the fre- 

 quency spectrum do not add up equally, but rather in accordance with 

 the function log coth | w | /2. This quantity, therefore, acts as a weight- 

 ing function. It is plotted in Fig. 2. As we might expect physically 



