130 BELL SYSTEM TECHNICAL JOURNAL 



characterized by the amplifying factor A which is independent of 

 time, and the network by the real function G{t) which is the response 

 caused by a unit impulse applied at time / = 0. The combined effect 

 of the amplifier and network is to convert a unit impulse to the 

 function AG{t). Both these characterizations will be used. 



The restrictions which are imposed on the functions in order that 

 the subsequent reasoning may be valid will now be stated. There is no 

 restriction on A other than that it should be real and independent of 

 time and frequency. In stating the restrictions on the network it is 

 convenient to begin with the expression G. They are 



It may be shown ^ that under these conditions G{t) may be expressed 

 by the equation 



where 



G{t) =^. f J(io:)e"^'d(io:), (5) 



JM = r G{t)e-''''dt. (6) 



ty — Qo 



These expressions may be taken to define /. The function may, 

 however, be obtained directly from computations or measurements; in 

 the latter case the function is not defined for negative values of co. It 

 must be defined as follows to be consistent with the definition in (6) : 



/(— iw) = complex conjugate of J{iw). (7) 



While the final results will be expressed in terms of AJi^iix)) it will be 

 convenient for the purpose of the intervening mathematics to define an 

 auxiliary and closely related function 



uiz) = -^ r^i:^Mrf(ico), <x < ^, (8) 



2irtjj ico — Z 



where 



z = x + iy (9) 



and where x and y are real. Further, we shall define 



2v(iv) = lim w{z). (10) 



z— »-0 

 ^ See Appendix II for fuller discussion. 



