200 BELL SYSTEM TECHNICAL JOURNAL 



M = - f" F(i^) cos [co/ + ^(co)]Jw, (1) 



and that the transfer impedance of the transmission system is given by 



Z(fco) = |Z(zco)|g''^('->, 

 where co/2x is the frequency. The received current is, then, 



^W =- r ^}^ cos [co/ + e{o:) - B{co)]dco. (2) 



7^(co) exists for all values of co from zero to infinity but, practically, 

 F{oj) is negligible except over a finite range which is determined by 

 the nature of the signal. For program transmission, for example, the 

 essential frequencies are now considered to lie in a band from about 

 100 to 5,000 cycles, while for slow speed telegraphy they lie in a 

 band between zero and 10 or 20 cycles per second. If we suppose, then, 

 that the essential frequency band extends from ui/lw to wijlir, we may 

 replace equations (1) and (2) by 



fit) = - I F{cc) cos [co^ + d{co)]dc^ (3) 



and 



^W = - Pj^J^cos [c^t + 0(co) - 5(a;)]Jco. (4) 



Now suppose that within the band of essential frequencies, wi 

 < CO < 0)2, we have 



|ZMl = i? (5) 



and 



B{o}) = COT zL nir, 



where R and r are constants and w = 0, 1, 2, • ••. Then we may 

 write 



/(/) = dz~ P ^(co) cos [co(/ - r) + 0(co)]f/co, (6) 



/(/) being positive or negative according to whether n is even or odd. 

 Whence the received current is proportional in amplitude to the 

 applied signal and merely delayed in time by the 'transmission time' 

 T. Thus the received current has the same wave form as the applied 



