770 



THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



where g is the rms deviation in a(ui) over the transmission band as 

 given by 



ni/2 



a = 



1 f y \ 1 

 — / a \w) aoi 



OJi Jo 





1 r ^ o 

 Ji -^0 



1/2 



(8.08) 



The rms amplitude deviation expressed in db is 



a'' = 20 logio(l + g) 



^ 8.69 g when g < 0.1 (8.09) 



A corresponding analysis can be made for fine structure imperfections 

 in the phase characteristic. The deviation /3(aj) = ypio:) — t/'o(w) from a 

 prescribed phase characteristic i/'o(co) may in this case be represented by 

 a sine Fourier series since the phase characteristic is an odd function of 

 co: 



/3(co) = 61 sin COT + 62 sin 2cor + • • • + 6», sin ??icor + 

 The resultant peak intersymbol interference becomes 



^6 = 1 &1 1 + 1 62 1 + • • • + I &« 1 + • • • 



and the rms intersjrmbol interference 



U, = 



\tK 



1/2 



= h, 



(8.10) 

 (8.11) 



(8.12) 



where h is the rms phase deviation in radians as given by 



[1 /•"! "]l/2 



- / |8'(co) do: , 

 COl Jo J 



(8.13) 



In the above derivation, the amplitude and phase deviations were 

 assumed independent of each other. The resultant rms intersymbol 

 interference from both is in this case 



u = {u: + u^Y' 



(g^ + i')' 



(8.14) 



This relationship, applying to an ideal transmission characteristic, has 

 been established by a different method in a paper by W. R. Bennett. 



