772 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



If a and /3 are random variables representing fine structure deviations 

 uniformly distributed over the transmission band, it is permissible to 

 simplify (8.18) to: 



U = v(-^T {a+hy'\ (8.19) 



where 



1 / /•Wmax \l/2 



a = I / adc^), and (8.21) 



/ 1 /""niax \l/2 



b = 1 / /3' dco , (8.22) 



\Wmax •'O / 



where Wmax is defined as in Fig. 30 and coi is the bandwidth at the half 

 amplitude point. 



For a transmission characteristic with linear phase shift, aside from 

 small random imperfections as considered here: 



P(0) = - / ^0 do. (8.23) 



T Jo 



For the particular case of a transmission characteristic with constant 

 amplitude between co = and coi = Wmax , 17 = 1. Pulses would in this 

 case be transmitted at intervals n = tt/oji so that tt/ojiti = 1 and (8.19) 

 is identical with (8.14). 



For a transmission characteristic of the type sho\vn in Fig. 13, pulses 

 would also be transmitted at intervals n = tt/wi so that tt/coiti = 1. 

 In this case comax = 2wi , and evaluation of (8.20) gives 77 = 3^'^/2 = 

 0.866. Rms intersymbol interference is thus reduced by the factor 0.866, 

 for the same values of g and b. However, these are now the rms devia- 

 tions taken over a band which is twice as great as with a sharp cut-off 

 at coi . 



Expressions (8.14) and (8.19) can also be applied to localized imper- 

 fections in the amplitude and phase characteristics confined to a narrow 

 portion of the transmission band. This follows when it is considered 

 that such deviations can be represented by Fourier series containing a 

 large number of coefficients, so that the resultant intersymbol inter- 

 ference can attain a great number of different values depending on the 

 sequence of transmitted pulses. A particular case of a localized imperfec- 

 tion in the amplitude characteristic in the form of a low-frequency cut-off 

 is considered in the following section. 



