THEORETICAL FUNDAMENTALS OF PULSE TRANSMLSSION 



771 



From (7.05) it will be seen that with minimum phase shift relation- 

 ships a small cosine deviation of amplitude a„, in the amplitude charac- 

 teristic will be accompanied by a phase deviation bm = am. Hence in this 

 case (8.14) gives 



U = 2'" a (8.15) 



This also follows when it is considered that in this case all the pulse 

 echoes occur after the main pulse, and have ampUtudes Oi , a^ • • • a^ . 

 The root sum square of the amplitudes is in this case [X* '^''«T ^ which 

 is greater than Ua as given by (8.05) b}^ the factor 2"". 



The above analysis was based on an infinite sequence of pulse echoes, 

 which combine to give the proper pulse distortion but may be regarded 

 as fictitious in nature. The assumption of an infinite secjuence of pulse 

 echoes can be avoided by a different method of analysis outlined below, 

 which does not involve the assumption that the coefficients are known 

 from a Fourier series analysis, and furthermore, does not assume an 

 ideal amplitude characteristic with a sharp cut-off as above. 



Let Ae and Aoe~**° designate tw^o transmission — ^ frequency charac- 

 teristics, where A, Ao , \j/ and i/'o are functions of w, which for con- 

 ^'enience is omitted in the following. The squared absolute value of the 

 difference in the transmission frequency characteristics is then 



A,e-'^' 1' = 



I A e"' 



^0 [2(1 - cos ^) (! + «) + a], (8.16) 



where a = a^u) = {A — .4o)A4o represents the deviation in the ratio 

 of the amplitude characteristics from unity and /3 = /3(co) = xp — \po the 

 deviation in the phase characteristic. 



Let P and Po designate the impulse characteristics corresponding to 

 the above transmission frequency characteristics, and let AP = P — Po • 

 Assume that unit impulses of varying polarity are transmitted at uni- 

 form intervals rj . The rms value of AP over the interval n in relation to 

 the maximum amplitude P(0) of the received pulses, or the rms inter- 

 symbol interference U, is then given by 



U 



— [ A'[2(l - cos i3)(l + a) + a] do: 

 rri Jo 



1 



P(0) L^rri Jo 



(8.17) 



For small values of a and /3, this expression becomes 



u = 



(8.18) 



