THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 701) 



pulses and will thus varj' with time. It can have any value assumed by 

 the expression 



UM) = ±|-^±|^±|?±|-.. ±|:±^+ ••• (8.03) 



The maximum possible intersymbol interference will thus be the sum 

 of the absolute values of the coefficients a„, . 



f'a = I «! ! + 1 ao I + I as I + • • • + I o,„ I + • • • (8.04) 



In certain pulse systems, such as PAM time division systems, rms 

 intersymbol interference is of main importance, while in others, such as 

 PCM or telegraph systems, peak intersymbol interference is of principal 

 interest. If the fine structure imperfections are regarded as of random 

 nature, in the sense that they are not predictable and vary between 

 systems having the same nominal transmission characteristics, peak 

 intersjTnbol interference can be estimated from rms interference by 

 applying a peak factor of about 4. With random variation in the ampli- 

 tude of intersymbol interference, the probability of exceeding 4 times 

 the rms value is in accordance with the normal law about 5 X 10~^ Peaks 

 in excess of 4 times the rms value will thus be so rare that they can for 

 practical purposes be neglected. 



The rms intersymbol interference is equal to the root mean square 

 of all the different values which can be assumed by expression (8.03). 

 This turns out to be equal to the root sum square of the amplitudes 

 am/2 and —am/2 of the pulse echoes, or 



Ua = 



1/2 





When Urn are the various coefficients in the Fourier representation of 

 a{u) over the frequency band from — cci to coi , the following relation 

 holds. 



^ 2_/ ^m = o~ / a (w) dco = ~- / a^((jo) do 



I 1 ZCOl J-ui COi Jo 



(8.06) 



where a(w) in the present case is given by (8.02) and represents the 

 departure in the ratio /l(aj)/^lo(co) from unity. 



With (8.06) in (8.05) the following expression is obtained for rms 

 intersymbol interference due to amplitude deviations a:(co) not accom- 

 panied by phase deviations 



C/a = g (8.07) 



