TIME DIVISION MULTIPLEX SYSTEMS 217 



and hence that for purposes of estimating requirements we may replace 

 I CjK- I" in (45) by its average value over the band. We may then write (45) 

 in the form: 



X 



Q.?= "^' -f/,. (46) 



Sj I WjK (wy) dwj 



The value of the right-hand member, which we have designated by the 

 symbol U jk , is either known or can be determined for the particular type of 

 signal. Hence our problem is reduced to finding the allowable departures 

 in transmission which keep the mean square absolute value of Cju from 

 exceeding a prescribed maximum value. 



We note that Cjk is the sum of TV complex quantities, each of which is 

 restricted to a range of values determined by the transfer impedance of the 

 line in an individual band of frequencies. A convenient simplification may 

 be made by regarding the N complex quantities as N independent chance 

 variables. This is tantamount to assuming that departures in one band 

 do not afifect departures in any other band; the assumption is not strictly 

 true, but should lead to no important error. We may then make use of the 

 following theorem : If 



r = bizi + 62Z2 + • • • + bnZn , (47) 



where Si , Z2 , • • • z„ are n independent complex chance variables and 

 bi , b2 , ' • • bji are complex constants, 



I f f = 1 61 r 1 21 1'' + • • • + 1 6„ r I z„ r m 



Application of this theorem to (44) gives 



I Q^ f = ~ Z (I ^[ii^ -\-mq + CO,-)] \' + 1 z*[i{v -\- mq - coy)] f) 



1\ m=n 



= jTfVTv, (49) 



if the average square of the absolute value of the departure is the same in all 

 bands and is equal to ] 2 | , which we shall define as the average squared 

 absolute value of the departure for the entire line band used. 

 From (42) and (40), 



I z(io}) 1=1 — 2p(co) cos 6(00) + p (co) 



= 1 - 2.10"'"^''" cos 0(co) + nf^"'"' (50) 



8 R. S. Hoyt, B. S. T. J., Vol. XII, No. 1, Jan. 1933, p. 64. 



