552 BELL SYSTEM TECHNICAL JOURNAL 



Appendix II 

 A ttenuation 

 The following is a derivation of the standard deviation of the real 

 part of the echo currents (which are received in phase with the direct 

 transmission) over a circuit such as has been assumed in Appendix I. 

 Accordingly, the reflected wave at the junction of the kth and {k + l)th 

 homogeneous elementary lengths, for a current of unit value traveling 

 down the circuit at this point, is: 



hk — hk+i. (1) 



This wave returns toward the sending end and in turn suffers partial 

 reflections. Consider this secondary reflection at the point between 

 the jth and (j + l)th lengths where j < k. The wave arriving at the 

 point in question is 



(hk - hk+i)e-P^'-^-^'\ (2) 



The fraction of this wave which is reflected back again is 



- (hi - //./+i), (3) 



so that the wave which starts back from this point in the same direction 

 as the original wave is: 



- (hi - h,-^^)(hk - hk+i)e-P^'-^^i\ (4) 



In traveling to the junction of the ^th and (k -\- l)th lengths it is 

 again multiplied by e~^'-''~^'"^ so that the echo which is joined to the 

 unit wave is therefore given by 



- (hi - hi^,)(hk - //,+i)e-^(*-'). (5) 



U m = k — j, this echo is 



— (hj — hi+i)(hj+m — hj+„,+i)e~"'^ when m > 0. (6) 



The sum of all the echoes for a given value of m > is: 



n — m 



- e-'^P Z (hj - hi+i)(hi+m - hj+m+i) = - e-^^Hm- (7) 



3=0 



When w = 0, a slightly different treatment is necessary. Let the 

 circuit be represented as in Fig. 1. 



A unit current traveling down the circuit will suffer a reflection loss 

 at each junction so that the current passing through the junction is 

 (1 — r],) times the current entering. The ratio of the current received 



