IRREGULARITIES IN WIRE TRANSMISSION CIRCUITS 553 

 to the current that would be obtained without reflection loss is 



-^ = (1 - r7o)(l - 7?0(1 - 772)(1 - 7?3) • • • (1 - y)n), (8) 



-10 



where the double reflected echoes of the previous type (w > 0) are 

 omitted. The echo which is joined to the unit wave when m = is 



A/ / - Jo 



h h 



(9) 



Logey- = Loge j = -y , when A/ is small. (10) 



Since 



A7 " " 



-y- = Loge n (1 - r?,) = E Log. (1 - 7/,) (11) 



^0 ,=0 ;=0 



and 



Log. (1 - r?) = - 7? - vV2 - 7,3/3 • . •, (12) 



therefore the echo is given as follows in nepers: 



- L (vi + viV2+ ..-) 



J=0 



= — \i— hi + hi — hi + hi — h + hs ■ ■ • — hn + //n] 



- ^ Z (hj - h^+iY. (13) 

 ;=o 



The first term is zero. The sum of all the echoes is 

 -lliihj-h^+iy]- Ze-^H^ 



l ■^ j=0 J Jn=l 



= - n E (/^,- - h,+iy I - i'H^B-e^-t (14) 



I- "^ ;=0 -^ ».=1 



The in-phase component of these echoes is 



Ecr= -\IZ {hi - hi+iY I - L HmB^ cos w0, (15) 



assuming /?'s may be taken as real and as having a symmetrical distri- 

 bution curve about zero, the square of whose standard deviation may be 

 denoted by h^. 



We will consider the distribution curve of Hm, which also is real. 

 The average value of a function H{h) in a given distribution is equal to 



