172 BELL SYSTEM TECHNICAL JOURNAL 



The sign of nl is chosen so that | X,. | < 1, and this is the value to be used 

 in M\ However, there is an ambiguity in the sign of Xr which is inherent 

 in this method. A relation between A and M^ is > 



AP = (/ + A')(/ - A')-' (A3.5) 



Let Ur be proportional to any non-zero column and Wr be proportional 

 to any non-zero row of the matrix adjoint to [^J — ZnYn] and form the 

 matrices 



U = [til , U2 , ■ ■ ■ Um] 

 W = [Wl,W2, • • • W„,] 



(cf. equations (A2.6) and (A2.7) for S and T) where w,- is the column 

 obtained from Wr. 



The voltages and currents are given, as before, by 



v(n) - PA"o + PA~"a 



(A2.8) 

 i(n) = QA^'a — QA "a 



and there is again a number of ways in which P and Q may be chosen. In 

 all cases the rth column of P may be expressed as a,Mr and the rth column of 

 Q as ^rWr . The equations fixing Q when P is chosen and vice versa are, 

 from equations (A3.1) 



Q = YnPM^' 



(A3.6) 

 P = ZuQM^' 



where M ' is the inverse of M^. Equations (A3. 6) may also be obtained 

 from (A2.10). 



Suitable choices for P and Q are 



1. P = U, Q = YnUW^' = ZnUM^ 



2. P = UM', Q = YnU = ZnUM 

 S, Q = W, P = ZnWM~' = YnWM'^ 



■ 4. (3 = WM'% P = ZiiW = YnWM 



P'Q and U'W must be diagonal matrices. That the expressions for v(n) 

 and i{n) just derived satisfy the difference equations (A3.1) may be verified 

 by making use of 



UM = ZnYnU, WM == FiiZijr (A3.8) 



Equations (A3.8) follow from the properties of the individual columns of 

 U and W. The characteristic impedance and admittance matrices are 



(A3.7) 



