TIME DIVISION MULTIPLEX SYSTEMS 209 



carrier waves applied to a product modulator, and an electrical analogue 

 of time division may be realized therefore by applying signal and a suitable 

 carrier to a product modulator. Phase shifts in the carrier supply circuit 

 may be made to serve the same purpose as the angular displacements 

 between commutator segments. It is thus of interest to examine various 

 other possible forms of the function Fj{t) which are suitable for multii)lex 

 transmission and investigate methods by which they can be realized. 



We note that (26) is suitable for an odd number of channels because it 

 makes use of the direct signal component (or sideband on zero frequency) in 

 addition to the paired sidebands on harmonics of the switching frequency. 

 It seems reasonable to expect that systems for even numbers of channels 

 can be devised using only upper and lower sidebands on harmonics and 

 omitting the signal itself. Complete information for the separation of N 

 channels should be contained in any set of .¥ sidebands; hence we should 

 not be forced to start with the sidebands of lowest frequency, but be able 

 to use other sets with a more suitable place in the spectrum or with better 

 equalization of amplitudes. 



We shall derive an expression for a quite general switching function 

 meeting the desired conditions of freedom from crosstalk and economy of 

 band width for an even number of channels by assuming the following 

 forms for .1,,,; and 0,„y in (3), 



/A,n < m < n + N /2 - \ \ 



A,„ = [ ^ (27) 



\0, m < n, or m > « + N/2 - 1/ 



e,„, = (i - 1) (m + h)rp + a (28) 



The switching function assumed contains iV/2 harmonics and hence will 

 produce N sidebands. The values of n, h, a and \J/ are first assumed to be 

 arbitrary. At the receiving end, a switching function similar except for a 

 time displacement /o will be assumed. That is, in (9), we take 



/B, n < m < n + N /2 - 1 \ 



5,„,. = (29) 



\0, m < n or ;;; > ;/ + X /2 - 1/ 



^mk = (^ - 1) (m + h)yp + {mq -^ v)r + a (30) 



Transmiissicn over the line is assumed to be of the distortionless form ob- 

 tained by setting Z(,{m) = Zo , a constant, in (20). Substituting (27)-(30) 

 in (14), we then calculate 



fl, j = k 



NABZ,,e~'''"' \ 2 sin N{j - /fe)^/4 cos (in + 4h + N - 2) {j - k)^P/4: 



F;,= 



N sin ij - k)xl,/2 



j 9^ k (31) 



