SWITCHED NETWORK FOR TIME MULTIPLEX SYSTEMS 1413 



different form. Advancing the time function Si{t) by t/2 seconds, one 

 gets the function So{t) which is even in t. As a result its transform So{p) 

 is purely real, that is, 



Soip) = ^ ^° g cosh ^. 

 p -\-coo ^ 



From an analysis carried out in detail in Appendix IV we finally obtain 



^^°^^^= 2[Z(p)So(p)]* • ^'^^ 



It should be pointed out that (23) is still valid when r = 0. Equations 

 (20) and (23) give the zeroth approximation to the gain of the system 

 for any driving current nit). 



In many cases it is sufficient to know only the steady-state response 

 Eioip) to an input io{t) = /oe^"^' . The response Eio(p), as given by (23) 

 [or (20)] includes both transient and steady-state terms. Since hip) = 



^-r— equation (24) gives 



P - J^o 



(^ E Zuip + jnco.) ^ /" r-) So{p)Zn{p) 



^"^^^ 2[Z{p)So(p)]* • ^^"^ 



Since neither So(p) nor Ziiip) have poles on the imaginary axis, the 

 steady state includes only the terms corresponding to the imaginary 

 axis poles of the summation terms. Thus the steady-state response is of 

 the form 



where, from (25), 



2 X) Zijuo + j(k - ti)cos]So[j(jOQ + j{k - n)a).v] 



. _ IoZi2(,j^(i)So(juo — jnuis)Zr2U^o — Jnc*}s) .. 



^« - ■ +^ ■ . (26) 



V. TRANSMISSION LOSS 



A practically important question is to find out a priori whether a 

 switched filter necessarily introduces some transmission loss. 



The following considerations apply exclusively to the zeroth order 

 approximation. It will be shown that assuming ideal elements, the trans- 

 mission at dc may have as small a loss as desired. 



