SWITCHED NETWORK FOR TIME MULTIPLEX SYSTEMS 1421 



8.3 Convergence Proof 



Consider a vector X of bounded norm corresponding to a time func- 

 tion x{0 having the property that x(t) = for r ^ f ^ T and x(t) 9^ 

 for < ^ < T. In the above scheme, the vector .Y would be /r„ . Let us 

 define the vectors F, Z, IJ and F by the relations 



Y = EX, (33) 

 Z ^ GY, (34) 

 U = FZ, (35) 



V = 2GU, (36) 

 hence 



V = 2GFGEX. (37) 



We wish to show that N{V) ^ aN(X) with a < 1, since these inequali- 

 ties imply that the infinite series (32) converges. 



Since (a) A^i and A^ are low-pass filters with cutoff ^ x radians/sec, 

 (b) E{p) = 1 for p = 0, (c) E{p) ex l/LnCp' for p » 1, only a few of 

 the Ek's will be of the order of unity In most cases E-i , Eq , Ei will be 

 smaller than unity, thus, 



N(Y) ^ NiX). (38) 



In view" of the pulsating character of x{t) the power spectrum of x{t) 

 is almost constant up to frequencies of the order of tt/t radians/sec. 

 Because of the low-pass characteristic of E{p), the function y(t) associ- 

 ated with the vector Y is smooth in comparison to x{t), thus from (34), 



N{Z) = f 1 z{t) {' dt= f \ y{t) I' dt = arNiY), 

 Jo Jo 



where a = 0(1). 



Since | F, ] ^ 1 for all k% from (33), NiU) ^ N{Z), hence N{U) = 

 hTN{Y) with h = 0(1). 



N{U) = brNiY) with 6 = 0(1). 



From (36) we have 



N{V) = 2 [ \ u{t) I' f// ^ 2 f I u{t) \- dt = 2N{U). 



Jo Jo 



Thus we finally get 



N{V) = 2bTN{Y) where 6 = 0(1), (39) 



