SWITCHED NETWORK FOR TIME MULTIPLEX SYSTEMS 1419 



Since r/T <<C 1, and since the frequency has been normahzed so that 

 T = 1, we have r « 1. 

 Using the convolution in the frequency domain, we have 



k=—<x> \a=— 00 / 



If we introduce the infinite matrix G defined by 



Gik = A,_,. ii,k= - X, ... , -1, 0, +1, • • • x), 



the convolution may be represented by the product, GIno , where /„o is 

 the vector whose components are /„o, a(A" = • • ■ — 1, 0, 1, • • •). 



(d) Considering the network shown on Fig. 3, let E{p) be the ratio of 

 In'iv) to Iriv)- Taking into account the assumed identity between A'^i 

 and A^2 it follows that 



+ In{v) 

 Iriv) 



= ^ = E(V). 



70=0 Iriv) 



Using the system of (1) and, for example, by Neumann series expan- 

 sion of the inverse matrix, Ave get 



(e) Considering now the effect of init) and in it) on irit), (42) of 

 Appendix I gives Iriv) ^s a function of Iniv) ^^nd In'iv)- I^^ ^he present 

 discussion where we are interested in the steady state of irit) it is es- 

 sential to keep in mind that since the switch opens at i = r, the memory 

 of the resonant circuit extends only over an interval < / ^ r. To take 

 this into account we must modify the factor (coo /2)/(p" + wo") of (40), 

 because the impulse response (which represents this memory) must be 

 identically zero for t > t. The resulting new expression is 



2 



Fiv) =^.r^.e-'"^'~W-" + e--"'], 

 or 



V + coo- 2 



Since the time function whose transform is F(p) is non-negative for all 

 fs and since F(0) = 1, it follows that 



IF(ico) I ^ 1. (28) 



