1418 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



zeroth approximation in order to obtain the first approximation. Thus 

 adding init) to iroit) we get the first approximation iro{t) + iriit). More 

 generally the A-th approximation is ^„=otrn(0- The procedure will con- 

 verge if, in particular, the infinite series 22*=o^Vn(0 converges. 



8.1 Preliminary Steps 



(a) Let us normalize the frequency (and consequently the time) so 

 that the switching period T is unity. Since the networks A''! and N2 must 

 have high insertion loss for co > h{2Tr/T) = tt, the pass band of A'^i and 

 N2 must be the order of 1 radian/sec. As a result the element values of 

 the capacitor C and the inductance L„ (see Fig, 3) are also 0(1). 



(b) For the excitation io = e'" , the zeroth approximation derived 

 above may be written in terms of Fourier components: 



■ /.\ iwt V~* 7- iiirkl 



Uo{t) = e 2^ lro,ke , 



k=-K, 

 +00 



tnoW = e 2^ ino.ke 



Let Iro denote the complex conjugate of iVo , then 



+00 



iroit)lro(t) = XI IrO,kIrO,(e^''' " ^\ 



k,(=-ao 



Since the functions e'"' \k — • • • — 1, 0, 1, • • • ] are orthonormal over 

 the interval (0, 1) and form a complete set, we have from Bessel's equal- 

 ity: 



f I iM \'dt = E I ho,k I'- = A^(/.o), 



Jo A=-oo 



where N{Iro} denotes the norm of the vector Iro which is defined by its 

 components /,o,/t(/v = • ■ • — 1, 0, +1 • • • )• Similarly, 



/.I +00 



/ I ino(t) f dt = ^ \ InO,k 1' = .V(/„o). 

 Jo k=-ac 



(c) Since the switch is periodically closed we shall be interested in 

 the Fourier series expansion of A(/) : 



A(0 = u(t) - u(t - r) = i: A,c'''''\ 



—00 



where Ao = r and 



+00 



