1412 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



If/(0+) is defined by lim/(e'), then, provided /(0 + ) = 0,* 



«^o 



F*{p) = £ 



E/Wsa- nT) 



n=0 



(20) 



Going back to the system of Fig. 4 we get^ 



J, ( ^ _ [Zn{p)hip)]*CS,{p)Zr2{p) 



""''^^^ 1 + 2C[Sr{p)Z{p)]* 



(21) 



and 



Iroip) = 



[Zn(p)h{p)]*CS,{p) 



(22) 



1 +2C[5i(p)Z(p)]*' 



where according to the notation defined by (18) 



1 " 

 [Z nip) hip)]* = Tj, 12 Znip + jnois)hip + jnc^s), 



1 71= — ZC 



-I +00 



[Slip) Zip)]* = 7p S Slip -\- Jnws)Zip + >co,). 



It should be stressed that (21) and (22) are not vaHd when r is made 

 identical to zero. When r = 0, Siip) = 1 for all p's and since Zip) '^ 

 1/Cp as p — > 3c the time function whose transform is Zip)Siip) is differ- 

 ent from zero at / = 0. In such a case (20) does not hold. From a physi- 

 cal point of view, the feedback loop of Fig. 4 is unstable when r is identi- 

 cally zero since an impulse generated by the impulse modulator 

 produces instantaneously^ a step at the input of the impulse modulator. 

 This step causes an instantaneous jump in the measure of the impulse 

 at the output of the impulse modulator and so on. In short the feedback 

 loop is unstable. 



It should be pointed out that if the power density spectrum of h is 

 zero for frequencies higher than cos/2, (21) reduces to 



h 



1 



CSiip)Ziiip) 



for \p\ < 



COs 



(23) 



T\ + 2C[Siip)Zip)]* '"- '" ' ^ 2 

 For certain applications it is convenient to rewrite (21) in a slightly 



by 



* When/(0), as defined above, is different from zero, (20) should be replaced 



£ 



23 /(/) i{t - nT) 



n=0 



= F*ip) +-/(0+). 



