SPECTRA OF QUANTIZED SIGNALS 469 



We may write an expression for the squared samples as a limit of the 

 product of the squared signal and a periodic switching function of infinitesi- 

 mal contact time, thus 



/(«/o) = Lim/(/)5(r, /) (I— 1) 



T-»0 



where: 



/I, -r/1 <t < r/2 \ 

 S{r, = (1-2) 



\0, r/2 < / < r - t/2/ 



S(t, t+T) = S(t, /), w = 0, d= 1, =t 2, . . • (1—3) 



By straightforward Fourier series expansion: 



Sir, = ; + t l^^L^^^Il/I COS 2m.f. I. (1-4) 



mir 



The mean square value of the samples is the limit of the average value of 

 pS taken over the contact intervals of duration r. The average value of 

 pS taken over all time, including the blank intervals, is in the limit a fraction 

 t/T of the average over the contact intervals only. Therefore 



f{nk) =Um-f{t)S{T,t) 



T— T 



= Lim fit) + ± ^^'^'"^'/T fQ) eo3 2m^f, t (1-5) 



T-»0 TO=1 tniTT 



= m+Limt ^^''""""'^^ fit) cos Imrf.i. 

 T-*o m=i tmrr 



Now the long time average value oip{t) cos Irmrfst must vanish unless /^(O 

 contains a component of frequency mfa . This could not happen except 

 where /(/) itself contains a component of frequency mfs/2 or two components 

 /i and/2 such that 



|/i±/2| = m/« (I--6) 



WTien no such relation of dependency exists: 



fink) = fit). (1-7) 



As pointed out before if /(/) contains no frequencies above fc , the response 

 of the ideal low-pass filter to the samples is/(0, and/(w/o) represents the 

 samples of f(t). If/(/) does contain frequencies exceeding/c , the response of 

 the filter is <j>{t), where </>(/) is wholly confined to the band to fc and yields 

 the same samples as/(0, i-c, 



<f>{nto) = /(«/o), » = 0, ±1, ±2, • • • (1—8) 



