492 BELL SYSTEM TECHNICAL JOURNAL 



expressed by equation (24). The Integral becomes 



■Dm'n' 



4ab 



. / m — m' , n — n' , , 

 exp 47r I w H 7 — z; ) t 



' exp -iTr(^ + ^\ d^dtdr]. (35) 



It will be noted that the exponential function of / is periodic in t, one 

 of the periods being to = 2b /v. Furthermore, this is just the difiference 

 between the upper and lower limits in /. Hence the integral in / may 

 be written 



I exp i 



Jo 



tir I II -\ ; V ) tat. 



a 



This integral is zero except when 



ni — m' , n — n' . .^^, 



u -\ r — y = 0, (36) 



a 



in which case 



I = to. (37) 



The meaning of these last few equations is clear. It is, as would 

 be expected, that a signal component m, n does not give rise to all 

 components m', n' in the final received picture, but that these latter 

 components are in general zero unless m' and n' satisfy a definite rela- 

 tionship with m and n, expressed by equation (36). A somewhat un- 

 expected result is, however, that equation (36) allows some other 

 w', w' components besides the normal one for which w' = mandn' = n. 

 That is to say, a given signal component m, n in the line will reproduce 

 in the final picture not only a corresponding m, n component, but as has 

 been foreshadowed in the discussion on confusion in the signal, it will 

 also reproduce certain other components with different indices. 



Let us consider first, however, the reproduction of the normal 

 component for which m' = m and n' = n, which is obviously allowed 

 by equation (36). The amplitude Bmn is then, neglecting constant 

 factors,^ 



Bmn = A„,nYi(m, fi) Yi{m, n), (38) 



where 



F2(m, n) = r^ r^ T,{^, r,) exp -iV ( ^ + ^ ) d^dr,. (38') 



^ The constant factor neglected as compared with equation (35) is to/i'iab). The 

 /o is the period of image repetitions (or "frame period"). It appears here because 

 the brightness of a single image depends on how quickly it is reproduced. 



