A THEORY OF SCANNING 467 



theory of the scanning processes used in telephotography and television 

 and led to the study outlined in the following pages. Since this study 

 will be confined to characteristics of the scanning processes all other 

 processes in the system, wherever used, will be assumed to be perfect 

 and cause no distortion. 



The general trend of this more complete theory can be foreseen when 

 it is considered that to obtain an adequate reproduction of the original 

 it is necessary to scan with a large number of lines as compared with 

 the general pictorial complexity of this original. This means that for 

 any original presenting a large scale pattern (as distinguished from a 

 random granular background) the signal pattern along successive 

 scanning lines will, in general, differ by only small amounts. Thus, the 

 signal wave throughout a considerable number of scanning lines may 

 be represented to within a small error by a function periodic in the 

 scanning frequency. Since such a function, developed in a Fourier 

 series, is equal to the sum of sine waves having frequencies which are 

 harmonics of the scanning line frequency, it will be natural to expect 

 the total signal wave to have a large portion of its energy concentrated 

 in the regions of these harmonics. 



Furthermore, the existence of signal energy at odd multiples of half 

 the scanning frequency will indicate the existence of a characteristic in 

 the picture which repeats itself in alternate scanning lines. It is to be 

 expected that such detail in a picture cannot be transmitted without 

 accurate registry between it and the scanning lines and that when the 

 detail spacing or direction or both differ somewhat from the scanning 

 line spacing and direction, beat patterns between the two will be pro- 

 duced in the received picture which may be strong enough to alter con- 

 siderably the reproduction of the original. 



These phenomena are exactly what is observed, and will be treated 

 in more quantitative fashion in the discussion below.^ 



An Image Field as a Double Fourier Series 



Let us first consider the usual expression of the image field as a 

 single Fourier series. The picture will be considered as a "still" so 

 that entire successive scannings are identical. Then if the long strip 

 corresponding to one scanning extends from — L to +L, the illumina- 



1 In the following treatment an effort has been made to confine the necessary 

 mathematical demonstrations almost exclusively to two sections entitled, respec- 

 tively, "Effect of a Finite Aperture at the Transmitting Station," and "Reconstruc- 

 tion of the Image at the Receiving Station." Even in these sections a number of 

 conclusions are explained in text which do not require reading the mathematics if 

 the demonstrations are taken for granted. The occasional mathematical expressions 

 occurring in the earlier sections are very largely for the purpose of introducing 

 notation. 



