A THEORY OF SCANNING 487 



tinuous). Each of the curves is of the equation 



. 2irNc I - mu 

 sin — ^ W ■" ^ 

 Y\{m, n) = Fj(m) 



u \ 2a j 



and therefore has a peak of the value Y^(m) at the point where n = 

 or/ = mu/2a, and trails ofif from the peak in each direction according 

 to a curve of the same shape as curve "A" In Fig. 12. The successive 

 curves are all of identical shape, but each one is to a reduced scale of 

 ordinates as compared with the preceding (In the useful frequency 

 range) as Imposed by the factor Y^im). 



The peaks, it will be noted, occur at the frequencies occupied by 

 what have been called the fundamental components (as distinguished 

 from the satellite lines) In the discussion above on the frequency spec- 

 trum of the signal. 



Assuming N to be 100 and for simplicity taking the factor u/2a as 

 equal to 1, a plot is shown in Fig. 13a of Yi{m, n) over a very limited 

 region near the upper end of the useful frequency range. The curve 

 shown In a solid line represents Fi(m, n) for m = 45, and the dotted 

 curves on either side represent the function for m = 44 and 46, 

 respectively. 



The function has been redrawn for the complete useful range of 

 frequencies and a little beyond, in Fig. 13b, with the frequencies to a 

 logarithmic scale. This logarithmic plot opens out the scale at the low 

 frequencies and enables the fine structure of the function to be indi- 

 cated there, and still enables the complete range of useful frequencies 

 to be shown without requiring a prohibitive size of drawing (it has, 

 however, the disadvantages that the distortion in the frequency scale 

 then masks the symmetry of the individual curves around the funda- 

 mental lines, the similarity of shape of these individual curves, and 

 also the constant frequency separation between the successive funda- 

 mental lines). 



The function Fi(m, n), as Is clear from equation (22) and Figs. 13a 

 and b, consists of a sort of envelope function Fj(w), "modulated" by 

 a fine structure function Y,,(n). The latter function has the value 

 unity at the positions of the fundamental lines in the frequency spec- 

 trum of Fig. 7 and diminishes for the satellite lines in the same way 

 that the envelope function diminishes for the fundamental lines 

 away from zero frequency. It will be seen that the envelope function is 

 the only one obtained by the simple one-dimensional analysis. The 



