468 BELL SYSTEM TECHNICAL JOURNAL 



tion £ as a function of the distance x along the strip may be expressed 

 as the sum of an infinite number of Fourier components, thus: 



E{x) = £ a„ cos i -^ + <Pn] . (1) 



In this summation a„ represents the intensity of the wth component 

 and (fn its phase angle. The complete array of these for all components 

 will vary if the picture is changed. 



The cosine series above is very convenient for physical interpretation. 

 It will be simple, however, for some of the later mathematical work to 

 use the corresponding exponential series. The cosine series can be 

 returned to, each time> as physical interpretation is required. That is, 

 since 



(TTX \ 



-7^ + <i5 ) = (ae^^)e(^'^^/'^> + (oe-i^)e(-»'^/^> (2) 



the series in equation (1) can be written 



+00 



E(x) = X! Anexpiir(nx/L) (3) 



ra= — 00 



if we make 

 and 



An = (l/2)a„exp (*>„) 

 A-n = (l/2)a„ exp (-*>„) (4) 



and if we use the notation exp d = e^. 



In this new summation the complex amplitude An represents both 

 the absolute intensity and the phase angle of the wth component. 

 The complex amplitude of the corresponding component with a nega- 

 tive subscript is merely the conjugate of this. 



As has already been noted, however, and as might readily be ex- 

 pected, the single Fourier series in equations (1) or (3) above do not 

 always represent a two-dimensional picture with sufficient complete- 

 ness. In order to consider the two-dimensional field more in detail, 

 let us assume that Fig. 2 represents such an image field of dimensions 

 2a and 2b, and take axes of reference x and y as indicated. The 

 brightness or illumination of the field is a function E{x, y) of both x 

 and y. Along any horizontal line (i.e., in the x direction, constantly 

 keeping y = yi) the illumination may be expressed as a single Fourier 

 series 



+ 00 



E(x,yi) = XI Amexpi(nix/a). (5) 



