. The Bell System technical journal . s boththe absolute intensity and the phase angle of the wth 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 notalways 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 dimensions2a and 2b, and take axes of reference x
. The Bell System technical journal . s boththe absolute intensity and the phase angle of the wth 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 notalways 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 dimensions2a and 2b, and take axes of reference x and y as indicated. Thebrightness or illumination of the field is a function E{x, y) of both xand y. Along any horizontal line (, in the x direction, constantlykeeping y = yi) the illumination may be expressed as a single Fourierseries + 00 E(x,yi) = XI Amexpi(nix/a). (5) A THEORY OF SCANNING 469 Along any other line in the x direction a similar series holds withdifferent coefficients, that is, the ^s are functions of y. They may. Fig. 2—Scanned field and Image,therefore each be written as a Fourier series along y Am = 12 AmnexpiTr(ny/b). (6) n=—OT Substitution in equation (5) gives the double Fourier series, E{x, y) = E E Amnexpiir I— +^)- (7) For purposes of physical interpretation, as in the case of the simpleFourier series, it is desirable to combine the -\-m, -\-n term with the— m, —n term (giving the single (m, +w)th component) and similarly 470 BELL SYSTEM TECHNICAL JOURNAL the +m, -w with the -m, -\-n terms (giving the single (m, -n)th com-ponent). This brings equation (7) back to a cosine series, nix ny . (8) 00 +00 E{x,y) = Y. H CLrnn COSm = 0 n= —00 when Amn = (l/2)a;„„exp (^V^n)and A-m-n = (l/2)a„„exp {-icpmn) and where a„in is always a real quantity. Each term of this seriesrepresents a real, two-dimensional, sinusoidal variation in brightnessextending across the image field. The image is built up of a superposi-tion of a series of such wa
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