. The Bell System technical journal . {-m, n) = I T^(^) exp {iTrm^/a)d^ 1 ^,,(17) exp (iirnr]/a)dr] »^aperture •-aperture = Y^im) . Y,(n) (22) and Fj and F, are each one-dimensional integrals of the type illustratedin Fig. 12. The rectangular aperture is a simple case of this type. Assumethe field to be scanned in N lines and take the dimensions of the aper-ture, 2c and 2d parallel to the x and y axes, respectively, as Then ,, / s sin Trmc/aY^{m) = and wmc/a sin TTfid/b■wnd/h and the frequency corresponding to a given signal component mn is,from equation (11) Thus, Yi{m, n) considered as a fun
. The Bell System technical journal . {-m, n) = I T^(^) exp {iTrm^/a)d^ 1 ^,,(17) exp (iirnr]/a)dr] »^aperture •-aperture = Y^im) . Y,(n) (22) and Fj and F, are each one-dimensional integrals of the type illustratedin Fig. 12. The rectangular aperture is a simple case of this type. Assumethe field to be scanned in N lines and take the dimensions of the aper-ture, 2c and 2d parallel to the x and y axes, respectively, as Then ,, / s sin Trmc/aY^{m) = and wmc/a sin TTfid/b■wnd/h and the frequency corresponding to a given signal component mn is,from equation (11) Thus, Yi{m, n) considered as a function of the signaling frequencycorresponding to each component of indices mn, consists of a succes-sion of similar curves u/2a cycles apart, corresponding to the successiveintegral values of m (these curves are themselves really not continuousbut consist of a succession of points u/{2aN) cycles apart. For con-venience, however, the drawings will always show the curves as con- 486 BELL SYSTEM TECHNICAL JOURNAL m = 45n=o m=46 y\M. 45 46 FREQUENCy Fig. 13a—Detail of equivalent transfer admittance of aperture for two-dimensional scanning. A THEORY OF SCANNING 487 tinuous). Each of the curves is of the equation . 2irNc I - musin —^ W ■ ^Y\{m, n) = Fj(m) u \ 2a j and therefore has a peak of the value Y^(m) at the point where n = 0or/ = mu/2a, and trails ofif from the peak in each direction accordingto a curve of the same shape as curve A In Fig. 12. The successivecurves are all of identical shape, but each one is to a reduced scale ofordinates as compared with the preceding (In the useful frequencyrange) as Imposed by the factor Y^im). The peaks, it will be noted, occur at the frequencies occupied bywhat have been called the fundamental components (as distinguishedfrom 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 asequal to 1, a plot is shown in Fig. 13a of Yi{m, n) ov
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