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BELL SYSTEM TECHNICAL JOURNAL 



Turning back to the ideal system comprising a very large number of 

 closely spaced unit antennas, which yields the single lobed pattern, let 

 us divide the antennas into N groups with n antennas in each group. 

 Calling the group spacing "a" and the phase shift between adjacent 

 antennas <^ the application of (3) gives, dropping the exponential 

 factor, 



— ZTT - (f — COS 5) 



n J 



sm- 



. 1 

 sm- 



4> — 2t~ (v — cos b) 

 n J 



Multiplying numerator .and denominator by 



results in 



li'^ - '^''n^' 



cos 5) 



(4) 



A' 



. n 

 sin 2 



sin- 



l-w-iv — cos 5) 

 n 



— 2ir— iv — cos 5) 

 n 



N, 



sin— [«^ — 2ira(v — cos 5)] 



X— J -• (5) 



sin-[w0 — 2Tra{v — cos 5)] 



Equation (5), which appears as the product of two array factors, is 

 merely another way of writing the array factor for the large number 

 (Nn) of unit antennas. The first factor represents an array of n "sub- 

 unit" antennas of spacing a/n and phase shift 0. The second repre- 

 sents the array of these arrays with a spacing of a\ and a phase shift ncj). 

 We now proceed to treat these two factors independently and assign 

 the values ^/ and </>„ to replace </> and n<l), respectively. Figure 6 

 depicts such an array of arrays. If now we regard the array of n sub- 

 units as constituting a fixed unit antenna and adjust it to receive at 

 zero angle (by putting 6/ = 27ra/«(u — 1)) in accordance with the 

 lower limit of the useful range, we obtain 



A" = 



sm 



27r- (1 — COS 8) 

 n 



27r - (1 — COS 5) 

 n 



N 

 sin — [01, — 2Tta{v — cos 5)] 



x-^ (^) 



sin -[</)„ — 2Tra(v — cos 8)'] 



The first factor in (6) represents the pattern of the unit antenna. It 

 is a relatively broad single lobed pattern with maximum response at 



