98 BELL SYSTEM TECHNICAL JOURNAL 



{m + 1) terms of (30). Dr. T. C. Fry of these Laboratories has suggested 

 that leaving ( undetermined and fixing the number of elements, an array 

 could be designed which would have the best fit to the prescribed pattern. 

 In this connection, the "best fit" means the least mean square deviation 

 of the approximating pattern from the given pattern. 



If -v/^ is given as a function F{d) of 6, then by virtue of the definition of 

 )/' we can write 



F{d) = F(cos-'t^^ =/(^). (31) 



Let us now consider a simple example for the sake of illustrating the 

 method. Let f{\f/) be defined by 



m = 0, < ,A < TT, 



(32) 



= 1, TT < 1^ < 2t. 



We shall assume that the separation between the elements is one-half 

 wavelength. This makes the range of \p equal to It. It is also seen that 

 regarded as a function of 6, f(\J/) retains its essential characteristic : being 

 equal to zero over one-half of the range and to unity over the remaining 

 half. 



Expanding (32) into a Fourier series we have 



2 IT ^-=l Zk — 1 



(33) 



Consequently 



Ao =- ^, Ak = ii k 9^ 0; 



Bk = 0, if ^ is even; (34) 



Bk = — 7— , if ^ is odd. 



Figure 19 shows several approximations to/(i/') by means of a finite num- 

 ber of elements. The curve Sm corresponds to an approximation by the 

 finite series (29). If S9 is deemed to be a sufficiently good approximation 

 to the given directive pattern, then 



v^ = - 



I|l2|l4|l6, S.^Vg 

 — -T — Z -f- — Z +—2 -r2 -4- — S 



97 5 3 2 



10 1 12 1 14 1 16 1 18 

 2 — 2 — 2 2 — ■ 2 



3 5 7 9 



(35) 



