Fig. 3. The anti-symmetric part 7^(9) of /(0). 

 Zero has been displaced outward to the dashed 

 circle to avoid plotting negative values. 



/s(e) = 



/(e) + /(-e) 



(1) 



and the anti-symmetric part (Fig. 3). 



h<Q) 



1(6) -n-d) 



(2) 



Obviously, the function I{6) is the sum of its anti-symmetric and symmetric part: 



lid) = I^id) + laid). 



(3) 



Insismuch as we shall be considering linear space processors, it will be worthwhile to expand the in- 

 tensity distribution, 1(6), over an orthogonal basis. Inasmuch as 1(6) is a periodic function of 6 with 

 period 2tt, the natured expansion is in a Fourier series: 





n=l 



In terms of the Fourier coefficients, the symmetric and anti-symmetric part are, respectively, 



W) 



and 



Ia(^) 



-^ -^ l^ "n cos n6 

 n=l 



7 b„ sin n0. 



n=l 



(4) 



(5) 



(6) 



With this dissection it is easy to see that the linear array responds only to the symmetric part of the 

 field, Ig(6), and is totally insensitive to the anti-symmetric portion of the field, /q(0), where 6 is the 

 angle measured wi. respect to the axis of the array. 



892 



