A MATHEMATICAL THEORY OF LINEAR ARRAYS 97 



Consider an array with an odd number of elements n = 2m -\- 1. Since 

 the modulus of z is unity the polynomial (3) can be divided by z" without 

 aflfecting V'i'; thus 



V^ = I doz-"' + aiS-'^+i + C2S-"'+2 -\- ... 



(26) 



+ Qm + am+lZ + • • • + ChmZ"" \. 



Let us now assume that the coefficients equidistant from the ends of the 

 polynomial are conjugate complex; then the polynomial is real and we can 

 drop the bars. Thus setting 



dm — Ao, a,nj^k = Ak — iBk , k > Q, am-k = cim-ik, (27) 

 where the -4's and B's are real; we have 



am+kz'' + a,„-AS-^ = {Ak - iBk)e'''l' + {Ak + iBk)e-''^ 



(28) 

 = 2Ak cos kx/y + 2Bk sin kx//. 



Consequently, (26) becomes 



V$ = S ^k{Ak cos kxl^ + Bk sin kxp), (29) 



k=0 



where ek is the Neumann number.'" 



If now we wish \/$ to be a prescribed function f{\p) of the variable ^, 

 we need only expand this function in a Fourier series 



00 



V^ = /("A) = S ^ic{pk cos bp + Qk sin ^i/'), (30) 



k=0 



and approximate it with any desired accuracy by means of a finite series 

 (29). Once the /I's and 5's are known, we calculate the a's from (27). 



It must be remembered that the real independent variable is not \p but 

 6 and the directive pattern is to be assigned as a function of 6. Besides 

 being dependent on d, \p is a. function of the distance (■ between the succes- 

 sive elements of the array. Since 6 varies from 0° to 180°, the range of \p 

 is ^ = 2/3f. The function /(>/') is prescribed within this range. .On the 

 other hand the period of the expressions (29) and (30) is 27r. This means 

 that if ^ > 27r, that is if f > X/2, it is impossible to obtain the desired direc- 

 tive pattern with our scheme, because the pattern repeats itself automatically 

 asi/' increases or decreases by 27r. But if f < X/2, we have a considerable 

 latitude in the design; outside the range of \p, we can supplement /(^) by an 

 arbitrary function of i/'. It is only when C — X/2 that there is a unique class 

 of linear arrays that will produce a directive pattern given by the first 



" eo = 1, ti = 2 when k 7^ ^. 



