88 



BELL SYSTEM TECHNICAL JOURNAL 



In general, if an array represented by 



f{z) = Go + fliz + a^z'^ + • • • + dn-lZ"-^ (15) 



is taken as the element of an array given by 



F(z) ^ bo+ hz + h^z' + • • • + hm-iz"^-\ (16) 



then the resulting array of arrays is represented by 



J{z)F{z) = bofiz) + hzfiz) + b2Z%z) + • • • + 6„._is-y(s). (17) 



Decomposition Theorem 



Consider now a pair of non-directive sources with strengths proportional 

 to 1, — /; then 



V^= \z-t\. (18) 



Geometrically, the complex number s — ^ is represented by a line drawn 

 from point / to point z (Fig. 7A). Accordingly, the radiation intensity 



(A) 



(B) 



Fig. 7 — The radiation intensity of a linear array is represented by the square of the product 

 of the lines joining the null points of V$ to a point z on the unit circle. 



of the pair of sources is represented by the distance between t and z. If v $ 

 vanishes for some particular direction in space, it vanishes for all direc- 

 tions making the same angle with the line of sources; these directions form 

 a cone of silence of the radiation system. Obviously, a radiating couplet 

 has a cone of silence if and only if the null point of \/$ is in the range of z; 

 in particular, there can be no cone of silence unless the null is on the unit 

 circle. 



By the fundamental theorem of algebra a polynomial of degree {n — 1) 

 has (w — 1) zeros (some of which may be multiple zeros) and can be fac- 

 tored into (« — 1) binomials; thus 



Vi = 1 (2 - /l)(z - /2) • • ■ (Z - tn-l) |. (19) 



