358 BELL SYSTEM TECHNICAL JOURNAL 



For the purpose of this discussion, it is sufficiently accurate to 

 assume that the propagation of space waves over the area covered by 

 the array only involves phase retardation, i.e., 



t' = j^'. (201) 



The output of the ^'th antenna is transmitted through a linear trans- 

 ducer having a transfer constant Pk to a common point where it is 

 combined with the outputs of the other antennas of the array. The 

 current from the ^'th antenna at the point of combination is therefore 

 /*e= 7e,e-^-^'t^./F'icosw-0,)^-p.^ ^202) 



where 



ek= e - vk (203) 



and 



X' 



(204) 



The total current received from the n antennas of the array is equal 

 to the sum of the currents received from the individual antennas, or 



fc=ra 



Je^J2 /e,e-^i2'^'^/^'i'=°^^^-**>e-^*. (205) 



k=l 



Equation (205) gives the total current received from any array of 

 antennas for any direction of wave propagation in a horizontal plane. 

 This general expression is not adapted to ready determination of 

 directional characteristics of antenna systems, but it may be simplified 

 by placing the following restrictions on the individual antennas forming 

 the array and their space relations in the array: 



(1) The antennas are all alike. This restriction may be defined by 

 the expression : 



lek — Ie(k+i)- 



(2) The axes of the antennas are parallel, as defined by the expression 



r]k = or X. 



(3) The initial ends of the antennas are equally spaced along straight 

 lines in each subgroup and the subgroups are equally spaced along 

 straight lines. All of the subgroups are identical. 



The general antenna array conforming to these restrictions is shown 

 in Fig. 19. In this figure, there are q groups of antennas equally 

 spaced by the distance a along a line 90 deg. from the zero axis. In 

 each of these g groups of antennas, there are p antennas, divided into 



