A Mathematical Theory of Linear Arrays 

 By 



S. A. SCHELKUNOFF 



A MATHEMATICAL theory, suitable for appraising and controlling 

 directive properties of linear antenna arrays, can be based upon a 

 simple modification of the usual expression for the radiation intensity of a 

 system of radiating sources. The first step in this modification is closely 

 analogous to the passage from the representation of instantaneous values 

 of harmonically varying quantities by real numbers to a symbolic repre- 

 sentation of these quantities by complex numbers. The second step con- 

 sists in a substitution which identifies the radiation intensity with the 

 norm of a polynomial in a complex variable. The complex variable itself 

 represents a typical direction in space. This mathematical device permits 

 tapping the resources of algebra and leads to a pictorial representation of 

 the radiation intensity. 



An antenna array is a spatial distribution of antennas in which the in- 

 dividual antennas are geometrically identical, similarly oriented, and 

 energized at similarly situated points. The first and the last properties 

 insure that the form of the current distribution is the same in all the ele- 

 ments of the array and that consequently the array is composed of antennas 

 with the same radiation patterns. The difference between individual ele- 

 ments consists merely in the relative phases and intensities of their radiation 

 fields. The second property means that the radiation patterns of the 

 individual elements are similarly oriented and that consequently the radia- 

 tion pattern of the array is the product of the radiation patterns of its typical 

 element ajid the ^^ space factor''. The space factor of an array is defined as 

 the radiation pattern of a similar array of non-directive elements. Hence in 

 studying the effect of spatial arrangement of antennas, we may confine 

 ourselves to non-directive elements and thus materially simplify the analy- 

 sis. 



An array is linear if points, similarly situated on the elements, are colinear. 

 In this paper we are concerned mostly with linear arrays of equispaced 

 sources although in conclusion we shall have an occasion to say a few words 

 about more general types. 



1 The norm of a complex number is the square of its absolute value. 



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