82 BELL SYSTEM TECHNICAL JOURNAL 



Hence the amplitude \/i> of the field strength is the absolute value of 

 (2); thus' 



/~ I I 12 1 I n— 2 1 71—1 I 



V* = Go + fllS + O2Z + • • • + 07,-23 + Z , 



•, (3) 



z = e'*, xP = ^( cosd - ^, am = A^e "'. 



In this equation: ao , ai , a2 , • • • an-2 , Qn-i — 1 are complex numbers repre- 

 senting the relative amplitudes of the elements of the array and the phase 

 deviations of these elements from a given progressive phasing. Thus if 

 all the coefficients are real and positive, they represent the relative ampli- 

 tudes of the elements of the array. If the algebraic sign of a particular 

 coefficient is reversed, the phase of the corresponding element is changed 

 by 180°; if some coefficient is multiplied by i or —i, the phase of the cor- 

 responding element is respectively accelerated or delayed by 90°; and in 

 general the phase acceleration is equivalent, in our scheme, to a multiplica- 

 tion by a unit complex number e'^ Some coefficients may be equal to zero 

 and the corresponding elements of the array will be missing. In view of 

 this possibility, we shall call ^ the "apparent" separation between the 

 elements; it is the greatest common measure of actual separations. When 

 the elements are equispaced the apparent separation is the actual separation. 



Thus we have the fundamental 



Theorem I: Every linear array with commensurable separations between 

 the elements can be represented by a polynomial and every polynomial can be 

 interpreted as a linear array. 



The total length of the array is the product of the apparent separation 

 between the elements and the degree of the polynomial. The degree of the 

 polynomial is one less than the "apparent" number of elements. The 

 actual number of elements is at most equal to the apparent number. 



The above analytical representation of arrays is accomplished with the 

 aid of the following transformation 



2=6'"'', (4) 



in which \p = fi( cos — ?? is a function of the angle 6 made by the line of 

 sources with a typical direction in space. Since \}/ is always real, the ab- 

 solute value of s equals unity and 2 itself is always on the circumference of 

 the unit circle (Fig. 2). As increases from 0° (which is in a direction of the 

 line of sources) to 180° (which is in the opposite direction), \p decreases and 



2 For brevity's sake, we shall call V* itself the "field strength." 



' Equation (3) could be derived directly from the physics of the situation in the same 

 manner as (1). The foregoing method of transition from (1) to (3) serves only the purpose 

 of showing the relationship between a less familiar formula and a very well known one. 



* If the separations are not commensurable the arrays are represented by an algebraic 

 function with incommensurable exponents. 



