A MATHEMATICAL THEORY OF LINEAR ARILWS 



103 



In order to find the relative amplitudes and phase deviations of the ele- 

 ments of the array represented by (37), we expand \/<P into a single poly- 

 nomial as follows^^ 





i + E(-) 



n-l 



i + Z(-) 



.-n+k\ fc(A.-+l) 



, (1 - r"+')(i - r^') •••(!- r^') -'^, 



t 



(1 - r^)(i - r^) ... (1 - r^) 

 (^ 



9 A; 



n-l 

 (/ ^ 



n-l 



2 i i \ kn 



_i 1 _!i t 



{t 2 - /2)(r' - ••• (/ 2 _ ^2) 



(50) 



i+Z(-) 



. (n — l)i/' . (w — k)\l/ 



it 2(« - 1) 2(w - 1) ^fc"'^ , 



^ '- ^ '- g2(n-l) 2«= 



Sin 



2{n - 1) 



sin 



krf/ 



2{n - 1) 



Hence the progressive phase delay from one antenna to the next is equal to 



^ ^ IT 



2{n - 1) 



and the amplitudes are in the ratio 



. (n - 1)^ . (n - 1)^ . (n - 2)^ 

 sin ^, —- sin i- ~ sin 



2(w - 1) 



l{n - 1) 2{n - 1) 



sm 



2(« - 1) 



sin 



sin 



2,A 



2{n - 1) 2(« - 1) 



. (n - l)\p . (n - 2)\p . (n - 3)^ 



sin ^, ^ sin "- ~ sin ^^ Vr 



2(« - 1) 2(w - 1) 2(w - 1) 



(51) 



sin 



sm 



2x1/ 



sm 



3xp 



2{n - 1) 2(n - 1) 2(w - 1) 



The amplitudes of the elements equidistant from the ends of the array are 

 equal. In the special case of an end-on array with nulls equispaced in the 

 range of s, ^ = 2(3f and ?? = (3f; hence the progressive phase delay from one 



antenna to the next is tt — 



w — 1 



While (50) serves well for finding the amplitude and phase distribution in 

 the individual elements of the array, another form is more general for cal- 

 culating the directive properties. In order to obtain this form we set 



11 Chrystal's Algebra, Vol. 2, p. 340, (1926). 



