A MATHEMATICAL THEORY OF LINEAR ARRAYS 



105 



Since the set of coefficients po , pi , p2 • • • pn-i is symmetric about the cen- 

 ter, we find 



$ = 2pl cos (n — 1)^ + 2(popi + pipo) cos (n — 2)<p 

 + 2{pop2 + /'i/'i + p2po) cos (» - 3).^ 

 + 2(poPi + /'i/'s + p2pi + Z's/'o) cos (n - 4)^ + • • • (58) 



+ 2(popn-2 + plpn-3 + /»2/'n-4 + ' " * + pn~2po) COS (f 

 + {popn-l + i!'l^»-2 + /'2/'n-3 + ' ' ' + pn-lpo) ■ 



Since 



/ ^ sin ede = --- I $ J(/?, 

 Jo P^ J vi 



z /(,« — 1; 



(59) 



we can write 

 / $ sin Bid = —. 



2pl sin [n — l)<p , 2{pQpi + ^i/'o) sin {n — 2)ip 



n — 1 



+ 



n — 2 



+ • • • + (pOpn-l + plp«-2 + • • • + pn-lpo)ip 



•Pi 



(60) 



For an end-on array with nulls equispaced in the range of z, (60) becomes 

 2 



/' 



Jo 



I3(i n - 1 



2(-)""'^o . , .,., (n- l)/3^ 



^ "^ -^ sm (w — l)j3c cos 



n — 1 



2(-y-\p,p, + ^,^o) . (/^ - 2)^< ^ 



+ -^ — ^^--^ sm (w — 2)pf cos — + • • • (ol) 



n — 2 n — 1 



+ (Popn-l + plpn-2 + • • • + pn-lpo)^t . 



Substituting in (54), we shall obtain the gain of the array. 



Similar expressions can be obtained for an end-on array in which the 

 amplitudes of the individual elements are equal. Thus we have 



$ = i- [e'("-^)^ + e''"-''^ 



+ •••+/' + !] 



+ e 



-i(n— 1)^1 



(62) 



= — [2 cos (n — \)\l/ -\- 4: cos (n — 2)\p 



-\- 6 cos (n — S)\p + • • • + 2{n — 1) cos \p + n], 



