APPENDIX B: DIRECTIVITY FUNCTION 
IN VECTOR NOTATION 
The directivity function for a discrete number of elements 
compensated for a direction (cosa), cos80, cosy.) anda 
finite distance Do, with coordinates (x0, Yo,Z0) is 
N 
R= ) £,expsklx(cosa- COSA )+y(cos8- cosBo )+z(cos y- cosy ) 
gol 
a i ha = 
p, (cosy, 1) p (cosy, 7 1)] 
This result can be written in vector notation as follows: 
Let 
v = (cosa, cos8, cosy) 
ee 
Vo = (cosdo, cos8o, cosyo) 
ie 
SO > (xo, Yo, Zo) 
Then the directivity function becomes 
i = v~ = o> |r |-7,-0 
R= ) F,expjh Ta (v-v0)+ |r-r, | ————-1 
,=] c ate 7 | 
U 
= = | 70 |-r., ~& 
2 |raoirs | u -1 
70-7, | 
U 
This can be simplified by multiplication inside the exponential 
function to 
M 
R= ) FE, expel |7|-|7 |- [Py | o> 
wall 
> 
|) 
