Directivity of an antenna is defined w.r.t. peak power direction. What if I want to know the directivity value at a specific coordinate point (X.Y or spherical coordinates) from the antenna. I want this directivity value to be plugged in the path loss equation, to find the path loss at a specific point from a BS. This point can be outside the main lobe of the antenna.
You need to know the polar diagram of your antenna and correct for variations from the quoted gain. If it is omnidirectional that is easy. If it isn't really omnidirectional (most omnis are only omni in the horizontal plane, and there will be some variation even there) then you need to know how the gain or directivity varies with angle.
You need to use gain rather than directivity, unless you have the efficiency already in your equation. Gain = directiviry * efficiency.
I think that approach of Malcolm White is correct. I used a similar concept in few of my publications, such as:
Conference Paper A METHOD OF MEASUREMENT OF DIRECTIVITY CHARACTERISTICS OF AN...
Article Partial characteristics measurement of the reception channel...
Directivity (D) and gain (G) are two different antenna parameters. The purpose of the former is to show how well the antenna suppresses the sidelobes (unwanted propagation) while the latter is for provided peak gain. Both are used in antenna design. They are related to each other with an antenna efficiency factor, i.e. G = e*D. Since e is almost 1 for the most current antennas, these two are used interchangeably in the literature.
The absolute antenna gains are already in the free space path loss equation. The received power is Pr = Pt + Gt + Gr - L where Pt is transmit power, Gt and Gr are the receiver and transmitter antenna gains and L is the path loss, all in dB. Here Gt and Gr depends on the angle. The correct usage should be Gt(p,q) where p and q are azimuth and elevation angles. Similarly, Gr(r,s). Assuming the local coordinate systems of transmitter and receiver perfectly aligned with global coordinate system, then Pr is a function of these 4 angles, i.e. Pr(p,q,r,s). If you know Gt(p,q) and Gr(r,s) for any angular point, Pr can be calculated. I believe that was your question.
Please see following for further details:
http://eceweb1.rutgers.edu/~orfanidi/ewa/ch16.pdf
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