Editing Direction finding (section)

==== Basic equations for two-port DF ====
If the main lobes of the antennas have a Gausian characteristic, then the output P<sub>1</sub>(φ), as a function of bearing angle φ, is given by<ref name = Lipsky />{{rp|238}}

:<math> P_1(\phi)= G_0.\exp \Bigr [ -A. \Big ( \frac{\phi}{\Psi_0} \Big )^2 \Bigr ] </math>

where

: G<sub>0</sub> is the [[antenna boresight]] gain  (i.e. when  ø = 0),
: Ψ<sub>0</sub> is one half the half-power [[beamwidth]]
: A = -\ln(0.5), so that P<sub>1</sub>(ø)/P1<sub>0</sub> = 0.5 when ø = Ψ<sub>0</sub>
: and angles are in radians.
The second antenna, squinted at Phi and with the same boresight gain G<sub>0</sub> gives an output

:<math> P_2 = G_0 .\exp \Bigr [ -A. \Big ( \frac{\Phi - \phi}{\Psi_0} \Big )^2 \Bigr ] </math>

Comparing signal levels,

:<math> \frac{P_1}{P_2} = \frac{\exp \big [-A.(\phi/\Psi_0)^2 \big ]}{\exp \Big [-A \big [ (\Phi - \phi)/ \Psi_0 \big ]^2 \Big ]}  =  \exp \Big [  \frac{A}{\Psi_0^2}.(\Phi^2 - 2 \Phi \phi) \Big ] </math>

The natural logarithm of the ratio is

:<math>\ln \Big ( \frac{P_1}{P_2} \Big ) = \ln(P_1) - \ln(P_2) = \frac{A}{\Psi_0^2}.(\Phi^2 - 2 \Phi \phi) </math>

Rearranging
:<math> \phi = \frac{\Psi_0^2}{2A.\Phi}. \big [ \ln(P_2) -\ln(P_1) \big ] + \frac{\Phi}{2} </math>

This shows the linear relationship between the output level difference, expressed logarithmically, and the bearing angle ø.

Natural logarithms can be converted to [[decibels]] (dBs) (where dBs are referred to boresight gain) by using ln(X) = X(dB)/(10.\log<sub>10</sub>(e)), so the equation can be written

:<math> \phi = \frac{\Psi_0^2}{6.0202 \Phi} . \big [ P_2(dB) - P_1(dB) \big ] +\frac{\Phi}{2} </math>