Editing Direction finding (section)

==== Basic equations for three-port DF ====
For a signal incoming at a bearing ø, taken here to be to the right of boresight of Antenna 1:

Channel 1 output is 
:<math> P_1 = G_T .\exp \Bigr [ -A. \Big ( \frac{\phi}{\Psi_0} \Big )^2 \Bigr ] </math>

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

Channel 3 output is 
:<math> P_3 = G_T .\exp \Bigr [ -A. \Big ( \frac{\Phi + \phi}{\Psi_0} \Big )^2 \Bigr ] </math>

where G<sub>T</sub> is the overall gain of each channel, including antenna boresight gain, and is assumed to be the same in all three channels.  As before, in these equations, angles are in radians, Φ = 360/N degrees = 2 π/N radians and A = -ln(0.5).

As earlier, these can be expanded and combined to give:

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

:<math> \ln(P_1) - \ln(P_3) = \frac{A}{\Psi_0^2}.(\Phi^2 + 2 \Phi \phi) </math>

Eliminating A/Ψ<sub>0</sub><sup>2</sup> and rearranging

:<math> \phi = \frac{\Delta_{1,2} -\Delta_{1,3}}{\Delta_{1,2} + \Delta_{1,3}}.\frac{\Phi}{2} = \frac{\Delta_{2,3}}{\Delta_{1,2} + \Delta_{1,3}}.\frac{\Phi}{2} </math>

where Δ<sub>1,3</sub> = \ln(P<sub>1</sub>) - ln(P<sub>3</sub>), Δ<sub>1,2</sub> = \ln(P<sub>1</sub>) - \ln(P<sub>2</sub>) and  Δ<sub>2,3</sub> = \ln(P<sub>2</sub>) - \ln(P<sub>3</sub>),

The difference values here are in [[nepers]] but could be in [[decibels]].

The bearing value, obtained using this equation, is independent of the antenna beamwidth (= 2.Ψ0), so this value does not have to be known for accurate bearing results to be obtained.  Also, there is a smoothing affect, for bearing values near to the boresight of the middle antenna, so there is no discontinuity in bearing values there, as an incoming signals moves from left to right (or vice versa) through boresight, as can occur with 2-channel processing.