Editing Direction finding (section)

=== Three-channel DF ===
[[File:Three-port_DF.png|thumb|Three-port DF, polar plot (normalized)]]
[[File:Three-port_DF,_log_scale.png|thumb|Three-port DF, log scale (normalized)]]

Improvements in bearing accuracy may be achieved if amplitude data from a third antenna are included in the bearing processing.<ref>Stott G.F., "DF Algorithms for ESM", Military Microwaves '88 Conference Proceedings, London, July 1988, pp. 463 – 468</ref><ref name = East />{{rp|157}}

For three-channel DF, with three antennas squinted at angles Φ, the direction of the incoming signal is obtained by comparing the signal power of the channel containing the largest signal with the signal powers of the two adjacent channels, situated at each side of it.

For the antennas in a circular array, three antennas are selected according to the signal levels received, with the largest signal present at the central channel.

When the signal is on the boresight of Antenna 1 (φ = 0), the signal from the other two antennas will equal and about 12&nbsp;dB lower. When the signal direction is halfway between two antennas (φ = 30°), their signal levels will be equal and approximately 3&nbsp;dB lower than the boresight value, with the third signal now about 24&nbsp;dB lower.  At other bearing angles, ø, some intermediate ratios of the signal levels will give the direction.

==== 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.