Editing Direction finding (section)

== DF by amplitude comparison ==
Amplitude comparison has been popular as a method for DF because systems are relatively simple to implement, have good sensitivity and, very importantly, a high probability of signal detection.<ref name = Tsui />{{rp|97}}<ref name = Lipsky />{{rp|207}}  Typically, an array of four, or more, squinted directional antennas is used to give 360 degree coverage.<ref name = East>East P.W., "Microwave System Design Tools with EW Applications", Artech House, 2nd Ed., 2008</ref>{{rp|155}}<ref name = Lipsky />{{rp|101}}<ref name = NAWC>National Air Warfare Center, "Electronic Warfare and Radar Systems", NAWCWD TP 8347, 4th Ed., 2013.  Find at: www.microwaves101.com/encyclopedias/ew-and-radar-handbook)</ref>{{rp|5–8.7}}<ref name = Tsui>Tsui J.B., "Microwave Receivers with Electronic Warfare Applications", Kreiber, Florida, 1992"</ref>{{rp|97}}<ref>Ly P.Q.C, "Fast and Unabiguous Direction Finding for Digital Radar Intercept Receivers", Univ. of Adelaide, Dec. 2013, p. 16. Find at: https://digital.library.adelaide.edu.au/dspace/bitstream/2440/90332/4/02whole.pdf</ref>  DF by phase comparison methods can give better bearing accuracy,<ref name = NAWC />{{rp|5–8.9}} but the processing is more complex. Systems using a single rotating dish antenna are more sensitive, small and relatively easy to implement, but have poor PoI.<ref name = Hatcher />

Usually, the signal amplitudes in two adjacent channels of the array are compared, to obtain the bearing of an incoming wavefront but, sometimes, three adjacent channels are used to give improved accuracy. Although the gains of the antennas and their amplifying chains have to be closely matched, careful design and construction and effective calibration procedures can compensate for shortfalls in the hardware.  Overall bearing accuracies of 2°  to 10° (rms) have been reported <ref name = NAWC /><ref>Blake B. (ed.), "Manta", " Sceptre" and " Cutlass" ESM Systems, Jane's Radar and Electronic Warfare Systems, 1st Ed., Jane's Information Group, 1989, pp.344 -345</ref> using the method.

=== Two-channel DF ===
[[File:Two-port_DF,_polar_plot.png|thumb|Two-port DF, polar plot (normalized)]]
[[File:Two-port_DF,_log_scale.png|thumb|Two-port DF, log scale (normalized)]]
[[File:Power_Dif._v._Bearing.png|thumb|Power Diff. (dB) v. Bearing]]

Two-channel DF, using two adjacent antennas of a circular array, is achieved by comparing the signal power of the largest signal with that of the second largest signal. The direction of an incoming signal, within the arc described by two antennas with a squint angle of Φ, may be obtained by comparing the relative powers of the signals received. When the signal is on the boresight of one of the antennas, the signal at the other antenna will be about 12&nbsp;dB lower. When the signal direction is halfway between the two antennas, signal levels will be equal and approximately 3&nbsp;dB lower than the boresight value. At other bearing angles, φ, some intermediate ratio of the signal levels will give the direction.

If the antenna main lobe patterns have a Gaussian characteristic, and the signal powers are described in logarithmic terms (e.g. [[decibels]] (dB) relative to the boresight value), then there is a linear relationship between the bearing angle φ and the power level difference, i.e. φ ∝ (P1(dB) - P2(dB)), where P1(dB) and P2(dB) are the outputs of two adjacent channels. The thumbnail shows a typical plot.

To give 360° coverage, antennas of a circular array are chosen, in pairs, according to the signal levels received at each antenna. If there are N antennas in the array, at angular spacing (squint angle) Φ, then Φ = 2π/N radians (= 360/N degrees).

==== 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>

=== 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.

=== Bearing uncertainty due to noise ===
Many of the causes of bearing error, such as mechanical imperfections in the antenna structure, poor gain matching of receiver gains, or non-ideal antenna gain patterns may be compensated by calibration procedures and corrective look-up tables, but [[thermal noise]] will always be a degrading factor. As all systems generate thermal noise<ref>Connor F. R., ''Noise'', Edward Arnold, London, 2nd ed. 1982, p. 44</ref><ref>Schwartz M., "Information Transmission, Modulation and Noise", McGraw-Hill, N.Y.,4th Ed., 1990, p.525</ref> then, when the level of the incoming signal is low, the [[signal-to-noise ratio]]s in the receiver channels will be poor, and the accuracy of the bearing prediction will suffer.

In general, a guide to bearing uncertainty is given by <ref name = NAWC /><ref>Al-Sharabi K.I.A. and Muhammad D.F., "Design of Wideband Radio Direction Finder Based on Amplitude Comparison", Al-Rafidain Engineering, Vol. 19, Oct 2011, pp.77-86  (Find at:  www.iasj.net/iasj?func=fulltext&aid=26752 )</ref>>{{rp|82}} <ref name = Wiley />{{rp|91}}<ref>Martino A. De, "Introduction to Modern EW Systems", 2nd Ed.,  Artech House 2012</ref>{{rp|244}}

:<math> \Delta \phi_{RMS} = 0.724 \frac{2. \Psi_0}{ \sqrt{SNR_0}}  </math>  degrees

for a signal at crossover, but where SNR<sub>0</sub> is the signal-to-noise ratio that would apply at boresight. 
	
To obtain more precise predictions at a given bearing, the actual S:N ratios of the signals of interest are used.  (The results may be derived assuming that noise induced errors are approximated by relating differentials to uncorrelated noise).

For adjacent processing using, say, Channel 1 and Channel 2, the bearing uncertainty (angle noise), Δø (rms), is given below.<ref name = Lipsky />{{rp||250}}<ref name = Wiley />{{rp|91}}<ref>East P, "Microwave Intercept Receiver Sensitivity Estimation", Racal Defence Systems Report, 1998</ref> In these results, square-law detection is assumed and the SNR figures are for signals at video (baseband), for the bearing angle φ.

:<math> \Delta \phi_{RMS} = \frac{\Phi}{2}.\frac{\Psi_0^2}{-ln(0.5).\Phi}.\sqrt{\frac{1}{SNR_1} + \frac {1}{SNR_2}} </math>  rads

where SNR<sub>1</sub> and SNR<sub>2</sub> are the video (base-band) signal-to-noise values for the channels for Antenna 1 and Antenna 2, when square-law detection is used.

In the case of 3-channel processing, an expression which is applicable when the S:N ratios in all three channels exceeds unity (when ln(1 + 1/SNR) ≈ 1/SNR is true in all three channels), is

:<math> \Delta \phi_{rms} = \frac{1}{-2.ln(0.5)}. \frac{\Psi_0^2}{\Phi^2}. \sqrt { \bigg ( \phi + \frac{\Phi}{2} \bigg ) ^2 .\frac{1}{SNR_2} + \frac{4. \phi ^2}{SNR_1} + \bigg ( \phi - \frac{\Phi}{2} \bigg ) ^2 .\frac{1}{SNR_3}} </math>

where SNR<sub>1</sub>, SNR<sub>2</sub> and SNR<sub>3</sub> are the video signal-to-noise values for Channel 1, Channel 2, and Channel 3 respectively, for the bearing angle φ.

=== A typical DF system with six antennas ===
[[File:Schematic_of_6-port_DF.png|thumb|right|upright=1.75|Six-port DF system]]

A schematic of a possible DF system,<ref name = Lipsky />{{rp|101}} employing six antennas,<ref>Blake B. (ed.), " Cutlass ESM Equipment", Jane's Radar and Electronic Warfare Systems, 3rd Ed., Jane's Information Group, 1991, p. 406</ref><ref>Streetly M., "SPS-N 5000 ESM System",  Jane's Radar and Electronic Warfare Systems, 10th Ed., Jane's Information Group, 1998, p. 396</ref> is shown in the figure.

The signals received by the antennas are first amplified by a low-noise preamplifier before detection by detector-log-video-amplifiers (DLVAs).<ref name = MITEQ>MITEQ, "IF Signal Processing Components and Subsystems", Application Notes" pp. 33-51, (2010), Find at: https://nardamiteq.com/docs/MITEQ_IFsignal_c17.pdf</ref><ref>Pasternack, " Broadband Log Video Amplifiers". Find at: www.pasternack.com/pages/Featured_Products/broadband-log-video-amplifiers</ref><ref>American Microwave Corporation, DLVA Model: LVD-218-50. Find at: www.americanmic.com/catalog/detector-log-video-amplifiers-dlva/</ref>  The signal levels from the DLVAs are compared to determine the angle of arrival. By considering the signal levels on a logarithmic scale, as provided by the DLVAs, a large dynamic range is achieved <ref name = MITEQ />{{rp|33}} and, in addition, the direction finding calculations are simplified when the main lobes of antenna patterns have a Gaussian characteristic, as shown earlier.

A necessary part of the DF analysis is to identify the channel which contains the largest signal and this is achieved by means of a fast comparator circuit.<ref name = East /> In addition to the DF process, other properties of the signal may be investigated, such as pulse duration, frequency, pulse repetition frequency (PRF) and modulation characteristics.<ref name = NAWC /> The comparator operation usually includes hysteresis, to avoid jitter in the selection process when the bearing of the incoming signal is such that two adjacent channels contain signals of similar amplitude.

Often, the wideband amplifiers are protected from local high power sources (as on a ship) by input limiters and/or filters.  Similarly the amplifiers might contain notch filters to remove known, but unwanted, signals which could impairs the system's ability to process weaker signals.  Some of these issues are covered in [[RF chain]].