Editing Direction finding (section)

=== 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.