One of the parameters characterizing the performance of every Direction Finding (DF) system is the ambiguity threshold. The ambiguity threshold is the Signal-to-Noise Ratio (SNR) below which, for a given number of samples, the probability of an ambiguous Direction-of-Arrival (DOA) result rises and thereby sets a limit on the performance of the DF system.
The value of the ambiguity threshold depends critically on the array geometry. In certain arrays, such as uniform linear arrays with half a wavelength spacing, the threshold occurs at relatively low SNR and low number of samples, while in other arrays, such as sparse arrays, the threshold occurs at relatively high SNR and high number of samples.
To illustrate the performance of a DF system below the ambiguity threshold, reference is now made to FIG. 1 which is a histogram illustration of maximum likelihood estimation for 400 DOA results for a source having SNR of -4 dB. The source impinges from 100.degree. on a sparse three-element linear array with inter-element spacing ration of 3:4 and aperture 3.lambda., where .lambda. is the wavelength. The DOA was estimated by the deterministic maximum likelihood (DML) estimator from 100 samples of the array output. The highest peak, referenced 12, occurs at the true DOA of 100 degrees. In addition, there are high ambiguous peaks at 150.degree. and at 60.degree., referenced 14 and 16, respectively. The ambiguous peaks 14 and 16 correspond to the secondary peaks of the multi-modal likelihood function which, as a result of the noise, occasionally exceed the main peak corresponding to the true DOA.
There are several approaches to the tracking of multiple targets. Various techniques for tracking the angles of arrival of moving targets have been proposed, for example, "An Efficient Algorithm for Tracking the Angles of Arrival of Moving Targets" by C. R. Sastry, E. W. Kamen and M. Simian, published by IEEE Trans on Signal processing, Vol.39:242-246 No. 1 January 1991. Further examples include: "Tracking the Direction of Arrival of Multiple Moving Targets" by C. R. Zao, C. R. Sastry and B. Zhou, published by IEEE Trans on Signal processing, Vol.42:1133-1144 No. 5 May 1994, and "Multiple Target Tracking Using Maximum Likelihood Principle" by A. SAtish and R. L. Kashyap, published by IEEE Trans on Signal processing, Vol.43:1677-1695 No.7 July 1995.
Reference is now made to FIGS. 2A-2F which are graphical illustrations of the likelihood function {L(.theta.)} of a typical source moving along the azimuth axis, .theta..
FIGS. 2A-2F are graphical illustrations of the likelihood function {L(.theta.)} of a typical source having a signal to noise ratio (SNR) of -4 dB moving from 85.degree. (FIG. 2A) through 90.degree. (FIG. 2B), 95.degree. (FIG. 2C), 100.degree. (FIG. 2D), 105.degree. (FIG. 2E) and 110.degree. (FIG. 2F) along the azimuth axis (.theta.). The source is received by a three-element linear array with inter-element ratio of 3:4 and aperture of 3.lambda.. Each graph illustrates the directions-of arrival (DOA) likelihood function {L(.theta.)} for a plurality of samples (approximately 100) below the ambiguity threshold.
The peaks of L(.theta.) represent the most likely directions-of-arrival for the given batch. Generally, above the ambiguity threshold, the peak corresponding to the correct DOA is the highest. However, below the ambiguity threshold, the height of the ambiguous peaks rise and may exceed the peak corresponding to the correct DOA.
As shown in FIG. 2A, there are two peaks, referenced 302, having a DOA of approximately 85.degree., and 304 (having a DOA of approximately 125.degree.). Peak 302 is lower than peak 304, having a likelihood value of just below 50. (The highest peak 304 occurs for a DOA of 125.degree. instead of the correct one of 85.degree..
FIG. 2B for an angle of 90.degree., has three peaks, 306, 308 and 310 indicating DOAs of approximately 50.degree., 90.degree. and 130.degree., respectively, having likelihood values of approximately 60, 65 and 60, respectively. FIG. 2C for an angle of 95.degree., has three peaks, 312, 314 and 316 indicating DOAs of approximately 55.degree., 95.degree. and 135.degree., respectively, having likelihood values of approximately 55, 55 and 50, respectively.
FIG. 2D for an angle of 100.degree., has three peaks, referenced 318, 320 and 322 having a likelihood value above 50, indicating DOAs of approximately 55.degree., 100.degree. and 120.degree., respectively. In addition, there are three other peaks, referenced 324, 325 and 326, having likelihood values slightly above and slightly below 50 and 60, respectively. In this case, the highest peak 318 occurs for a DOA of 55.degree. instead of the correct one of 100.degree. (peak 320).
FIG. 2E for an angle of 105.degree., has three peaks, referenced 128, 130 and 132, indicating approximate DOAs of approximately 65.degree., 105.degree. and 160.degree., respectively, having likelihood values of approximately 70, 75 and 65, respectively.
FIG. 2F for an angle of 110.degree., has a peak 334 (likelihood value 55) indicating a DOA of 110.degree.. In addition, there is a second peak, referenced 336, having a likelihood value of approximately 50 indicating a DOA of approximately 75.degree. and a potential third peak 338 indicating a DOA above 170.degree..
Thus, as illustrated in FIGS. 2A-2F, hereinabove, prior art systems which are based on using the most likely DOA at each measurement point are likely to be confused by similar peaks and even incorrectly calculate the DOA (FIG. 2A and 2D). For the angles of 85.degree., 95.degree. and 105.degree., a second peak occurs close to but slightly higher in likelihood value than the correct peak, thus making a correct assessment difficult.
A disadvantage of these prior art techniques is that the direction-of-arrival tracking algorithms cannot handle the frequent, large and non-random errors resulting from operation below the ambiguity threshold.