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Definition 2 The probability of target detection and reliable tracking P dT , is defined as the probability of making a correct target detection and of recovering a target trajectory such that each retrieved state, ( iˆ ˆ k, lk) , k = 1,... K , is within ε r cells in range and ε d cells in Doppler from the actual location, and the estimated ambiguity number sequence ˆ K Μ = ( mˆ ˆ 1,..., m K ) is equal to the true one. Hence PdT , = Pr ⎛( max K( K, mK) V ⎞ ⎜ Γ x > T ∪ dT , ⎟ ⎝ xK, m ) E (57) K ⎠ where the event E dT , is expressed as abs( iˆ k −ik) ≤ ε r, abs( lˆ k −lk) ≤ εd, k = 1,... K , Μˆ K = Μ K (58) It’s evident that the overall system performance depends on the choice of the tolerance factors: ε r and ε d . The smaller the tolerance factors, the larger the SNR is required in order to maintain a certain value of detection probability P d and P under a preset probability of false alarm rate. In the dT , following, the detection and tracking performance with respect to average SNR and tolerance factors are investigated. We first consider the case that the nuisance parameters are known in advance; the corresponding −6 implementation of the DP recursive is given in (33). In Fig.9, we set P fa = 10 , and plot P d and P versus average SNR under different settings of tolerance factor. The tolerance dT , factor ( ε r, ε d) from up to down is (4,2) , (2,1) and (0,0) . ( ε r, ε d) = (0,0) means correct detection and estimation. We note that as the tolerance factor becomes smaller (i.e., the estimation accuracy becomes higher), both P d and P dT , decreases considerably. We can also notice that compared to P dT , , P is less sensitive to the tolerance factor. Satisfactory detection and tracking performance can be d achieved when the tolerance factor is (4,2) . Fig. 9. P and d P dT , versus the average SNR in range-Doppler domain, with known nuisance parameters, ( ε , ε ) = (4, 2), (2, 1), (0, 0) from up to down r d We now investigate the situation when the nuisance parameters are unknown; the DP recursive with −6 unknown nuisance parameters is given in (50). In Fig. 10, we set P fa = 10 , and report P d 24
and P dT , versus the average SNR under several values of tolerance factor. For the sake of comparison, the tolerance factor ( ε r, ε d) from up to down is also set to (4,2) , (2,1) and (0,0) . Similar result to Fig. 10 is obtained: both P d and P decreases considerably with the decreasing of tolerance factors. By dT , inspection, we note that P d and P dT , suffer an approximately 1dB loss due to the prior uncertainty as to the nuisance parameters. Fig. 10. P and d P versus the average SNR in range-Doppler domain, with unknown nuisance dT , parameters, ( ε , ε ) = (4, 2), (2, 1), (0, 0) from up to down r d 7. Conclusions In this paper we have investigated the TBD procedures with reference to a low PRF surveillance radar framework. We give physical and statistical models for the received power measurements on the ambiguous range-Doppler maps. In order to avoid the nonlinear transformation between the target’s state coordinates and measurement coordinates (defined by range and apparent Doppler); we model the target’s dynamics directly in the measurement coordinate. Exploiting the target dynamics in the ambiguous range-Doppler maps as a hybrid system, we have proposed DP based methods for joint MAP estimation of target’s trajectory in the ambiguous range-Doppler maps and the corresponding ambiguity number sequence. The detection and tracking performance of the proposed procedure is studied using two measures: 1) detection probability P d , 2) detection and reliable tracking probability P dT , . Simulation results show satisfactory detection and tracking performance is achieved when the tolerance factor is set to within four cells in range and two cells in Doppler from the true location. The impact of the uncertainties in the nuisance parameters (target power and noise variance) has also been studied. Simulation shows that both P d and P dT , suffer an approximately 1dB loss due to the prior uncertainty of the nuisance parameters. 25
Page 2 and 3: The 3 rd VACPS Research Workshop Pr
Page 4 and 5: Organizing Committee Victorian Asso
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