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( xˆ , m ˆ ) = argmax Γ ( x , m ) (51) k k k k k xk, mk The previous states and modes are found by tracing backwards from ( xˆ , mˆ ). k k 6. Simulation and Results In our simulation, an approaching target, referring back to Fig. 2, travels at initial range r 0 = 300 km, with velocity v = 2.8 Mach, and headingα = 225 deg. Since the target is not travelling along the radial direction, the Doppler is inevitably time-varying. We set the Doppler rate increment variance 2 2 Q = (6Hz/s ) .Fig. 5 and Fig.6 show the trajectories of target range and Doppler, respectively. The target Doppler information is represented by the observed apparent Doppler and corresponding ambiguity number. Fig. 5. target range trajectory Fig. 6. target Doppler trajectory, represented by apparent Doppler trajectory and ambiguity number trajectory The ambiguity number transits for i to j = i − 1 when the guard condition is satisfied. To account for the random variations in Doppler increment, δ ij is modeled as a Gaussian distributed variable with 20
mean δ ij = 50 Hz and variance Σ ij = 16 , denoted byδij N ( δij ;50,16), and then, as discussed in Section 3, the state-dependent ambiguity number transition probability is ⎧Φ( Cx ij k −δij ; Σ ij ) if j = i ⎪ πij ( xk ) = ⎨1 −Φ( Cijx k −δij; Σ ij ) if j = i−1 (52) ⎪ 0 else ⎪⎩ A maximum target velocity v max = 3Mach is assumed in our simulation. We consider here a low PRF surveillance radar using a carrier frequency of 1.2 GHz and transmitting trains of N d = 64 rectangular pulses with pulse width τ = 1.2 μs and PRF=860 Hz. Radar revisit interval is T R = 4 s . For the reader’s sake, the main surveillance radar parameters are summarized in Table I .After discretization and Doppler processing, the received data are transformed into the ambiguous range-Doppler maps. For low PRF surveillance radars, the measured range is unambiguous while the measured Doppler is ambiguous. Since we are primarily concerned with early warning of approaching targets, the range cells under consideration are limited to the furthest N r = 256 cells. Thus, after each visit, we obtained a 256× 64 ambiguous range-Doppler map. In the simulation, the unknown target’s power u k at each visit is randomly drawn from an exponential distribution with average return power u . Note that we ignore the power variation caused by the changing range since the range variation is small in comparison to the distance between the target and radar. The SNR is defined in range-Doppler domain. The SNR of the target at kth visit is expressed 2 as (SNR) = 10log( u / σ ) . k k Table I Radar System Parameters It’s clear that SNR experienced by the target varies at each visit due to the variation of target power. In the following, in order to characterize the detection performance of the proposed TBD algorithms, we define an average SNR as 2 SNR = 10log( u / σ ) (53) Notice that, since we assume the number of coherently processed pulses N d = 64 , the average received SNR per pulse is obtained by subtracting 10log 64 18dB from the reported values in range-Doppler domain. Fig.7 shows a mesh plot of the reflected power in ambiguous range-Doppler maps. The target is located at cell (231, 14). When the SNR is high (13dB) and it’s easy to discern the target as a high peak (Fig.7(a)). When the SNR is relative low (10dB), it’s not easy to distinguish the target from the noise (Fig.7(b)). TBD procedures provide a way to improve the detestability of low SNR target by jointly processing several range-Doppler maps. In the following simulation, we set the number of jointly processed ambiguous range-Doppler maps to K = 6 . 21
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