ADAPTIVE WAVEFORM SELECTION AND TARGET TRACKING BY WIDEBAND MULTISTATIC RADAR/SONAR SYSTEMS

Transcription

1 ADAPTIVE WAVEFORM SELECTION AND TARGET TRACKING BY WIDEBAND MULTISTATIC RADAR/SONAR SYSTEMS Ngo Hung Nguyen, Kutluyıl Doğançay, and Linda M. Davis Institute for Teleommuniations Researh, Shool of Engineering University of South Australia, Mawson Lakes SA 5095, Australia ABSTRACT An adaptive waveform seletion algorithm for target traking by multistati radar/sonar systems in wideband environments is presented to minimize the traking mean squared error. The proposed seletion algorithm is developed based on the minimization of the trae of error ovariane matrix for the target state estimates (i.e. the target position and target veloity). This ovariane matrix an be omputed using the Cramér- Rao lower bounds of the wideband radar/sonar measurements. The performane advantage of the proposed adaptive waveform seletion algorithm over the onventional fixed waveforms with minimum and maximum time-bandwidth produts is demonstrated by simulation examples using various FM waveform lasses. Index Terms adaptive waveform seletion, wideband, multistati, radar/sonar, target traking. 1. INTRODUCTION Adaptive waveform seletion has been shown to signifiantly improve the traking performane of radar/sonar systems [1 5], where the trak information obtained from the past measurements is utilized for seleting the waveform of the next signal transmission in order to ahieve optimal traking performane. The early work in adaptive waveform seletion onsidered target traking in a one-dimensional lutter-free environment [1]. The authors then extended their work to a luttered senario [2]. The problem of adaptive waveform seletion was further investigated for two-dimensional target traking in different senarios (single or multiple targets, narrowband or wideband environments, and lutter-free or luttered environments) [3 5]. All of these works were onduted for monostati systems, in whih the transmitter and reeiver are olloated. In many appliations, it has beome apparent that multistati radar/sonar systems an provide superior performane over monostati radar/sonar systems [6]. Adaptive waveform seletion, whih has been addressed for the monostati ase, is a promising approah to improve the traking performane of multistati radar/sonar systems. However, there are signifiant differenes between monostati and multistati radar, partiularly in terms of performane. The multistati radar performane depends not only on transmitted waveform but also on radar geometry [7, 8]. Thus it is not readily obvious how to apply adaptive waveform seletion to the multistati ase. In our previous work [9, 10], we have suessfully demonstrated the superior performane of adaptive waveform seletion over onventional fixed waveforms in a multistati setting. However, the work in [9, 10] is restrited to narrowband signals. The narrowband ondition T B /ṙ an be easily satisfied in radar appliations beause the speed of light is very large ompared to the speed of typial targets. Here T B is the time-bandwidth produt, and r is the total bistati target range. However, this narrowband ondition is generally very hard to satisfy in sonar appliations, where the speed of sound ( s 1500 m/s) is omparable to the speed of underwater targets (on the order of 10 m/s) and the time-bandwidth produt of the sonar signal is usually large (T B > 100) [5,11]. In this senario, wideband signal models are ommonly employed whereby the reeived signal is not simply a time-shifted and frequeny-shifted opy of the transmitted signal with no time saling as in the narrowband ase. The Cramér-Rao lower bounds (CRLB) of the wideband radar/sonar measurements are also different to those of the narrowband ase [12]. In this paper, we onsider the problem of adaptive waveform seletion for target traking by wideband multistati radar/sonar systems. We first derive a new wideband bistati signal model. Note that a bistati signal model was first introdued by Tsao et al. [13], but for narrowband signals only. A bistati signal model for wideband radar signals was then derived in [14]. However, this model does not work well in sonar appliations, where the speed of sound is omparable to the speed of typial underwater targets. By following the derivation in [13], we form a wideband bistati signal model whih an be employed for both radar and sonar appliations. The target traking problem by wideband multistati radar/sonar systems is then defined utilizing the new derived signal model. The adaptive waveform seletion algorithm is developed to minimize the traking mean squared error of the target state estimate, i.e. target position and veloity, by equivalently minimizing the trae of the traking error ovariane matrix. The superior performane of the proposed

2 waveform seletion algorithm vis-à-vis onventional fixed waveforms is demonstrated by simulation examples. 2. WIDEBAND BISTATIC SIGNAL MODEL Consider a bistati hannel between the transmitter Tx, the reeiver Rx, and the target Tgt. Let s(t) denote the transmitted bandpass signal. Then the reeived signal is given by r(t) = As[t τ(t)] + n(t) (1) where τ(t) is the total travel time of the signal from Tx to Tgt and to Rx, A is the signal attenuation, and n(t) is the noise at the reeiver. Denote the target-transmitter distane and the target-reeiver distane by R T and R R, respetively, with R = R T +R R giving the total distane. Using the results derived in [13] without narrowband approximation, we have τ(t) = τ a + Ṙ(t ) + ṘR(t ) (t τ a) (2) where t is the time instant when the transmitted signal reahes Tgt, and τ a is the atual total time delay given by τ a = R(t ) = R T (t ) + R R (t ). (3) Using (2) and (3), the reeived signal r(t) in (1) beomes r(t) = As[β(t τ a )] + n(t) (4) where β is the Doppler streth fator given by β = 1 Ṙ(t ) ( + ṘR(t )). (5) Note that, sine ṘR in radar appliations, we have β 1 Ṙ(t )/ [14]. In sonar appliations, where ṘR is omparable to the speed of propagation of sound, this approximation does not hold. Note that a wideband environment is haraterized by β 1 ompared to β = 1 for narrowband environments. 3. TARGET TRACKING BY MULTISTATIC RADAR/SONAR SYSTEM We onsider an ative multistati radar/sonar system with a single dediated transmitter at the origin [0, 0] and N reeivers loated separately at [x i Rx, yi Rx ], i = 1, 2,..., N for traking a single target in a two-dimensional wideband environment as shown in Fig. 1. The onsidered traking senario is assumed to be lutter-free with moderately high signalto-noise ratio (SNR) at the reeivers. We also assume that ommuniation links are available between the transmitter and reeivers with negligible time-synhronization errors. Fig. 1. Illustration of multistati radar/sonar geometry. At eah reeiver, the measurements of time delay, Doppler streth fator, and arrival angle are available. These measurements from all reeivers are sent to a entral proessor loated at the transmitter site and used for target traking and waveform seletion. To deal with the nonlinearity between the target state vetor x k and the measurement vetor z k, an extended Kalman filter (EKF) is employed for traking the target. Note that in this setion we utilize the wideband signal model in Setion 2 to formulate the measurement model of the target traking problem. Let x k = [x k, y k, ẋ k, ẏ k ] T denote the target state vetor at time k = 0, 1,..., with x k, y k orresponding to the target position and ẋ k, ẏ k orresponding to the target veloity in Cartesian oordinates. The target dynamis are modelled by a nearly onstant veloity model given by x k+1 = Fx k + w k (6) where w k N (0, Q) is the proess noise. The matries F and Q are given in [15]. The measurement equation for entralized target traking is given by z k =[τ 1 k, β 1 k, θ 1 k,..., τ N k, β N k, θ N k ] T =h(x k ) + n k =[h 1 τ (x k ), h 1 β(x k ), h 1 θ(x k ),.., h N τ (x k ), h N β (x k ), h N θ (x k )] T + [n 1 τk, n 1 βk, n 1 θk,..., n N τk, n N βk, n N θk] T (7) where τk i, βi k, θi k are the measurements of time delay, Doppler streth fator, and arrival angle at the i-th reeiver, respetively, and n k N (0, N k ) is the measurement error at time k. Using the wideband bistati signal model in Setion 2, i.e. equations (3) and (5), we have h i τ (x k ) = [x k, y k ] + [x k, y k ] [x i Rx, yi Rx ] Ṙ i h i β(x k ) = 1 + Ṙi R = 1 ṘT + Ṙi R + Ṙi R (8a) (8b)

3 where Ṙ T =ẋkx k + ẏ k y k [x k, y k ] (9a) Ṙ i R =ẋk(x k x i Rx ) + ẏ k(y k y i Rx ) [x k, y k ] [x i Rx, yi Rx ]. (9b) Furthermore, h i θ(x k ) in (7) is given by ( h i yk y i ) Rx θ(x k ) = atan2 x k x i Rx (10) where atan2 denotes the four quadrant artangent. In this paper we assume, as ommonly done, that the measurement errors n k an ahieve their CRLB. Thus N k equals to the CRLB C k of the measurement vetor z k that onsists of all measurements in the system. As the measurement errors at different reeivers are statistially independent, we have N k = C k = diag(c 1 k, C 2 k,..., C N k ) (11) where C i k is the CRLB of the measurements (τ k i, βi k, θi k ) at the i-th reeiver. Sine the CRLB Cθ i k orresponding to θk i is independent from the CRLB C i (τ k,β k ) orresponding to (τk i, βi k ) [16], we have Ci k = diag(ci (τ k,β k ), Ci θ k ). Moreover, Cθ i k only depends on SNR at reeiver [16], thus an be modelled by Cθ i k = σθ 2/SNRi k where σ θ is a onstant. C i (τ k,β k ), on the other hand, an be evaluated from the wideband ambiguity funtion of the transmitted waveform [12]. In fat C i (τ k,β k ) is the inverse of the Fisher information matrix J i (τ k,β k ). For a general form of the transmitted bandpass signal given as s(t) = a(t) exp j(ψ(t) + 2πf t) (12) where a(t) and ψ(t) are the amplitude and phase modulation funtions, respetively, and f is the arrier frequeny, we have J i (τ k,β k ) = 2SNRi k I (τ,β) with the elements of I (τ,β) given by [12] I 1,1 = I 1,2 =I 2,1 = I 2,2 = (ȧ 2 (t) + a 2 (t)ψ 2 (t))dt [ 2 λ/2 a (t)ψ(t)dt] 2 (13a) t(ȧ 2 (t) + a 2 (t)ψ 2 (t))dt a 2 (t)ψ(t)dt t 2 (ȧ 2 (t) + a 2 (t)ψ 2 (t))dt [ ] 2 λ/2 ta 2 (t)ψ(t)dt 1 4 ta 2 (t)ψ(t)dt (13b) (13) where Ψ(t) = ψ(t) + 2πf. Note that the ambiguity funtion and CRLB with respet to the target-to-reeiver range and bisetor veloity is often onsidered in multistati radars [8, 13, 17]. This CRLB an also be utilized for the onsidered traking problem. However, it is to be emphasized that, in the ontext of target traking, the ultimate objetive is to minimize the mean squared error of the target state estimate, i.e. target position and veloity (see Setion 4). We an show that the traking error ovariane an be equivalently omputed using either the CRLB of time delay and Doppler or the CRLB of reeiver-to-target range and bisetor veloity. Moreover, using the CRLB of reeiver-to-target range and bisetor veloity requires more omputations as this CRLB is needed to ompute from the CRLB of time delay and Doppler using the derivative hain rule [8]. Therefore, based on these onsiderations, the CRLB of time delay and Doppler streth is employed here. 4. ADAPTIVE WAVEFORM SELECTION As an be seen in Setion 3, the CRLB for the measurements of time delay and Doppler streth are dependent on the transmitted waveform; where the elements of I (τ,β) are omputed from a(t) and ψ(t). As a result, the measurement error ovariane N k+1, is also dependent on the transmitted waveform. In other words, N k+1 (Ω k+1 ) an be expliitly shown to be a funtion of the transmitted waveform parameters Ω k+1. Note that the waveform parameters might be different depending on waveform lasses. By using the EKF s update equations, the error ovariane matrix P k+1 k+1 of the target state estimate at time k + 1 an be omputed prior to the signal transmission at time k + 1 as follows [15]: P k+1 k = FP k k F T + Q S k+1 (Ω k+1 ) = H k+1 P k+1 k H T k+1 + N k+1 (Ω k+1 ) K k+1 (Ω k+1 ) = P k+1 k H T k+1s k+1 (Ω k+1 ) 1 P k+1 k+1 (Ω k+1 ) = [I K k+1 (Ω k+1 )H k+1 ]P k+1 k where H k+1 is the Jaobian matrix of h(x k+1 ) [15], whih is straightforward to derive from (8) (10). P k+1 k+1 now beomes a funtion of the parameters Ω k+1 of the waveform to be transmitted at time k + 1: P k+1 k+1 (Ω k+1 ). The objetive of the proposed adaptive waveform seletion sheme is to minimize the total traking mean squared error of the target state estimate in both target position and veloity. This an be ahieved by minimizing the trae of P k+1 k+1 (Ω k+1 ). Therefore, the optimal transmitted waveform is obtained by Ω k+1 = arg min Tr(P k+1 k+1 (Ω k+1 )). (15) Ω k+1 Library Note that Ω k+1 is found using a grid searh over a finite searh spae (i.e. a waveform library). The waveform library

4 Fig. 3. A pattern of seleted waveform parameters for LFM. Fig. 2. Traking performane for LFM waveform. may ontain a single waveform lass with varying parameters or a number of different waveform lasses. This method of disrete grid searh is omputationally heaper than gradientbased methods and perform well in pratie [3]. However, the omputation of disrete grid searh is quite expensive if the waveform library is large, thus a losed-form or approximation solution is needed. This is out of the sope of the present paper and will be onsidered in our future work. 5. SIMULATION EXAMPLES In this setion, we onsider a multistati sonar system with a dediated transmitter and four reeivers traking an underwater target in a wideband environment, where the transmitter is loated at the origin [0, 0 m], and the reeivers are loated at [500, 0] m, [250, 500] m, [250, 200] m, and [0, 250] m. The arrier frequeny of the transmitted waveform is f = 25 khz. The speed of sound under water is assumed to be onstant at 1500 m/s. The initial position and veloity of the target are [600, 400] m and [ 4, 2] m/s, respetively. The sampling interval is T = 1s, and the onstant assoiated with the maneuver level of the target in the proess noise ovariane matrix Q is q = The SNR at the i-th reeiver is modelled by SNR i k = R0/(R 4 T k RRk i )2, where R 0 = 1000 m. The onstant σ θ assoiated with Cθ i is set to 0.01 rad. We onsider three different waveform lasses, viz. the linear FM (LFM), hyperboli FM (HFM), and exponential FM (EFM) [4]. Their phase funtions ψ(t) are given in Table 1 Waveform Phase funtion ψ(t) F LFM 2πb[t/γ + (t + λ/2) 2 /2)] bλ λ HFM 2πb[ln(t + γ + λ/2)] b γ(γ+λ) b EFM 2πb[exp( (t + λ/2)/γ)] γ (e γ 1) Table 1. FM waveforms onsidered in the simulation. Fig. 4. Traking performane of different waveform lasses. with referene time t r = 1. A trapezoidal envelope is used for the amplitude modulation of the transmitted waveform given by [4] a(t) = α t f (t Ts 2 t f ), T s /2 t f t < T s /2 α, T s /2 t < T s /2 α t f ( Ts 2 + t f t), T s /2 t < T s /2 + t f where α is hosen so that s(t) in (12) has unit energy, and t f T s /2 is the rise/fall time. We hoose t f = 0.001λ, where λ = T s + 2t f is the hirp duration. The hirp duration λ and the frequeny sweep F are the waveform parameters with λ {0.05, 0.1, 0.15, 0.2, 0.25, 0.3} s and F {400, 900, 1400, 1900} Hz. The averaged root mean squared error (averaged-rmse) obtained from 500 Monte Carlo simulation runs is used for evaluating traking performane. It should be noted that this averaged-rmse inludes traking errors in both target position and veloity. Fig. 2 ompares the traking performane of the proposed adaptive waveform seletion sheme with those of the onventional fixed waveform shemes with the maximum and minimum time-bandwidth produt (BT max or BT min ) using the LFM waveform lass. It is apparent in Fig. 2 that the proposed

5 adaptive waveform signifiantly outperforms the fixed BT min and BT max waveforms. Fig. 3 shows a pattern of the seleted waveforms for a single Monte Carlo simulation run. Fig. 4 ompares the traking performane of the adaptive waveform seletion algorithm orresponding to three different waveform lasses (LFM, HFM, EFM). Note that the same parameter ranges of the hirp duration λ and frequeny sweep F are used for these three waveform lasses. It an be seen that the traking performane of the adaptive HFM and EFM waveforms are signifiantly better than the adaptive LFM waveform s performane while the adaptive EFM waveform performs the best among the three waveforms for the simulated traking senario. We observed that the proposed algorithm works well in high SNR onditions where CRLB an be used to approximate the radar measurement errors. However, CRLB is not a tight bound in low SNR onditions [18]. Therefore, our assumption on the measurement error ahieving their CRLB is not valid for the ase of low SNR, hene other lower bounds may need to be derived. Moreover, EKF may diverge if SNR is low. 6. CONCLUSION We presented an adaptive waveform seletion algorithm for target traking by wideband multistati radar/sonar systems to minimize the traking mean squared error. A wideband bistati signal model for both radar and sonar appliation was derived and utilized to formulate the wideband target traking problem. The proposed waveform seletion algorithm is developed by seleting the waveform whih yields the smallest trae of the error ovariane matrix of the target state estimate. This error ovariane is omputed using the CRLB of the wideband measurements of time delay, Doppler streth, and arrival angle. Our simulation examples demonstrate that the traking performane of wideband multistati radar/sonar systems an be signifiantly improved by the proposed adaptive waveform seletion algorithm. The onsidered problem an be further extended to multiple target traking and/or with the presene of lutter. This extension will be onsidered in our future work. REFERENCES [1] D.J. Kershaw and R.J. Evans, Optimal waveform seletion for traking systems, IEEE Trans. Inf. Theory, vol. 40, no. 5, pp , [2] D.J. Kershaw and R.J. Evans, Waveform seletive probabilisti data assoiation, IEEE Trans. Aerosp. Eletron. Syst., vol. 33, no. 4, pp , [3] S.P. Sira, A. Papandreou-Suppappola, and D. Morrell, Dynami onfiguration of time-varying waveforms for agile sensing and traking in lutter, IEEE Trans. Signal Proess., vol. 55, no. 7, pp , [4] S.P. Sira, A. Papandreou-Suppappola, and D. Morrell, Advanes in Waveform-Agile Sensing for Traking, Morgan and Claypool, [5] S.P. Sira, A. Papandreou-Suppappola, and D. Morrell, Waveform sheduling in wideband environments, in Pro. IEEE ICASSP, 2006, pp. V1121 V1124. [6] E. Hanle, Survey of bistati and multistati radar, IEE Pro., vol. 133, no. 7, pp , [7] M.C. Wiks, E.L. Mokole, S.D. Blunt, R.S. Shneible, and V.J. Amuso, Eds., Priniples of Waveform Diversity and Design, SiTeh Publishing, Raleigh, NC, [8] M.S. Greo, P. Stino, F. Gini, and A. Farina, Cramér- Rao bounds and seletion of bistati hannels for multistati radar systems, IEEE Trans. Aerosp. Eletron. Syst., vol. 47, no. 4, pp , [9] N.H. Nguyen, K. Doganay, and L.M. Davis, Adaptive waveform seletion for multistati target traking, IEEE Trans. Aerosp. Eletron. Syst., under review. [10] N.H. Nguyen, K. Doganay, and L.M. Davis, Adaptive waveform sheduling for target traking in lutter by multistati radar system, in Pro. IEEE ICASSP, 2014, pp [11] Q. Jin, K.M. Wong, and Z.Q. Luo, The estimation of time delay and doppler streth of wideband signals, IEEE Trans. Signal Proess., vol. 43, no. 4, pp , Apr [12] D.W. Riker, Eho Signal Proessing, Springer, [13] T. Tsao, M. Slamani, P. Varshney, D. Weiner, H. Shwarzlander, and S. Borek, Ambiguity funtion for a bistati radar, IEEE Trans. Aerosp. Eletron. Syst., vol. 33, no. 3, pp , [14] M.G.M. Hussain, Ambiguity funtions for monostati and bistati radar systems using UWB throb signal, IEEE Trans. Aerosp. Eletron. Syst., vol. 47, no. 3, pp , [15] Y. Bar-Shalom and W.D. Blair, Multitarget-Multisensor Traking: Appliations and Advanes, Volume III, Arteh House, Boston, [16] A. Dogandzi and A. Nehorai, Estimating range, veloity, and diretion with a radar array, in Pro. IEEE ICASSP, 1999, vol. 5, pp [17] P. Stino, M.S. Greo, F. Gini, and A. Farina, Posterior Cramér-Rao lower bounds for passive bistati radar traking with unertain target measurements, Signal Proess., vol. 93, no. 12, pp , [18] H.L. Van Trees and K.L. Bell, Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Traking, Wiley-IEEE Press, 2007.