Waveform Design and Scheduling in Space-Time Adaptive Radar

Transcription

1 Waveform Design and Sheduling in Spae-Time Adaptive Radar Pawan Setlur and Natasha Devroye Eletrial and Computer Engineering University of Illinois at Chiago Chiago, IL Muralidhar Rangaswamy US Air Fore Researh Laboratory Sensors Diretorate, RYAP WPAFB, OH Abstrat Waveform design and waveform sheduling are addressed in the ontext of spae time adaptive proessing (STAP) for radar. An air-borne radar with an array of sensors is assumed, whih interrogates ground based targets. The designed waveform is assumed to be transmitted over one oherent proessing interval (CPI). The waveform design and waveform sheduling problems are formulated with a ost funtion similar to the Minimum Variane Distortionless Response (MVDR) ost funtion as in lassial radar STAP. Least-squared solutions for the designed waveform are obtained. It is shown that both the designed waveform and the sheduled waveforms will depend on the spatial and Doppler responses of the desired target; in partiular, its spatial and temporal steering vetors. The fous of this paper will be the performane of the designed and sheduled waveforms for unknown orrelation matries but estimated from the training data, and will be addressed via simulations. I. INTRODUCTION The objetive of this paper is to address waveform design and waveform sheduling via spae time adaptive proessing (STAP) in radar [1] [5]. An air-borne radar is assumed with an array of sensor elements observing a moving target on the ground. We will assume that the waveform design and sheduling are performed over one CPI rather than on an individual pulse repetition interval (PRI). Traditional STAP involves multidimensional adaptive filtering whih ombines signals from several antenna elements and from multiple waveform repetitions to suppresses lutter, interferene and noise [2] in both spae and time. Although detetion is not the fous here, it is well known that STAP improves detetion of targets in both mainlobe and sidelobe lutter and in jamming interferene environments [1] [4]. To failitate waveform design and sheduling, we develop a STAP model onsidering the fast time samples along with the slow time proessing. This is different from traditional STAP whih generally onsiders the data after mathed filtering [1], [2]. Nonetheless STAP researh efforts have been proposed whih onsider inlusion of fast time samples in spae time proessing, see for example [1], [6], [7] and referenes therein. It is shown that by spatio-temporal proessing prior to mathed filtering, the spatio-temporal steering vetor is also a funtion of the waveform transmitted. The minimum variane distortionless response (MVDR) optimization problem [8] for STAP seeks to minimize the undesired response from noise, lutter and interferene while simultaneously preserving the target response [1] [4]. In line with traditional STAP, we formulate both the waveform design, and the waveform sheduling problem as an MVDR type optimization. The noise, lutter, and interferene are modeled stohastially and are assumed to be mutually unorrelated. Clutter is assumed from ground refletions and hene is assumed to be persistent in most range gates. The lutter orrelation matrix is a funtion of the waveform, and the orrelation matrix of the ombined noise, interferene and lutter is hene also a funtion of the waveform transmitted. In this ase, a losed form solution to the waveform design problem is not tratable. For simpliity in the analysis, we ignore the dependeny of the waveform in the lutter orrelation matrix, and derive a suboptimal least square (LS) solution. Not surprisingly then, it is then shown that both the waveform design and sheduling riteria depend on the spatial and temporal steering vetors of the desired target. The paper is organized as follows: the model utilizing the fast time as well as slow time is presented in Setion II. The waveform design and sheduling problems are formulated in Setion III, simulations are presented in Setion IV. In Setion V, the onlusions are drawn based on the analysis and simulations. In pratie, the designed and sheduled waveforms will depend on the orrelation matries of noise, interferene and lutter whih are unknown. The major fous of this paper is to investigate the impat of using the estimated orrelation matrix from training data to examine the performane loss, and will be addressed in the Setion IV. II. STAP MODEL The radar onsists of an air-borne linear array omprising M sensor elements. Without loss of generality, assume that the first sensor in the array is the phase enter, and ats as both a transmitter and reeiver, the rest of the elements are purely reeivers. Further assume that the array is alibrated and eah element in the array has an idential antenna pattern. The first sensor is loated at x r 2 3 and the ground based point target at x t 2 3. The radar transmits the burst of pulses: u(t) = LX s(t lt p )exp(j2 f o (t lt p )),t2 [0,T) (1) l=1

2 where, f o is the arrier frequeny, and T p = 1/f p is the inverse of the pulse repetition frequeny, f p. The pulse width is denoted as T =1/B and is the inverse of the bandwidth, B. Hene the oherent proessing interval (CPI) onsists of L pulses, eah of width equal to T. The geometry of the sene is shown in Fig. 1, where t and t denote the azimuth and elevation, both of whih will be useful subsequently when introduing the spatial steering vetor. The radar and target are both assumed to be moving. To develop the model, we ignore the noise, lutter and interferene for the time being and assume a non-flutuating target. Then the desired target s reeived signal for the l-th pulse, and at the m-th sensor element is given by s ml (t) = t s(t lt p m )exp(j2 (f o +f dm )(t lt p m )) (2) where the target s observed Doppler shift is denoted as f dm, and its omplex bak-sattering oeffiient as t. Assume that the array is along the loal x axis as shown in Fig. 1. Then, the oordinates of the m-th element is given by x t +md, d := [d, 0, 0] T,m=0, 1, 2...,M 1, where d is the inter-element spaing. The delay m ould be re-written as m = x r x t / + x r + md x t / s xr xt xr xt = + 1+ md 2 (a) xr xt + xr xt =2 xr xt x r x + 2mdT (x r x t) t 2 x r x t 2 (3) 1+ mdt (x r x t) x r x t 2 + mdt (x r x t), (4) x r x t where in approximation (a), the term / md 2 was ignored, i.e. it is assumed that d/ x r x t << 1, and then a binomial approximation was employed. It is useful now to introdue the azimuth and elevation angles, where, by geometri manipulations, we have: x r x t x r x t =[sin( t)sin( t ), sin( t ) os( t ), os( t )] T. Using the above equation in (4), the delay m,m = 0, 1,...,M 1 an be rewritten as m =2 x r x t + md sin( t)sin( t ). (5) The Doppler shift, i.e. f dm is omputed as (ẋ r ẋ t ) T (x r x t ) f dm =2f o (6) x r x t md T apple ẋ r ẋ t (x r x t )(ẋ r ẋ t ) T (x r x t ) + f o x r x t 2 k x r x t 3 where ẋ ( ) is the vetor differential of x ( ) w.r.t time. In pratie d is a fration of the wavelength, and assuming that d/ x r x t << 1 we approximate the seond term in (6) as 0. The Doppler shift is no longer a funtion of the sensor index, m, and is rewritten as f dm = f d =2f o (ẋ r ẋ t ) T (x r x t ) x r x t Assumption A1: From here onwards, the standard narrowband assumption is invoked [1], i.e. the signal propagation time aross the array is assumed to be muh smaller than the inverse of the signal bandwidth. This then implies that s(t 2 x r x t / md sin( t )sin( t )/) s(t 2 x r x t /). Using this and substituting (5) and (7) in (2), and downonverting to baseband we obtain, s ml (t) = t s(t lt p t )e j4 (fo+f d) t e j2 md sin( t ) sin( t ) o e j2 f d md sin( t ) sin( t ) e j2 f d(t lt p) where t =2 x r x t / and o is the operating wavelength. Some pratial approximations an now be made on (8). Assumption A2: For arguments sake let d = o /p, where p is an arbitrary positive integer (p =2is the ritial spatial Nyquist). Then, exp( j2 f d md sin( t )sin( t ) )=exp( j2 mf d sin( t )sin( t ) pf o ) 1 It is easily shown that this assumption is valid in most general ases. However, this is invalidated for long array apertures (M >100), whih in the first plae ould be impratial for air-borne radar systems. Assumption A3: We assume that the phase from the Doppler is insignifiant within the fast time, i.e. t. In other words, we assume that exp(j2 f d t) 1,t 2 [0,T). For pratial Doppler shifts this is reasonable. These assumptions are now enfored in (8), without expliitly stating them in the rest of the paper. Examples validating A1-A3 are subsequently disussed in Setion II-B. A. Vetor signal model Let s(t) be sampled disretely resulting in N disrete time samples. Consider for now the single range gate orresponding to the time delay t. Then after a suitable alignment to a ommon loal time (or range) referene, (8) may be rewritten in a vetor defined as y l 2 NM, and given by y l = t s a( t, t)exp( j2 f d lt p ) (9) s := [s(0),s(1),...,s(n 1)] T 2 N a( t, t) :=[1,e j2 #,e j4 #,...,e j2 (M 1)# ] T 2 M where # := d sin( t )sin( t )/ o is defined as the spatial frequeny. Further it is noted that in (9), the onstant phase terms have been absorbed into t. Considering the L pulses together, i.e. onatenating the desired target s response for the entire CPI in a tall vetor y, is defined as y 2 NML =[y 1 T, y 2 T,...,y T L ] T y = t s a( t, t) v(f d ) (10) v(f d ):=[1,e j2 f dt p,e j4 f dt p,...,e j2 f d(l 1)T p ] T The vetor y onsists of both the spatial and the temporal steering vetors as in lassial STAP, as well as the waveform dependeny, via waveform vetor s. (7) (8)

3 At the onsidered range gate, the measured snapshot vetor onsists of the target returns and the undesired returns, i.e. lutter returns, interferene and noise. The ontaminated snapshot at the onsidered range gate is then given by ȳ =y + y i + y + y n (11) =y + y u where y i, y, y n are the ontributions from the interferene, lutter and noise, respetively, and are assumed to be statistially unorrelated with one another. The ontribution of the undesired returns are treated in detail, starting with the noise as it is the simplest. Noise: The noise is assumed to be zero mean, identially distributed aross the sensors, aross pulses, and in the fast time samples. The orrelation matrix of y n is denoted as R n 2 NML NML. The simplest example is when the noise is independent aross the sensors, the pulses, and the fast time samples, i.e. R n / I, where I is the identity matrix of appropriate dimensions. Interferene: The interferene onsists of jammers and other intentional / un-intentional soures whih may be ground based, air-borne or both. Let us assume that there are K interferene soures. Further, sine nothing is known about the jammers waveform harateristis, the waveform itself is assumed to be a stationary zero mean random proess. Consider the k-th interferene soure in the l-th PRI, and at spatial o-ordinates ( k, k). Its orresponding snapshot ontribution is modeled as, y kl = kl a( k, k),k =1, 2,...,K,l =1, 2,...,L (12) where kl =[ kl (0), kl (1),..., kl (N 1)] T 2 N is the random disrete segment of the jammer waveform, as seen by the radar in the l-th PRI. Staking y kl for a fixed k as a tall vetor, we have y k = k a( k, k) =[y T k1, y T k2,...,y T kl] T 2 NML (13) k :=[ k1 T, k2 T,..., kl T ] T 2 Using the Kroneker mixed produt property, (see for e.g. [9]), the orrelation matrix of y k is expressed as {y k y H k } = R k a( k, k)a( k, k) H where, { k H k } := R k. For K mutually unorrelated P interferers, the orrelation matrix is R i = K {y k yk H }, and is simplified as R i = = KX R k a( k, k)a( k, k) H k=1 k=1 NL KX (I NL a( k, k))r k (I NL a( k, k) H ) k=1 = A(, )R A(, ) H (14) where R := Diag{R 1, R 2,...,R K } 2 A(, ) 2 NML NLMK NMLK NMLK :=[I NL a( 1, 1), I NL a( 2, 2),...,I NL a( K, K)], for I NL the identity matrix of size NL NL, and Diag{,,..., } the matrix diagonal operator whih onverts the matrix arguments into a bigger i diagonal matrix. For example, Diag{A, B, C} = h A B C Clutter: The ground is a major soure of lutter in air-borne radar appliations and is persistent in all range gates upto the gate orresponding to the platform horizon. Other soures of lutter surely exist, suh as buildings, trees, as well as other uninteresting targets. We will ignore the other soures of lutter and treat ground lutter stohastially. Let us assume that there are Q lutter pathes indexed by parameter q. Assume that the q-th lutter path is at ( q, q),q = 1, 2...,Q, with a orresponding o-ordinate vetor denoted as x q. Eah of these lutter pathes are omprised of say P satterers. Assuming that the satterers do not sintillate in the PRI s, the radar return from the p-th satterer in the q-th lutter path is given by pqs a( q, q) v(f q ) where pq is its random omplex refletivity, and f q is the Doppler shift observed from the q-th lutter path. It is impliitly assumed that the satterers in a partiular lutter path have idential Doppler as they are in the same range gate. Furthermore, it is also impliitly assumed that due to the far-field assumptions, the satterers are in the same azimuth resolution ell. In other words the spatial responses of satterers in the same lutter path are idential to one another. The Doppler f q is given by,. f q := 2f oẋ T r (x r x q ). (15) x r x q Sine the lutter path is stationary, the Doppler is purely from the motion of the airraft as seen in (15). The ontribution from the q-th lutter path to the reeived signal is given by y q = PX p=1 with orresponding orrelation matrix pqs a( q, q) v(f q ), (16) R q := B q R pq B q H (17) where, B q =[s a( q, q) v(f q ),...,s a( q, q) v(f q )] 2 NML P and R pq is the orrelation matrix of the random vetor, [ 1q, 2q,..., Pq] T. Assuming that a partiular satterer from one lutter path is unorrelated to any other satterer belonging to any other lutter path, P we have the net ontribution of lutter y = Q y q, with q=1

4 orresponding orrelation matrix given by R = QX R q. (18) q=1 B. Assumptions on the parameters Assume f o = 10 GHz, B =1/T = 50 MHz, M = 10, t = 60 o, t = 40 o, that the radar platform has a veloity vetor given by ẋ r = [100, 0, 0] T m/s, likewise the target s veloity vetor is ẋ t = [60, 0, 0] T miles per hour. Then the propagation time aross the array is 4.5e-10 assuming the inter element spaing is o/2, whih is learly muh less than the inverse of the bandwidth. Hene the narrowband assumption i.e. A1 is satisfied. Using these values of the radar parameters, we obtain the target Doppler, f d =2.713 khz. Substituting these values, we find that A2 is also satisfied for p =2, 3,... Next, we find that exp(j2 f d T )= j, learly then for t apple 1/B, assumption A3 is also satisfied. III. WAVEFORM DESIGN AND WAVEFORM SCHEDULING The radar return at the onsidered range gate is proessed by a filter haraterized by a weight vetor, w, whose output is given by w H ȳ. The objetive of STAP is to obtain the desired w suh that the power from the undesired response is minimized, while leaving the target response as is. Sine the waveform s prominently figures in the steering vetor, say for example in (10), our objetive is to both design the waveform as well as obtain the desired weight vetor, w. Mathematially, we may formulate this problem as: min w,s { w H y u 2 } (19) s. t w H (s a( t, t) v(f d )) = 1 Solving (19) jointly over the optimization variables proves diffiult. However, the method of onentration as applied to maximum likelihood problems, proves useful. In other words, solving the minimization problem w.r.t to w by initially treating s as a onstant, the solution to (19) is well known, and expressed as Ru 1 (s a( t, t) v(f d )) w o = (s a( t, t) v(f d )) H Ru 1 (s a( t, t) v(f d )) (20) where R u = R i + R + R n. We further emphasize that the the weight vetor is an expliit funtion of the waveform. Now substituting w o bak into the ost funtion in (19), the minimization is purely w.r.t s, with the onstraint already being satisfied 8s. In other words, the new minimization problem is unonstrained, and ast as, min (s a( t, t) v(f d )) H Ru 1 (s a( t, t) v(f d )) (21) s 1 In the presene of lutter, whih is assumed here, the orrelation matrix R u is a funtion of s, although not expliitly stated but whih an be seen from say (16) and (17). In the absene of lutter but presene of noise and interferene, this is not true. Solving (21) while enforing the dependeny of R u on s is intratable. Rather, a suboptimal solution ignores the impliit dependeny of R u on s is advoated. Then, the solution to (21) an be formulated as Rayleigh-Ritz optimization [9], resulting in the solution: s a( t, t) v(f d )=µ min (R u ) (22) where µ min (R u ) is the eigenvetor orresponding to the minimum eigenvalue of R u. This tensor equation impliitly defines the optimal s; whether this equation may be met with equality depends on the dimensions and values of a( t, t), v(f d ), and R u. In general the system is overdetermined and we solve this equation approximately via least squares (LS) [10], as desribed next. A. Waveform design solution Define := a( t, t) v(f d )=[ 1,..., ML] T, likewise define s i := s(i) to simplify notation. Then, from (22), the following NML equations are obtained: s i j = µ h,i=1, 2,...,N,j =1, 2,...,ML (23) h =(i 1)ML+ j where µ h is the h-th element of vetor µ min (R u ). The system of equations in (23) may be written as a linear matrix equation, µ min (R u )=Hs H := NML N..... (24) where 0 is a olumn vetor of dimension N, onsisting of all zeros. A LS solution is employed to solve (24), with the orresponding ost funtion and solution readily given by min s µ min (R u ) Hs 2 (25) ŝ =(H H H) 1 H H µ min (R u ) (26) where ŝ is the LS estimate of s. After some straightforward matrix algebra, the solution to (25) is simplified further, i.e. ŝ i = H µ h H,i=1, 2,...N (27) µ h =[µ (i 1)ML+1,µ (i 1)ML+2,...,µ iml ] T 2 ML It is readily seen that ŝ i are solutions to the individual LS optimization osts, min µ h s i 2. In other words, the LS s i ost in (25) deouples into N separable LS osts. It is noted that the waveform solutions are unonstrained, the solutions will hange when we put additional onstraints, for example, onstant modulus, whih is not the fous of this paper. In pratie it is noted that the matrix R u is unknown and must be estimated from the STAP data ube shown in Fig. 2. Typially several range ells are used to estimate the

5 z Range gates Air-borne array v r x r t Pulse index t y Sensor index Guard gate x x t Target Guard gate Gate under onsideration with N fast time samples Fig. 1. Radar sene onsidering the ground based target at azimuth ( t), elevation ( t). The (x, y, z) axis are loal to the airraft arrying the array. Fig. 2. STAP data ube before mathed filtering or range ompression, depiting the onsidered range gate/ell and fast time slies (dashed lines). undesired orrelation matrix whih are not in the immediate viinity of the range ell under onsideration. This is done to prevent self-nulling of the hypothesized target responses from either the main lobe or via the sidelobe responses. If ȳ r 2 NML,r =1, 2,...,R denotes the radar returns from R range gates onsisting of only undesired returns (target free), then the following sample matrix estimate of R u is used: ˆR u = RX ȳ r ȳ r H /R (28) r=1 Therefore to ensure invertibility in (21), R NML is needed. The effet of using (28) will have an impat on the designed as well as sheduled waveforms, and is addressed in the simulations setion. Waveform sheduling: When waveform sheduling rather than design is desired, then (21) may be used diretly by minimizing over the waveform library given by the set S = {s 1, s 2,...,s U }. For example, if a target of interest is being traked, then sheduling is envisioned by using the previously obtained estimate of R u from the prior CPI to shedule for the future CPI s. Typially CPI s are in the order of milliseonds (or lower). Hene it may be reasonable to assume that the orrelation matrix of the undesired radar returns remain approximately stationary for a few ontiguous CPIs. It is further noted that waveform design may aid in waveform sheduling, i.e. the waveform library ould be made dynami by inorporating some of the previously designed waveforms into the waveform library, on-the-fly. IV. SIMULATIONS In pratie, the designed and sheduled waveforms will depend on the orrelation matries of noise, interferene and lutter whih is unknown. The major fous of this paper is to investigate the impat of using the estimated orrelation matrix from training data to examine the performane loss, and is addressed via a numerial simulation. The noise orrelation matrix was assumed to be a saled identity matrix assuming an SNR of 20dB. The arrier frequeny was hosen to be 10GHz, and the radar bandwidth was 50MHz. To redue omputation omplexity in inverting large matries and their eigen-deompositions,we onsidered M = 5,L = 32,N = 5. The element spaing i.e. d = o /2. Two interferene soures were onsidered at ( = /3, =5 /2) and at (2 /3, 5 /2). Both these interferene soures had idential disrete orrelation funtions given by 0.8 n,n=0, 1, 2,..., in other words omprising the appropriate elements in matries R 1 and R 2. The interferene orrelation was onstruted using (14). To simulate lutter we onsidered two lutter pathes, onsisting of four satters eah. To keep the analysis simple, we assumed that the lutter satters are unorrelated in their respetive pathes as well as aross them. In other words, R pq = I8(p, q). The two lutter pathes were assumed to be at angle o-ordinates given by ( = /4, = /4) and (2 /5, /4), respetively. The veloity ẋ r = [100, 0, 0] T. The lutter Doppler an now be omputed from say (15), and the orresponding lutter orrelation matrix may be omputed from (18). The loss of performane an now be omputed and is defined as the ratio of the variane of the Capon using the true orrelation matrix to the ratio of the variane of the apon using the estimated orrelation matrix, see also [2]. To estimate the orrelation, we sampled a multivariate Gaussian distribution using the true parameters, namely zero mean and orrelation given by R u. Then the estimate used in (28) was used. The results are shown in 3. It is seen that to be lose to the 3dB tolerane, we must have R 2NML, to get in the proximity of 1dB to the optimal performane we need 3NML apple R apple 4NML, whih may be prohibitive in ertain airborne appliations. V. CONCLUSIONS AND FUTURE DIRECTIONS Waveform design and waveform sheduling were addressed for spae time adaptive proessing (STAP) in an airborne radar. An linear array of radar sensors was assumed, interrogating

6 Loss of performane (db) [5] A. Farina, A. Saverione, and L. Timmoneri, The MVDR vetorial lattie applied to spae-time proessing for AEW radar with large instantaneous bandwidth, IEE Pro. Radar, Sonar and Navigation (Pt. F), vol. 143, no. 1, pp , Feb [6] D. Madurasinghe and A. P. Shaw, Mainlobe jammer nulling via tsi finders: a spae fast-time adaptive proessor, EURASIP J. Appl. Signal Proess., vol. 2006, pp , Jan [7] Y. Seliktar, D. B. Williams, and E. J. Holder, A spae/fasttime adaptive monopulse tehnique, EURASIP J. Appl. Signal Proess., vol. 2006, pp , Jan [Online]. Available: [8] J. Capon, High-resolution frequeny-wavenumber spetrum analysis, Proeedings of the IEEE, vol. 57, no. 8, pp , Jun [9] R. Horn and C. Johnson, Topis in Matrix Analysis. Cambridge University Press, [10] G. Strang, Linear Algebra and Its Appliations. Thomson, Brooks/Cole, NML (samples) Fig. 3. Loss of performane in dbs vs samples ground based targets. The waveform design and waveform sheduling problems were formulated with a ost funtion similar to the MVDR ost funtion as in lassial radar STAP. It is shown analytially derived that both the designed waveform and the sheduled waveforms will depend on the spatial and Doppler responses of the desired target. A numerial result was shown that demonstrates that when the ovariane matrix of the undesired responses are estimated, the loss of performane is inversely proportional to the number of samples used in estimation of the ovariane matrix. The analysis in this paper thus far ignored the signal dependeny of the lutter orrelation matrix, resulting in the well known Rayleigh-Ritz optimization problem leading to the eigenvetor solution. Future diretions along this line of researh may inlude this signal dependeny of lutter. Further, possible investigative diretions may also inlude adding additional radar waveform speifi onstraints suh as peak sidelobe levels, onstant modulus, Doppler tolerane levels et.. Nonetheless, it remains to be seen if suh solutions result in minimizing the MVDR variane to appreiably lower levels than the suboptimal eigenvetor solution. ACKNOWLEDGEMENT This work was sponsored by US AFOSR under award FA ; no offiial endorsement must be inferred. REFERENCES [1] R. Klemm, Priniples of Spae-Time Adaptive Proessing. Institution of Eletrial Engineers, [2] J. Ward, Spae-time Adaptive Proessing for Airborne Radar, ser. Tehnial report (Linoln Laboratory). Massahusetts Institute of Tehnology, Linoln Laboratory, [3] J. Gueri, Spae-Time Adaptive Proessing for Radar. Arteh House, [4] L. E. Brennan and L. S. Reed, Theory of Adaptive Radar, IEEE Transations on Aerospae and Eletroni Systems, vol. AES-9, no. 2, pp , Mar