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Computationally-Efficient Range-Dependence Compensation Method for Bistatic Radar STAP Fabian D. Lapierre* and Jacques G. Verly† *Royal Military Academy, Departement of Electrical Engineering, B-1000 Brussels, Belgium †University of Liège, Department of Electrical Engineering and Computer Science, B-4000 Liège, Belgium
[email protected],
[email protected]
Abstract— We address the problem of detecting slow-moving targets using space-time adaptive processing (STAP). The construction of the optimum weights at each range implies the estimation of the interference-plus-noise covariance matrix. This is typically done by straight averaging of snapshots at neighboring ranges. However, in most bistatic configurations, snapshot statistics are range dependent. Straight averaging results thus in poor performance. In an earlier paper, we proposed a range-dependence compensation method that provides optimum performance. However, this implies a large computational cost. In this paper, we adapt the method to provide optimum performance at low computational cost. Index Terms— Radar, space-time adaptive processing, bistatic
I. I NTRODUCTION We consider a space-time adaptive processing (STAP) radar using an array of N antenna elements and a coherent train of M pulses and operating in a bistatic (BS) configuration, i.e., with physically-separated transmitter and receiver. BS radar systems are currently receiving increasing attention [2], [3]. The data collected by a STAP radar can be viewed as a sequence of M N × 1 vectors, called “snapshots.” Implementing the optimum STAP processor generally involves inverting the covariance matrix (CM) of the snapshots. This matrix is typically estimated by averaging the CMs of single-snapshot at neighboring ranges. However, in virtually all BS configurations, the snapshots’ statistics change with range. This results in a loss of performance. The variation of the snapshots’ statistics with range is referred to as the “range-dependence (RD) problem.” One of the most visible manifestations of the RD problem is the deformation with range of the 2D clutter power spectrum density (PSD). The support of this clutter ridge is the “direction-Doppler (DD)” curve. In [4], we proposed a registration-based RD compensation method that computes the CM at a reference range gate l by averaging properly-transformed single-snapshot CMs at a series of neighboring range gates k. This transformation, guided by analytical formulas describing DD curves, aims at bringing the clutter ridge at k into registration with that at l. We distinguished between “true-parameters (TP)” methods, which assume exact knowledge of the configuration parameters, and “estimated-parameters (EP)” methods, which estimate the parameters from the data. The methods in each class rely on a common “registration-based compensation (RBC)”
Fig. 1. BS radar configuration. (a) Transmitter (T ) - receiver (R) scatterer (S) geometry and related parameters. (b) Receiver antenna (A) and related angles.
module. The EP methods rely on an additional “configurationparameters estimation (CPE)” module. For further information on the CPE module, the reader is referred to [5], [4], [6], where two different realizations of the CPE module are described. The realization of the RBC module proposed in [4], [7] was shown to provide optimum detection performance. However, achieving such performance implied a high computational load. In this paper, we propose a new realization of the RBC module that provides optimum detection performance with a low computational load compared to that of the method proposed in [4], [7]. II. B ISTATIC R ADAR - MEASURMENT CONFIGURATION The BS configuration geometry is shown in Fig. 1. The transmitter T is at the center of an (x, y, z) coordinate system. The x-axis points in the same direction as the velocity vector v T of T . The z-axis points vertically up. The receiver R is located at (xR , yR , zR ). Its velocity vector v R is assumed to be horizontal and to make an angle αR with respect to (wrt) v T . The horizontal linear antenna A is located at R and makes an angle δ wrt v R . The BS range Rb to some scatterer S is the distance from T to S to R. The angular position of S wrt to s is denoted by ξs . The ground is assumed to be a horizontal plane at z = −H. All scatterers corresponding to ground clutter are thus located in this plane. The magnitudes of v T and v R are denoted by vT and vR , respectively. Any BS configuration is fully characterized by the vector of parameters θ = (xR , yR , zR , αR , δ, vT , vR , H).
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Fig. 3. Example of 2D flow lines obtained by projecting 3D flow lines in the (νs , νd )-plane for the configuration of Fig. 2(d).
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Fig. 2. Example DD curves for four BS configurations and four ranges (170, 210, 250, and 400 km). Parameters specific to each configuration are listed at the top of the graphs. Units of xR , yR , and zR are km; units of αR and δ are degrees. Common parameters are H = −50 km and vR = vT = 90 m/s.
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III. D IRECTION -D OPPLER CURVES AND SURFACES The three important physical parameters associated with each scatterer S are Rb , ξs , and the relative velocity vr . The related parameters that are more directly measured from the radar returns are the roundtrip delay τrt , the spatial frequency fs , and the Doppler frequency fd . For a stationary S, τrt = Rb /c, fs = cos ξs /λc , and fd = vr /λc , where λc is the carrier wavelength and c is the speed of light. A. 2D “direction-Doppler (DD)” curves All scatterers S characterized by the same R b are located on an isorange surface, which is an ellipsoid of revolution with foci at T and R. The intersection of this surface with the ground is an isorange curve, which is an ellipse (parameterized with the polar angle ψ). For any given configuration and range, all stationary scatterers S at this range map onto a curve showing the relation between fs and fd for any such S. This curve is called a “direction-Doppler (DD)” curve. DD curves are typically represented in terms of the normalized spatial frequency νs = (λc /2)fs and the normalized Doppler frequency νd = (λc /2(vR + vT ))fd . Figure 2 shows that BS DD curves vary significantly with configuration and range. The variation of these curves with range for any particular configuration is one of the most visible manifestations of the RD problem. To derive the equations of BS DD curves, we express ν s and νd in terms of ψ, i.e., νs = g1 (ψ) and νd = g2 (ψ). The derivation of the gi (ψ)’s is lengthy and thus omitted. B. 3D flow lines Using the pair of equations νs = g1 (ψ) and νd = g2 (ψ), we can compute the “flow line” corresponding to each discrete
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Fig. 4. Example DD surfaces generated using (a) DD curves and (b) flow lines, for BS Config. of Fig. 2(d). Rb varies from 152 km to 350 km.
value of ψ. Figure 3 shows example projection of 3D flow lines in the (νs , νd )-plane. As Rb increases, we move along each particular flow line. C. 3D “direction-Doppler (DD)” surface The DD surface is the surface obtained by stacking DD curves for successive values of Rb , as shown in Fig. 4(a) for the configuration of Fig. 2(d). The DD surface can also be generated by “sweeping” the 3D (νs , νd , Rb )-space with the flow lines for all values of ψ, as shown in Fig. 4(b). IV. S NAPSHOTS AND OPTIMUM PROCESSOR In each coherent processing interval, a coherent train of M pulses is transmitted from T . The returns are sensed at each of the N elements of the linear antenna array A at R. Finally, the sensed returns are sampled at a number of discrete ranges (called range gates) covering the range interval of interest. Ranges are indexed with l ∈ L = {0, 1, . . . , L − 1}. We regard the data as a sequence of M × N 2D arrays (snapshots) at successive ranges l. The snapshot corresponding to a specific l or Rb and to a single scatterer S characterized by specific parameters (Rb , νs , νd ) can be written as the M N × 1 vector [8] (2) y(νs , νd ) = βr v(νs , νd ),
where |βr | is found from the radar equation and v(νs , νd ) is the M N × 1 steering vector v(νs , νd ) = b(νd ) ⊗ a(νs ),
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where ⊗ is the Kronecker product, a(νs ) is the N × 1 spatial steering vector, and b(νd ) is the M × 1 temporal steering vector. For uniform linear arrays, we have a(νs ) = b(νd ) =
(1 . . . ej2πνs n . . . ej2πνs (N −1) )T (1 . . . ej2πνd m . . . ej2πνd (M −1) )T .
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The target snapshot y t is given by Eq. (2), where we use the appropriate target parameters (Rbt , νst , νdt ). The clutter snapshot y c is found by integrating y(νs , νd ) over the isorange curve, i.e., Z 2π y c (νs , νd ) = βc (ψ) v(νs (ψ), νd (ψ)) dψ. (6) 0
Since βc (ψ) is a stochastic process, y c is a stochastic vector. We assume that it is wide-sense stationary wrt space and time. It is thus characterized by a constant CM R c = E{y c y †c }. Jammer snapshots are not considered here. The noise snapshot y n is assumed to be spatially and temporally white. The important quantity in STAP is the interference-plus-noise (I+N) snapshot y q = y c + y n , characterized by the I+N CM Rq = E{y q y †q }. The weight vector that maximizes the output signal-tointerference-plus-noise (SINR) is [9] wo (νs , νd ) = αR−1 v(νs , νd ), q
(7)
where α is an arbitrary complex constant. The output of the processor is the complex scalar z = w †o (νs , νd ) y, where y is the received snapshot for a particular range. Its magnitude is compared to a threshold to determine whether a target is absent or present at the triplet (νs , νd , Rb ) of interest. In practice, Rq has to be estimated from the data. A typical estimate is [8] X ˜ (k) = y (k)y † (k), (8) ˜ (k) with R ˆ (l) = 1 R R q q q q q Nl k∈Sl
where Nl is the number of snapshots used for estimation, S l is the set of snapshot indices used for the estimation, and y q (k) ˆ (l) is “maximum likelihood” is the I+N snapshot at range k. R q
only if the y q (k)’s are independent and identically distributed (i.i.d.) wrt range and each have complex Gaussian probability density functions [10]. Unfortunately, in virtually all BS configurations, the y q (k)’s are not i.i.d. wrt range! One of the clearest manifestations of this is the variation with range of the clutter ridge in the clutter PS. If no RD compensation is performed, the estimate of Eq. (8) provides poor performance. The goal of any RD compensation method is to produce the best possible estimate of R q (l) based on the available data. The performance of a processor using a weight w is measured by the SINR loss defined as [8] † 2 w v SINR = , (9) SINRL = † SINR0 (w Rq w)(v † v)
where SINR0 is the SINR in the absence of clutter. Optimum performance is achieved for w = w o . Performance is degraded by losses due to the estimation of the I+N CM and to inappropriate handling of the RD problem. V. P RINCIPLE OF OUR REGISTRATION - BASED RANGE - DEPENDENCE COMPENSATION METHOD Since the RD is most visible in the spectral domain (via the deformation of the clutter ridge), we perform the RD compensation in this domain. The idea is to deform the clutter PSD Py,k (U, V ) at each range k ∈ Sl to bring its clutter ridge into registration with that of the PSD at reference range l. Because of the direct relation between clutter ridges and DD curves, we can also think in terms of DD curves. The PSD a estimate Py,l (U, V ) at l is found by averaging the properly transformed Py,k (U, V )’s at k ∈ Sl . We thus have 1 X 0 a Py,k (U, V ), (10) Py,l (U, V ) = Nl k∈Sl
where
0 P Py,k (U, V ) = Tkl [Py,k (U, V )] ,
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P where Tkl [ . ] is the transformation to be found that will bring the clutter ridge of Py,k (U, V ) into registration with that of Py,l (U, V ). Below, we do not provide an analytical expression P for Tkl [ . ]. Instead, we provide algorithms, described using P block diagrams, that implement Tkl [ . ]. The space-time equivalent of Eqs. (10) and (11), which ˆ a of R is obtained as folrelates the y k ’s to the estimate R l l ˆ a is obtained by applying an “inverse PSD” operation, lows. R l denoted by the PSD-1 { . } operator, to P a (U, V ) of Eq. (10),
i.e.,
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a ˆ a = PSD-1 Py,l R (U, V ) . l
A realization of the PSD-1 { . } operator is proposed in Section VII, This operator is chosen to be linear. The Py,k (U, V )’s are obtained by applying a spectral estimation technique, denoted by PSD{ . }, to the y k ’s, i.e., n o Py,k (U, V ) = PSD y k . (12) Finally, using the linearity of the PSD-1 { . } operator and Eqs. (12) and (10), we have 0 a 1 X ˆ a = PSD-1 Py,l R PSD-1 Py,k (U, V ) (U, V ) = l Nl k∈Sl
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X 0 ˆa = 1 ˆ R R l k Nl
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h n oio n h i P R ˆ 0 = PSD-1 Tkl PSD y R yk , = T kl k k
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R where applying Tkl [ . ] to y k , k ∈ Sl , has exactly the same P final effect as applying Tkl [ . ] to Py,k (U, V ), k ∈ Sl .
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(a) True-parameters (TP) methods (b) Estimated-parameters (EP) methods Fig. 5. Comparison of architectures of (a) true-parameters (TP) methods and of (b) estimated-parameters (EP) methods. Whereas TP methods use the true value of θ, EP methods use an estimate θˆ of θ. TP and EP methods both rely on a common registration-based compensation (RBC) module. EP methods also rely on a generic configuration-parameters estimation (CPE) module. y
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Fig. 6. Realization of the proposed in [7].
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We consider two classes of methods. The “true-parameters (TP)” methods assume that θ is known exactly and the “estimated-parameters (EP)” methods estimate θ from the snapshots. The general architecture of each class of methods is shown in Fig. 5. All methods rely on a common “registrationbased compensation (RBC)” module. This module performs R the transformation Tkl [ . ] of Eq. (14), using either θ or ˆ its estimate θ. EP methods also require a “configurationparameters estimation (CPE)” module to produce θˆ from all the snapshots. Realizations of the RBC module are presented in Sections. VI and VII. Two realizations of the CPE module are proposed in [5], [4], [6]. VI. THE “ PICKET- FENCE ” EFFECT R The realization of the Tkl [ . ] operator in Eq. (14) used in RBC module proposed in [7] is shown in Fig. 6 for one y k . Assume that the input y k is not padded with zeros. Observing the discrete output of the “periodogram” operation, denoted by Py,k (U, V ), is analogous to looking at the true PSD through a “picket-fence,” since we can observe the exact behavior of Py,k (U, V ) only at discrete points (Up , Vq ). The largest values of a particular spectrum could thus lie between two of these discrete frequencies. This is the picket-fence effect. The true frequencies where the major peaks lie are thus not garanteed to correspond to the discrete frequencies for which Py,k (U, V ) is computed.
In Fig. VI, this implies that the (Up , Vq ) peak locations at the output of the “interpolation” operation do not lie exactly on the corresponding DD curve at l. Indeed, these peaks are located on a N ×M grid and cannot thus be all located exactly on the corresponding DD curve. This induces errors in the estimation of Rl . To reduce the picket-fence effect, we can increase the number of discrete frequencies where the periodogram is computed by zero-padding. Even if this solution leads to optimum detection performance [4], it implies a huge computational cost. Below, we propose a more appealing solution. VII. N EW COMPUTATIONALLY- EFFICIENT RBC MODULE Figure 7 shows the block diagram of the new realization of the RBC module. Below, we describe the operations involved in this block diagram. a) Tuned periodogram: At the input of the RBC module we can compute the corresponding DD curve at each range, ˆ We thus know the exact since we know θ or its estimate θ. or approximate locus of the PSD peaks prior to computing the PSD estimate! We can thus compute a 2D “discrete-time” Fourier transform (DTFT) tuned to a number of frequency pairs (UC , VC ) located exactly on the DD curve. We thus have
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N −1 M −1 1 X X Py,k (UC , VC ) = yk (n, m) e−j2π(nUC +mVC ) , NM n=0 m=0 (15)
where (UC , VC ) is a point that lies on the DD curve C corresponding to θ and k. Thus, we compute a periodogram tuned to each (U, V ) point located on C, hence the term “tuned periodogram.” Equation (15) must then be computed for all points distributed along C. In practice, we only compute Eq. (15) at a number JC of (UC , VC ) points at each k ∈ Sl . Recall that any DD curve can be expressed parametrically in terms of ψ. Hence, we discretize the angle ψ into J C angles ψj = ψ0 + j∆ψ, j = 0, . . . , JC − 1, where ∆ψ = 2π/JC . The jth (UC,j , VC,j ) point is thus given by (UC,j , VC,j ) = (νs (ψj ), νd (ψj )).
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C The input of the tuned periodogram at k is thus the set S pl,k of JC DD curve locations (UC,j (k), VC,j (k)) where the PS is to be estimated, i.e., C Spl,k = {(UC,0 (k), VC,0 (k)), . . . , (UC,JC −1 (k), VC,JC −1 (k))} . (17)
The output of the tuned periodogram at k is the C set Spa,k of JC amplitudes of the extracted peak, i.e., Py,k (UC,j (k), VC,j (k)), corresponding to the locations C (UC,j (k), VC,j (k)) in Spl,k , i.e., C Spa,k = {Py,k (UC,0 (k), VC,0 (k)), . . . , Py,k (UC,JC −1 (k), VC,JC −1 (k))}
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The “tuned periodogram” operation at each k ∈ S l takes C C the set Spl,k as input and produces the set Spa,k as output.
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Fig. 7. Generic block diagram of the new computationally-efficient realization of the RBC module. Range gate l 1 corresponds to the index of the smallest range gate in Sl . l− and l+ are the range gates in Sl just below and just above the guard cells, respectively. lNl corresponds to the index of the longest range gate in Sl .
C b) Mapping to l + averaging:: The Nl pairs of sets Spl,k C and Spa,k are the input of the “mapping to l + averaging” operation. The output of this operation consists of the pair of C C sets Spl,l and Spa,l that contain the peak locations and the peak amplitudes at l, respectively. C Consider the jth point (UC,j (k), VC,j (k)) in Spl,k for each k ∈ Sl . These Nl points thus correspond to the same value of ψ, say ψj , and they are thus located on the same flow line. The C jth value in the set Spl,l at l, denoted by (UC,j (l), VC,j (l)), is thus also on the flow line corresponding to ψj , but is located at l instead of at k. C We denote by Py,l (UC,j (l), VC,j (l)) the jth element in Spa,l . This value denotes the amplitude associated with the jth C peak in Spl,l , i.e., (UC,j (l), VC,j (l)). Py,kl (UC,j (l), VC,j (l)) is obtained by averaging the peak amplitudes that are located on the flow line corresponding to ψj and that correspond to the k’s in Sl . Thus, Py,l (UC,j (l), VC,j (l)) is obtained by averaging C the jth element in the sets Spa,k for all k ∈ Sl . We thus have 1 X
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C a C set Spl,l . The value of Py,l (U, V ) at a (U, V ) point in Spl,l , say (UC,j (l), VC,j (l)), is equal to the corresponding value in C the set Spa,l , i.e., Py,k (UC,j (l), VC,j (l)). ˆ a : This operation computes the d) Computation of R l,v estimate of the representation of the CM in the (U, V )-plane, a ˆ , from P a (U, V ), using a relation we have denoted by R y,l l,v derived, a PJC −1 1 a
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s N
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] /N M and ∆C is where γ = e−jπ[ j the distance, along C, between the point (UC,j−1 (l), VC,j−1 (l)) and the point (UC,j (l), VC,j (l)). The demonstration of this expression is lengthy and isa thus omitted. ˆ to R ˆ a : We convert R ˆ a to the e) Conversion from R l,v l l,v estimate of the CM in the space-time domain (or in the (n, m)ˆ a , using the following equation plane), i.e., R l ˆa = V † R ˆa V , R l l,v
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{(UC,0 (l), VC,0 (l)), . . . , (UC,JC −1 (l), VC,JC −1 (l))} {Py,k (UC,0 (l), VC,0 (l)), . . . , Py,k (UC,JC −1 (l), VC,JC −1 (l))} .
a c) Computation of Py,l (U, V ): Here, we compute the a C C estimate Py,l (U, V ) from the two sets Spl,l and Spa,l . In fact, a Py,l (U, V ) is zero except for the (U, V ) points that are in the
VIII. P ERFORMANCE EVALUATION Below, we evaluate the performance of the TP method using the new realization of the RBC module of Section VII. This method is denoted by E-TP with “E” standing for efficient. The TP method using the realization of the RBC module of Section VI is simply denoted by TP.
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Fig. 9. Illustration of the influence of the number JC of (νs , νd ) samples where the “tuned periodogram” is computed for the E-TP method. Each row corresponds to an SINR curve (at a constant νs ) shown in terms of color. Successive rows, from top to bottom, correspond to increasing values of J C . All plots were generated with 16 snapshots, i.e., Nl = 16.
A. End-to-end performance We begin by considering the end-to-end performance of the E-TP and TP methods in terms of SINR-loss curves. These curves are shown in Fig. 8. We see that E-TP has much better performance than that of TP. Indeed, the SINR-loss curves for the E-TP method and the OP are undistinguishable. The E-TP method thus provides near-optimum detection performance. This increase in performance results from the suppression of the influence of the picket-fence effect discussed in Section VI.
registration-based method based on the registration of clutter ridges at each range. In [4], we showed that this method provides optimum detection performance. However, achieving such performance induced a high computational load. In this paper, we adapted the method of [4], [7] to achieve optimum detection performance with a low computational load as compared to that of the method of [4], [7]. The performance of the method was evaluated in detail and found to be excellent.
B. Influence of the number JC of (νs , νd ) points taken along the DD curve
R EFERENCES
A parameter that influences the performance of the E-TP method is the number JC of (νs , νd ) points where the “tuned periodogram” is computed. Recall that these J C points are taken along the DD curve. Below, we study the effect of the value of JC on the performance of the E-TP method. Since an increase in JC implies an increase of the computational load, it is crucial to find the minimum value of J C that provides optimal detection performance! Figure 9 shows the effect of JC on the performance of the E-TP method. We see that the minimum value of JC providing performance that is undistinguishable from that of the OP is approximately 50. C. Influence of the size Nl of the sample support Simulations show that the value of Nl has no influence on the performance of the E-TP method. Optimum detection performance is thus possible with only two snapshots! IX. C ONCLUSION The range-dependence (RD) problem in STAP originates from the dependence of the snapshots’ statistics upon range. Its clearest manifestation is the deformation with range of the power spectrum. The usual maximum-likelihood estimate of the interference-plus-noise covariance matrix thus leads to poor detection performance. Therefore, RD compensation methods have to be developed. In [4], [7], we introduced a
[1] F. Lapierre and J. Verly, “Registration-based method for rangedependence compensation in bistatic radar STAP operating on simulated stochastic snapshots,” Accepted for publication in EURASIP JASP Journal, 2004. [2] R. Klemm, Principles of space-time adaptive processing. IEE Radar, Sonar, Navigation, and Avionics 9, 2002. [3] S. Kogon and M. Zatman, “Bistatic STAP for airborne radar systems,” in Adaptive Sensor Array Processing Workshop, MIT Lincoln Laboratory, Lexington, MA, 13-14 March 2000. [4] F. Lapierre, X. Neyt, and J. Verly, “Performance evaluation of a registration-based range-dependence compensation method for bistatic STAP radar using simulated snapshots,” in IEEE International Radar Conference 2004, Toulouse, France, 19-21 October 2004. [5] F. Lapierre, M. Van Droogenbroeck, and J. Verly, “New solutions to the problem of range dependence in bistatic STAP radars,” in IEEE National Radar Conference, Huntsville, AL, 5-8 May 2003. [6] X. Neyt, F. Lapierre, and J. Verly, “Automatic configuration-parameters estimation and range-dependence compensation using simulated, singlerealization, random snapshots in STAP,” in Adaptive Sensor Array Processing Workshop, MIT Lincoln Laboratory, Lexington, MA, 16-18 March 2004. [7] F. Lapierre and J. Verly, “Registration-based solutions to the rangedependence problem in STAP radars,” in Adaptive Sensor Array Processing Workshop, MIT Lincoln Laboratory, Lexington, MA, 11-13 March 2003. [8] J. Ward, “Space-time adaptive processing for airborne radar,” MIT Lincoln Laboratory, Lexington, MA, Tech. Rep. 1015, 1994. [9] L. Brennan and L. Reed, “Theory of adaptive radar,” IEEE Transactions on Aerospace and Electronic Systems (AES), vol. 9, no. 2, pp. 237–252, 1973. [10] N. Goodman, “Statistical analysis based on a certain multivariate complex gaussian distribution (an introduction),” Annals of Mathematical Statistics, vol. 34, pp. 152–177, 1963.
