Knowledge-aided Array Calibration for Registration-based Range-dependence Compensation in Airborne STAP Radar with Conformal Antenna Arrays Philippe Ries #1 , Marc Lesturgie ∗2 , Fabian D. Lapierre ◦3 , Jacques G. Verly #4 #
University of Li`ege, Department of Electrical Engineering and Computer Science Sart-Tilman, Building B28, B-4000 Li`ege, Belgium 1
∗
[email protected],
4
[email protected]
ONERA, Chemin de la Huni`ere et des Joncherettes, 91120 Palaiseau, France 2
◦
[email protected]
Royal Military Academy, Department of Electrical Engineering Avenue de la Renaissance 30, B-1000 Brussels, Belgium 3
[email protected]
Abstract— We consider space-time adaptive processing (STAP) when the radar returns are recorded by a conformal antenna array (CAA). The statistics of the secondary data snapshots used to estimate the optimum weight vector are not identically distributed with respect to range, thus preventing the customary STAP processor from achieving its optimum performance. The compensation of the range-dependence of the secondary data requires the precise knowledge of the space-time steering vector. We propose a new knowledge-aided method based on the eigenstructure of the space-time covariance matrix for calibrating the gain and phase of each sensor in the CAA. Based on the calibrated space-time steering vectors, we can perform an accurate range-dependence compensation to obtain a valid estimate of the covariance matrix. End-to-end performance analysis in terms of signal to inference-plus-noise ratio loss shows that the method yields promizing performance.
I. I NTRODUCTION Space-time adaptive processing (STAP) is the technique of choice for detecting slow-moving targets against a strong interference background [6], [13], [3]. The data recorded by STAP radars can be regarded as a sequence, in range, of N M x1 vectors, called snapshots, where N is the number of sensors of the antenna array and M is the number of coherent pulses. For each range, the optimum processor computes the optimum weighted linear combination of the snapshot elements to determine whether a target is present or not at that range. In practice, the interference-plus-noise (I+N) covariance matrix (CM) needed for the computation of the optimum filter is not known and must be estimated from the snapshots at neighboring ranges; these snapshots constitute the secondary data. Unfortunately, for conformal antenna arrays (CAAs), the secondary data snapshots are typically not identically dis-
tributed with respect to range. It is thus not acceptable to estimate the I+N CM by directly applying to the secondary data the estimator proposed in [11]. Indeed, the estimator proposed in [11] is maximum likelihood only if the secondary data snapshots are independent and identically distributed. The range-dependence compensation method for CAAs we proposed in [1], [9] allows one to obtain an I+N CM estimate not suffering from range-dependence. However, this method assumes that the space-time steering vector is known exactly. The present paper proposes a new method based on the eigenstructure of the space-time CM for calibrating the gain and phase of each sensor in the CAA of interest. Based on the calibrated space-time steering vector, one can then perform a range-dependence compensation of the snapshots to obtain an accurate estimate of the clutter CM tuned to the actual clutter locus. The method presented in [2] gives good performance for estimating the sensors’ gains and phases, but it is limited to the case where (a) the number of incoming signals is smaller than the number of sensors and (b) the signal environment is stationary. On the contrary, our new method does not require any such assumptions. It operates in the presence of any given number of incoming signals and when the data recorded by the sensors is not identically distributed. Section II presents the radar measurement geometry. Section III introduces the snapshot model used in this paper. Section IV discusses the range-dependence phenomenon for CAAs. Section V describes our calibration algorithm. Section VI indicates how the registration-based range-dependence compensation (RBC) method of [9] is extended to work with an estimated space-time steering vector. Section VII gives results. Section VIII concludes.
z
Hersey et al. suggest to model a clutter snapshot as the coherent sum of the returns of Nc statistically independent clutter patches the scattering centers of which are located on the isorange under consideration. The clutter return for range k is thus written as
y R Rs
φ
H
yc,k =
θ
i=0
δ vR
x S
Fig. 1.
Monostatic measurement configuration: radar R, scatterer S.
II. R ADAR M EASUREMENT G EOMETRY Figure 1 shows a canonical monostatic (MS) measurement configuration, with a radar R and a scatterer S (target or clutter patch). The origin of the coordinate system (x, y, z) is chosen to coincide with R. The x-axis makes a crab angle δ with the platform velocity vector v R and the z-axis points vertically up. The range Rs is the distance between R and S. III. CAA
SIGNAL MODELLING
In this paper, we use a modified version of the signal model presented by Hersey et al. [4]. A single scatterer S characterized by a direction of arrival (DoA) given by (θ, φ) and by a normalized Doppler frequency νd [6] gives rise to the M N x1 space-time snapshot y = β st (νd ) ⊗ (γ ss (ν s )), where is the element-wise product and ⊗ is the Kronecker product. β is the complex-valued random amplitude return. γ = [α0 ejψ0 . . . αN −1 ejψN −1 ]T is called the sensors’ gain and phase vector and models the gains αi and phases ψi , i = 0, . . . , N − 1, of the individual sensors, with T denoting transpose. st (νd ) = [1 ej2πνd · · · ej2πνd (M −1) ]T is the temporal steering vector. T T ss (ν s ) = [ej2πν s p(0)/(λc /2) · · · ej2πν s p(N −1)/(λc /2) ]T is the spatial steering vector when all sensors have unity gain and zero phase, where ν s is the normalized spatial frequency vector, which is linked to the DoA by νs
NX c −1
= 0.5(cos θ cos(φ − δ), cos θ sin(φ − δ), sin θ)T ,
p(i) is the position vector of the ith sensor, and λc is the wavelength at the carrier frequency. It is important to note that γ ss (ν s ) is the true steering vector and that ss (ν s ) is the model of the steering vector we use when we assume a unit gain and a zero phase for all sensors. We use two simplifying assumptions: all sensors are considered independent of each other and isotropic, and range ambiguities are ignored.
βk,i st (νd,i ) ⊗ γ ss (ν s,i ) ,
(1)
where i is the index of the ith clutter patch and βk,i is a random process indexed by i (for each k) and characterized ∗ 0 by E{βk,i } = 0 and E{βk,i βk,i 0 } = ηk,i δ(i − i ). The total snapshot is thus equal to y k = yc,k + n, where n is a zero-mean Gaussian random vector that models the receiver noise. This noise is assumed to be spatially and temporally white and not correlated with the βk,i ’s. The space-time CM Rk = {yk y†k } can thus be written as Rk = Rc,k +Rn , where Rc,k = E{yc,k y†c,k } is the clutter CM and Rn = σn2 I is the noise CM. IV. R ANGE
DEPENDENCE FOR
CAA S
To illustrate the range-dependence phenomenon for CAAs, we adopt the spectral representation for space-time signals recorded by CAAs that was introduced in [9]. The clutter patches contributing to the ground clutter in a snapshot at a given range Rs are located on the corresponding isorange. The signal reflected by each clutter patch along that isorange is characterized by a particular pair (ν s , νd ). Hence, the signal from each clutter patch corresponds to a particular point in the 4D clutter power spectrum (PS) or, equivalently, to a particular point in the 4D spatio-temporal frequency domain (ν s , νd ) [9]. Each snapshot is the result of the contributions of all the clutter patches along the corresponding isorange. Hence, the support of the clutter PS in the 4D spatio-temporal frequency domain (ν s , νd ) is represented by a continuous curve. For a range Rs , this 4D curve is called the 4D clutter PS locus [9]. This curve is thus a valuable tool for investigating the behavior of the clutter signature with range. This is shown in Fig. 2 for a MS flight configuration. We clearly see that the clutter snapshots are range dependent, which gives rise to severelybiased estimates of the clutter CM when one simply uses an average of sample CMs. The problem addressed here is to estimate, from the secondary data, the true spatial steering vector, i.e. the gain and phase vector γ, in order to apply the RBC algorithm of [9] when the true space-time steering vector γ ss (ν s ) cannot be assumed to be completely known. V. C ALIBRATION
OF SENSOR GAIN AND PHASE
The proposed method is based on the eigendecomposition of the space-time CM. The following property of the space-time CM Rk forms the basis for the eigenstructure-based approach. Property: Let the λk,i ’s and uk,i ’s, i = 0, . . . , M N − 1, be the eigenvalues (ordered by decreasing magnitude, i.e. λk,i ≥ λk,i+1 ) and the corresponding eigenvectors of Rk . It is well known that the clutter CM Rc,k is rank deficient
subsnapshot y J (j) is generated by using J consecutive pulses beginning with the jth pulse. Thus, the expression for the jth subsnapshot is
Rs
νd
νsz
Rs
yJk (j) =
NX c −1
βk,i ej2πνd,i j sJi + nJ (j),
(3)
i=0
νsy
νsx
νsy
νsx
Fig. 2. Evolution of the 4D clutter PS locus for increasing range in a monostatic configuration. The first graph is the projection of the 4D clutter PS locus in the (νsx , νsy , νd ) subspace; the second graph is the projection in the (νsx , νsy , νsz ) subspace.
where sJi = [1 · · · ej2πνd,i (J−1) ]T ⊗ ss (ν s,i ) and nJ (j) is the corresponding noise subvector. STEP 2: Averaging of outer products of subsnapshots: ˆ of R that does not suffer from range To get an estimate R k k dependence, we propose to average the outer products of the yJk (j)’s, which are all taken from the same range cell k, i.e. we form the sample CM [7] 0
[13], [6]. Hence, if qk designates the rank of Rc,k , the λk,i ’s obey the relation λk,0 ≥ λk,1 ≥ · · · ≥ λk,qk ≥ λk,qk +1 = · · · = λk,N M −1 = σn2 . The space-time steering vectors corresponding to the spatial and temporal frequencies of the scatterers at range k are orthogonal to the columns of the matrix U = [uk,qk +1 , . . . , uk,N M −1 ], the columns of which span the noise subspace [7], [12]. This property suggests that one should first estimate Rk and then obtain an estimate of γ by minimizing C=
L−1 X
ˆ kU
l=0
†
st (νd,l ) ⊗ γ ss (ν s,l ) k2 ,
(2)
ˆ is an estimate of U. In (2), the ν ’s and νd,l ’s where U s,l corresponding to the L scatterers in range cell k under consideration are obtained by using the priori knowledge that consists of (a) the terrain map (assumed flat for simplicity) and (b) the MS flight configuration parameters. To follow the trend in the literature, we refer to our calibration method as knowledge-aided array calibration [8]. However, the range dependence of the secondary data ˆ by simply makes it impossible to obtain a valid estimate R k averaging the sample CMs of different range cells (cf. Sect. IV). This is why the proposed method consists in two steps: (A) We use temporal smoothing [10] to obtain a CM estimate ˆ that does not suffer from range dependence, and (B) we R k use this CM estimate to estimate the gain and phase vector γ. Below we give a description of our method. ˆ A. Estimation of R k Our method is inspired from [2]. It constitutes an extension of [2] to the space-time case. This extension allows one to consider any number of incoming signals for calibration but requires one to deal with a non-stationary signal environment. Our method operates on a single snapshot y k of M 0 pulses to estimate Rk . Note that the number of pulses M 0 used to estimate γ is typically much larger than the number of pulses M used for the actual STAP processing. STEP 1: Generation of subsnapshots: The first step is to divide the N M 0 x1 vector yk into subsnapshots. The jth
ˆ = R k
M −J X 1 yJ (j)(yJk (j))† . M 0 − J + 1 j=0 k
ˆ and use We then perform the eigendecomposition of R k ˆ into a estimates of the λk,i ’s to split the eigenstructure of R k clutter subspace and the corresponding orthogonal subspace. ˆ of U and corresponds to The latter will be our estimate U the S (15 in our case) smallest eigenvalues. Due to the rank deficiency of the clutter part of the CM [13], it will always be possible to divide the signal space in a signal part and an orthogonal noise part. B. Estimation of γ ˆ we can start minimizing C. Now that we have obtained U, Noting that the vector st (νd,l ) ⊗ (γ ss (ν s,l )) in (2) can be written as (1J ⊗ γ) sl with sl = st (νd,l ) ⊗ ss (ν s,l ), where 1J is a column vector of length J and of all ones, (2) can be written as C=
L−1 X
†
ˆU ˆ ((1 ⊗ γ) s). ((1J ⊗ γ) sl )† U J
l=0
Noting further that (1J ⊗ γ) sl = ˜sl (1J ⊗ γ), where ˜sl is a diagonal matrix given by ˜sl = diag(sl ), we finally obtain ! L−1 X † ˆ ˆ† † ˜s U U ˜s (1 ⊗ γ). C = (1 ⊗ γ) (4) J
l
l
J
l=0
This minimization problem can be cast into the form of the following well known constrained optimization problem [12]: Minimize t† Q t subject to G† t = c.
P ˆU ˆ † ˜s , G must In our case, t = (1J ⊗ γ), Q = L−1 s†l U l=0 ˜ l ensure that t takes the structure of a Kronecker product, and c = [1 0 . . . 0]T . For example, if J = 3, G may be given by † [1 0 . . . 0] [0 . . . 0] [0 . . . 0] IN −IN 0N , G= 0N −IN IN where IN is the N xN identity matrix, and 0N is the N xN matrix of all zeros. The solution γˆ to our minimization
VI. E STIMATED SENSOR
GAIN / PHASE
REGISTRATION - BASED RANGE - DEPENDENCE COMPENSATION (SGP-RBC)
The registration-based range-dependence (RBC) compensation method introduced in [1] and extended to CAAs in [9] can now be applied using the estimated sensors’ gain and phase vector. The RBC method relies on the registration of the clutter PS locus at the different ranges and consists of the three following steps: (A) An analysis step, where the PS of the snapshot at each range is independently computed along the clutter PS locus at the corresponding range. (B) A registration step, where the PS at different ranges are averaged along socalled flow lines. (C) A synthesis step, where the CM at the range of interest is synthesized from the PS along the clutter PS locus at the range of interest. The analysis of the clutter PS along the clutter PS locus requires precise knowledge of the steering vector to obtain an accurate estimate of the clutter energy [12]. The synthesis step of the clutter part of the CM estimate also requires precisely known steering vectors for the clutter notch to be correctly placed in the space-time filter using this estimate. Finally, knowledge of the steering vector is needed in the expression ˆ −1 (1 ⊗ γˆ ) s. ˆk = R of the STAP processor, i.e. w M k VII. R ESULTS The end-to-end performance of our algorithm in terms of signal to inference-plus-noise ratio (SINR) loss has been tested on a N = 12 element circular antenna array using (M 0 , J) = (90, 3) to estimate γ and operating with M = 12 pulses for the actual STAP processing. We used a CNR of 30 dB. The curves of Fig. 3 are the result of 200 Monte-Carlo simulations. The continuous curves give the mean results, and the dotted curves are one standard deviation away from the corresponding continuous curves. We compare the performance of the optimum processor (OP) [6], the sample matrix inversion (SMI) algorithm [11], our RBC algorithm from [9], and our new SGP-RBC algorithm. We see that the performance curve of SGP-RBC is better than that of SMI, and RBC, and is close to that of the OP. We thus conclude that our algorithm yields promizing performance. VIII. C ONCLUSION We presented a new technique for estimating the unknown sensors’ gains and phases of an arbitrary antenna array in the presence of an arbitrary number of incoming signals and of a non-stationary signal environment. The knowledge of the sensors’ gain and phase vector allows us to accurately estimate the clutter CM in range-dependent configurations. Our method can handle arrays of arbitrary shape and thus conformal antenna arrays. End-to-end performance analysis shows that
0
OP −10 SINR Loss (dB)
problem is the first N elements of the optimum weight vector to , which is given by [5] −1 to = Q−1 G G† Q−1 G c.
RBC
SGP−RBC
−20
−30
SMI −40
−50 −0.5
0 ν
0.5
d
Fig. 3. SINR Loss curves for a 12-element circular antenna array operating in a monostatic configuration. We compare the performances of the optimum processor (OP) [6], the sample matrix inversion (SMI) algorithm [11], our RBC algorithm from [9], and our new new SGP-RBC algorithm.
our method provides promizing performance. Future work will look into extending the current method to more realistic antenna models. ACKNOWLEDGEMENTS This work was supported by a fellowship of the FNRS (Fonds National de Recherche Scientifique), Brussels, Belgium.
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