Manifold Optimization for Joint Design of
MIMO-STAP Radars
Jie Li
, Student Member, IEEE, Guisheng Liao
, Senior Member, IEEE, Yan Huang
, Member, IEEE,
and Arye Nehorai
, Life Fellow, IEEE
Abstract—In order to maximize the
signal-to-interference-plus-noise ratio (SINR) under a constant-envelope (CE) constraint, a fast and efficient joint design of the transmit waveform and the receive filter for colocated multiple-input multiple-output (MIMO) radars is essential. Conventional joint optimization is performed using nonlinear optimization techniques such as the semidefinite relaxation (SDR) algorithm. In this letter, we propose a novel manifold-based alternating optimization (MAO) method, which reformulates the waveform optimization subproblem as an un-constrained optimization problem on a Riemannian manifold. We present the geometrical structure of the feasible region and de-rive the explicit expressions for the Riemannian gradient and the Riemannian Hessian, thus the reformulated optimization could be solved by using the Riemannian trust-region (RTR) algorithm. Numerical experiments demonstrate that the proposed method has faster convergence with reduced computational cost compared with conventional SDR-based algorithm in Euclidean space.
Index Terms—Multiple-input multiple-output (MIMO) radar,
manifold optimization, waveform design, receive filter design.
I. INTRODUCTION
M
ULTIPLE-INPUT multiple-output (MIMO) radar hasbeen widely studied over the past decades [1]. For both colocated MIMO radars [2] and distributed MIMO radars [3], one of the most concerning problems is how to design probing signals appropriately. Existing approaches can be largely classi-fied into five categories: optimizing an estimation-oriented lower bound [4], optimizing the radar ambiguity function [5], achieving a desired beampattern [6], optimizing the detection performance based on information theory [7], and joint design of transmitter and receiver to maximize the signal-to-interference-plus-noise Manuscript received June 28, 2020; revised August 18, 2020; accepted August 31, 2020. Date of publication October 28, 2020; date of current version Novem-ber 13, 2020. This work was supported in part by the National Natural Science Foundation of China under Grant 61621005, in part by the National Natural Science Foundation of China under Grant 61901112, in part by the Natural Science Foundation of Jiangsu Province under Grant BK20190330, in part by the Advanced Research Foundation under Grant 61404130223, and in part by the Fundamental Research for the Central Universities under Grant 1104000399. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ioannis D. Schizas. (Corresponding authors: Jie Li;
Guisheng Liao; Yan Huang.)
Jie Li and Guisheng Liao are with the National Key Laboratory of Radar Signal Processing, Xidian University, X’ian 710071, China (e-mail: [email protected]; [email protected]).
Yan Huang is with the State Key Lab of Millimeter Waves, Southeast Univer-sity, Nanjing 210096, China (e-mail: [email protected]).
Arye Nehorai is with the Department of Electrical and Systems Engineer-ing, Washington University in St. Louis, MO 63130 USA (e-mail: nehorai @wustl.edu).
Digital Object Identifier 10.1109/LSP.2020.3022239
ratio (SINR) [8]. Moreover, it is known that the space-time adaptive processing (STAP) technology has a strong ability to suppress interference [9]. Compared with conventional single-input multiple-output (SIMO) STAP systems, MIMO-STAP sys-tems have much sharper clutter notches and improved minimum detectable velocity (MDV) performance [10].
In this letter, we focus on the joint design of transmit waveform and receive filter for airborne colocated MIMO-STAP radars to improve the detection performance of the low-velocity moving target. Counting in the constant-envelope (CE) constraint, the joint optimization problem is an intractable non-convex optimiza-tion problem. Most of the existing approaches for solving the interested joint optimization problem use the semidefinite relax-ation (SDR) followed by Gaussian randomizrelax-ation [11]. Whereas, the SDR-based algorithm is not scalable to large-scale antenna systems since the number of variables is quadratically dependent on the number of antennas. In addition, extracting a rank-one component from the optimum solution to the SDR problem is NP-hard in general.
The development of optimizations on matrix manifolds has burgeoned during the past decade [12]. When provided with underlying geometries of the feasible sets, using the geometri-cal structures can deliver high quality solutions to the NP-hard problem with much lower computational cost. Hence, by a fully consideration of the abovementioned difficulties, we propose a novel manifold-based alternating optimization (MAO) algorithm which tackles the optimization problem directly on a Riemannian manifold. We view the feasible sets, which are Hermitian, positive semidefinite (SDP), and fixed-rank matrices with a unit diagonal, as a Riemannian manifold. Then the geometrical structure of the feasible sets is established and the waveform optimization problem is reformulated as an unconstrained problem on this manifold. Next, we strictly derive the explicit expressions for the Riemannian gradient and the Riemannian Hessian. Thus, the Riemannian trust-region (RTR) method [12] can be developed to solve the optimization problem instead of using the algorithms on Euclidean space. As a result, the proposed algorithm achieves better performance and avoids the time-consuming randomization procedure.
Notations: {·}T,{·}∗, and {·}H denote, respectively, trans-pose, conjugate, and conjugate transpose. vec(·) stacks the columns of a matrix to construct a vector. tr(·) represents the trace of a matrix. IN denotes an identity matrix of size N× N. 0N denotes the N -dimensional zero vector. diag(A) gets the
diagonal elements of A to construct a vector. 1 indicates the all-ones vector.|a| denotes the modulus of a. ⊗ and denote, respectively, the Kronecker product and the Hadamard product. Re{x} indicates the element-wise real part of x.
Fig. 1. Illustrations of range-azimuth bins. II. SIGNALMODEL ANDPROBLEMFORMULATION Consider an airborne colocated MIMO radar with Nt trans-mitting antennas and Nr receiving antennas. Both of the transmitting and receiving arrays are equispaced with inter-element spacings dtand dr, respectively. The radar transmits a burst of K pulses in a coherent processing interval (CPI) at a constant pulse repetition frequency (PRF) fr. We assume that the waveform of each transmit antenna is repeated pulse to pulse. LetS ∈ CNt×Ldenote waveform matrix of the k-th pulse with L being the code length. At each receiver, the received signal is down-converted to baseband, and the received signal of a far-field moving target for the k-th pulse is given by
YT,k= αTej2π(k−1)fT 3˜ptb(ψT)aT(ψT)S, (1)
where αTis the target complex amplitude,a(ψT) andb(ψT) de-note, respectively, the transmit and the receive spatial steering vec-tors of the target,λ is the radar wavelength, ψT is the direction of angle (DOA) of the target, satisfied cos (ψT) = cos(φT) sin (θT) with φT and θT being the elevation and the azimuth of the target, and fT is the normalized target Doppler frequency. Stacking all the columns ofYT,k, we have
yT = αT(INrK⊗ST)vT, (2) wherevT =u(fT)⊗b(ψT)⊗a(ψT) denotes the virtual steer-ing vector of the target withu(fT) being the temporal steering vector of the target. Consider that the signal-dependent interfer-ence is the superposition of the echoes from different uncorrelated scatterers [13], as illustrated in Fig. 1, from a number of range rings. Specifically, if the target is at the r-th range ring, the received interference vector associated with the [r− P, r + P ] range cells is modeled by
yI = P p=−P Nc q=1 αI,p,qINrK⊗JTpSTvI,p,q, (3) where Nc is the number of discrete azimuth sectors in the in-terested range ring,vI,p,q=u(fI,p,q)⊗b(ψI,p,q)⊗a(ψI,p,q), and αI,p,q represent, respectively, the virtual steering vector, and the complex amplitude of the (r + p, q)-th interference source. ψI,p,q and fI,p,qdenote, respectively, the DOA and the normalized Doppler frequency of the (r + p, q)-th interference.
Jp∈ RL×L is the shift matrix [11]. Note that the target and interference information can be obtained from a prior knowledge by suing an environment dynamic database [14]. The noise vector
n ∈ CNrKL×1is circularly symmetric complex Gaussian with
zero mean and covariance Rn= σn2INrKL, where σn2 is the power of the noise. Herein, we assume that σ2n= 1. Thus, the
entire received signalsy within a CPI can be recast as
y = yT +yI+n. (4)
In order to maximize the output signal-to-interference-plus-noise ratio (SINR), we formalize the joint optimization of the transmit waveform and the receive filter under a practical con-straint. Assuming that the observationsy is filtered through a linear finite impulse response filterw ∈ CNrKL×1, the SINR can be expressed as
SIN R(w, S) = |αT|2wHSTvTvHTS∗w
wH(Φ + Rn)w , (5)
whereΦ =Pp=−PNq=1c σ2I,p,qS˜TpvI,p,qvHI,p,qS˜∗pis the
inter-ference covariance matrix with σI,p,q2 being the variance of the (r + p, q)-th interference, ˜S = INrK⊗S, and ˜Sp=INrK⊗
SJp.
In practical applications, the synthesized waveform should be constant modulus due to the restriction of the nonlinear radar amplifiers [15]. Thus, the constant-envelope (CE) constraint is incorporated in the joint optimization problem, which can be formulated as max w,R SIN R(w, R) = wHΘw wH(Φ + Rn)w = tr (ΓR) tr (ΣR), s.t. diag(R) = P0 NtL·1, rank(R) = 1, R 0, (6) where Θ = (ΥT ⊗ΛT)⊗ATPTR∗P AHT , (7) Φ = P p=−P Nc q=1
σI,p,q2 (ΥI,p,q⊗ΛI,p,q)
⊗AI,p,qPTR∗P AHI,p,q, (8) Γ = (W∗⊗I Nt)vTvHT(WT ⊗INt), (9) Σ = P p=−P Nc q=1
σI,p,q2 JpW∗(ΥI,p,q⊗ΛI,p,q)WTJ−p ⊗a(ψI,p,q)aH(ψI,p,q). (10)
withΥT =uTuHT,ΥI,p,q=uI,p,quHI,p,q,
AT =aT(ψT)⊗IL,AI,p,q=aT(ψI,p,q)⊗J−p,
ΛT =bTbHT,ΛI,p,q=bI,p,qbHI,p,q,P being a communication
matrix such that vec(S) = P (vec(ST)).R = s∗sTis the wave-form covariance matrix (WCM) withs = vec(S). P0is the total transmit power of Nt transmitters. Note that the CE constraint of waveform S is transformed to the rank constraint of the WCMR. Problem (6) is NP-hard in general [16], which has no close-form solution. Herein, we propose a novel manifold-based alternating optimization (MAO) algorithm which provides high quality solutions to the NP-hard problem (6).
A. Receive Filter Design
In the first alternating step, we deal with the receive filterw design for a fixed R. Specifically, we handle the optimization problem which is equivalently reformulated as
max
w SIN R(w) =
wHΘw
wH(Φ + Rn)w, (11)
The optimization problem in (11) is a generalized Rayleigh quo-tient [17], which has an optimal solution
wopt= (Φ + Rn)−1/2V (Φ + Rn)−1/2Θ (Φ + Rn)−1/2 , (12) whereV[(Φ + Rn)−1/2Θ(Φ + Rn)−1/2] indicates the princi-ple eigenvector associated with the largest eigenvalue of (Φ +
Rn)−1/2Θ(Φ + Rn)−1/2.
B. Manifold-Based Waveform Optimization
In the next alternating step, given the fixedw, the optimization problem (6) associated to the WCM can be equivalently recast as
max R SIN R(R) = tr (ΓR) tr (ΣR), s.t. diag(R) = P0 NtL·1, rank(R) = 1, R 0. (13) Notably, problem (13) is generally a NP-hard problem which can be approximatively solved by semidefinite relaxation (SDR). However, the SDR-based algorithm [11] is disadvantaged by the prohibitive computational burden. Therefore, we propose an efficient manifold-based method to solve problem (13).
The search region is a set of Hermitian, positive semidefinite (SDP), fixed-rank and fixed diagonal matrices, which suggests a quotient geometry for the complex fixed-rank elliptope. AllR feasible for (13) have a unit diagonal, so that the search space of (13) is compact. This restriction is easily enforced by factoring
R = Y YH, yielding an equivalent problem
min
Y ∈M f (Y ) = −
trΓY YH
tr (ΣY YH), (14)
with the search space M = Y ∈ CNtL: diag(Y YH) = P0 NtL·1 , (15) being a smooth and compact manifold. It can be linearized at each pointY ∈ M by a tangent space
TYM = ξY ∈ CNtL: diag(ξYYH+Y ξHY) =0NtL .
(16) Endowing the tangent spaces ofM with the Euclidean metric gY(ξ, η) = ξ, η = Re{ξHη} turns M into a Riemannian sub-manifold of Euclidean spaceCNtL. Herein, the smoothness of both the search space (15) and the cost function (14) make for straightforward second-order optimality conditions. Moreover, the retraction is a mapping fromTYM to M. Herein, we define the retraction as
RetrY(ξY) Y + ξY
Y + ξY . (17)
With the geometric structure ofM defined above, we can develop the Riemannian trust-region (RTR) algorithm to solve (14) in the following subsection.
C. Riemannian Trust-Region Algorithm
In order to minimize the cost function (14) onM, we need to compute the Riemannian gradient and the Riemannian Hessian of f (Y ). The Riemannian gradient gradf(Y ) satisfies
gY (ξY, gradf (Y )) = Df(Y ) [ξY] , ∀ξY ∈ TYM, (18) where Df (Y )[ξY] lim t→0 f (Y + tξY)− f(Y ) t . (19)
Specifically, the Riemannian gradient gradf (Y ) is the orthogo-nal projection of the Euclidean gradient of f (Y ) to the tangent spaceTYM [12], i.e.,
gradf (Y ) = ∇f − Y Re {∇f Y } , (20) where
∇f = −2
ρ2(ρΓY − μΣY ) (21)
is the Euclidean gradient of f (Y ) with respect to Y , with ρ = tr(ΣY YH) and μ = tr(ΓY YH). The Riemannian Hessian of f (Y ) at Y is a similarly restricted version of the Euclidean Hessian ehessf to the tangent space,
Hessf (Y ) [ξY] = ehessf−Y Re {ehessf Y } −ξY Re {ehessf Y } , (22) with ehessf (Y ) = − 2 ρ2 ρΓξY + tr(ΣξYYH)ΓY + tr(ΣY ξHY)ΓY − tr(ΓξYYH)ΣY − tr(ΓY ξH
Y)ΣY − tr(ΓY YH)ΣξY
+ 4
ρ3
ρΓY − tr(ΓY YH)ΣY
×trΣξYYH+ tr(ΣY ξHY) . (23) With the Riemannian gradient and the Riemannian Hessian at hand, we develop the Riemannian trust-region (RTR) algorithm to our problem. The trust-region algorithm solves the problem
min
ξY∈TYM f (Y ) + gY (gradf (Y ), ξY)
+1
2gY (Hessf (Y ) [ξY] ,ξY) ,
s.t. gY (ξY,ξY)≤ δ2, (24)
where δ is the trust-region radius.
TABLE I
JOINTOPTIMIZATION OFTRANSMITWAVEFORM ANDRECEIVEFILTERVIA
MAO ALGORITHM FORMIMO-STAP RADARS
Fig. 2. Output SINR against the number of iterations. The total computing time of the MAO algorithm and the SDR-based algorithm are, respectively, 46.97 s and 314.29 s.
The procedures of the joint optimization of transmit waveform and receive filter via the MAO algorithm are summarizes in Table I.
IV. SIMULATIONRESULTS
In this section, we provide several numerical examples to demonstrate the performance of the MAO algorithm. We used the MATLAB implementation of the RTR method [18] in our sim-ulations. Consider an airborne MIMO-STAP radar with Nt= 4, Nr= 4, and dt= 2λ and dr=λ/2. The carrier frequency and bandwidth of the waveform are f0= 1 GHz and B = 1 MHz. The altitude of the radar platform is 9 km and the platform is moving with a speed VP = 200 m/s. The number of pulses in a CPI is K = 16, and the code length is L = 16. PRF is set to fr= 4VP/λ and B = 1 MHz. The target is at range R = 13 km with φT = arcsin (H/R) and θT = 0◦. The normalized Doppler frequency of the target is fT = 0.3. The number of discrete azimuth sectors of the interested rang ring is Nc= 361 and P = 1. We assume that σI,p,q2 = 1. The total transmit power is P0= 100. The terminating criteria is ε = 10−3. For the RTR method, the upper bound for the trust-region radius ¯δ is chosen as [12]
¯
δ =NrL + KNrL. (25)
The initial trust-region radius is chosen as ¯δ/8.
First, we analyze the convergence of the MAO algorithm compared with the Algorithm 2 in [11], which is an SDR-based alternating algorithm. Fig. 2 shows the output SINRs of the two algorithms versus the number of iterations. Under the same terminating criteria, the MAO algorithm converges at the 22-th
Fig. 3. Spatial-temporal beampatterns of the MAO algorithm, the SDR-based algorithm, the SIMO-STAP, and the orthogonal waveform.
iteration, while the SDR-based algorithm converges at the 35-th iteration. Moreover, the total computing time of the MAO algo-rithm and the SDR-based algoalgo-rithm are, respectively, 46.97 s and 314.29 s. It is evident that the MAO algorithm converges faster than the SDR-based algorithm. In addition, the SINRs associated with the MAO algorithm and the SDR-based algorithm are, respectively, 40.63 dB and 38.79 dB. Thus, the MAO algorithm is computationally more efficient and achieves higher SINR value than the SDR-based algorithm.
In Fig. 3, we present the spatial-temporal beampatterns asso-ciated with the MAO algorithm, the SDR-based algorithm, the SIMO-STAP, and the orthogonal waveform. The corresponding receive filters of the coherent waveform and the orthogonal wave-form is calculated via (14). We can observe from Fig. 3 that all the four beampatterns have deep nulls in the clutter ridge and significant high gain in the target direction. The response in the target direction of the MAO algorithm is the strongest, followed by the SDR-base algorithm, the SIMO-STAP and the orthogonal waveform. The superiority of the MAO algorithm is once again highlighted in Fig. 3.
V. CONCLUSION
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