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 Open Access
Robust constrained waveform design for MIMO radar with uncertain steering vectors
 Xianxiang Yu^{1},
 Guolong Cui^{1}Email author,
 Marco Piezzo^{2},
 Salvatore Iommelli^{3} and
 Lingjiang Kong^{1}
https://doi.org/10.1186/s1363401604379
© The Author(s) 2017
 Received: 1 August 2016
 Accepted: 16 December 2016
 Published: 4 January 2017
Abstract
This paper considers the robust waveform design of multipleinput multipleoutput (MIMO) radar to enhance targets detection in the presence of signaldependent interferences assuming the knowledge of steering vectors is imprecise. Specifically, resorting to semidefinite programming (SDP)related technique, we first maximize the worstcase signaltointerferenceplusnoise ratio (SINR) over uncertain region to optimize waveform covariance matrix forcing a uniform elemental power requirement. Then, based on least square (LS) approach, we devise the waveform accounting for constant modulus and similarity constraints by the obtained waveform covariance matrix using cyclic algorithm (CA). Finally, we assess the effectiveness of the proposed technique through numerical simulations in terms of nonuniform pointlike clutter and uniform clutter.
Keywords
 Robust waveform design
 Multipleinput multipleoutput (MIMO)
 Signaldependent interferences
 Signaltointerferenceplusnoise ratio (SINR)
 Waveform covariance matrix
 Constant modulus and similarity constraints
1 Introduction
Recently, some advances in radar technology including digital arbitrary waveform generators, solid state transmitters, and highspeed and offtheshelf processors [1–3] have been greatly developed, making possible for modern radar systems to adaptively adjust the synthesized transmit waveform to detect environment. This adjustment of transmit waveform can be used to significantly enhance its ability of target detection, identification, and classification. Nevertheless, usually, these techniques suffer from inaccuracies on the knowledge of the actual target/clutter processes, which is necessary to ensure an effective adaptation.
Robust waveform design to resist uncertainty sets of target or clutter parameters (i.e., Doppler frequency, angle) has received considerable attention during the last decades [4–19]. According to the structure of radar systems, these works can be classified into two categories. The first one focuses on the robust design for monostatic radar systems. Specifically, in [4], the design of robust radar code under energy and similarity constraints is investigated to improve the worstcase signaltonoise ratio (SNR) over the possible target Doppler frequencies by using semidefinite programming (SDP)related technique. In [5], the robust approach based on SDP and randomization with respect to the target Doppler is proposed to synthesize the radar waveform accounting for a peaktoaveragepower ratio (PAR) and an energy constraint in order to the improvement of the worstcase SNR. For the detection problem of the extended target with the uncertain set on target impulse response (TIR), the robust transmit waveform accounting for constant modulus constraint and receiving filter have been designed jointly. In [6], considering worstcase SINR as the performance measure and assuming an interval of target Doppler shift are available, the robust design of transmit sequence and receive filter under energy and similarity constraints has been considered by exploiting related SDP relaxation. Considering the same criteria, the uncertain set, and waveform constraints, the related generalized Dinkelbach’s procedure [7] is developed to optimize radar waveform and doppler filter bank. For extended targets considering the uncertainties on the TIR and using the PAR and energy as signal constraints, in [8], a design procedure, based on SDP relaxation and randomization technique, is proposed to devise robust transmit code and receive filter aiming to improve the worstcase SINR. Assuming as figure of merit worstcase SINR, in [9], a maxmin approach against the uncertainly sets of TIR for extended targets and the secondorder statistics of the interference, is exploited to devise the waveform considering energy constraint.
The second category focuses on the robust waveform design for MIMO radar systems. Precisely, in [10], based on the criteria of the mutual information (MI) and minimum meansquare error (MMSE) estimation, the minimax robust waveform design has been addressed by leveraging the a priori knowledge of target power spectral density lying in an uncertainty class of spectrabounded and signalindependent interference statistics. In [11], assuming as figure of merit the cumulated power of probing signal, the worstcase scenario against the uncertain sets of targets locations have been considered to design the waveform covariance matrix forcing a uniform elemental power constraint into each transmitting antenna. In [13], a robust approach against uncertainties on steering vectors is proposed to design the robust transmit beampattern so as to minimize beampattern sidelobes considering power constraint and 3 dB mainbeam width constraint. In [14], using the imperfect clutter prior knowledge, the robust waveform design has been addressed for maximizing the worstcase SINR. The robust joint design problem of the spacetime transmit code (STTC) and the spacetime receive filter (STRF) for a moving pointlike target is considered in [15] assuming as figure of merit the worstcase SINR over the actual and signaldependent clutter statistics and considering both energy and similarity constraints on the sought code. In order to improve the worstcase output SINR over the unknown angle of the target of interest, the robust design problem of transmit waveform satisfying energy constraint and the receive filter is investigated under signaldependent interferences [16]. In particular, we here note that the signaldependent interferences are from the terrain and the objects of no tactical importance within the illuminated area, generated by the reflections of the signal and transmitted by the radar of interest [17]. In other words, this is a kind of selfinduced radar interference, usually referred to as the reverberation phenomenon, owing to the interaction of the transmitted waveform with the scattering environment. These interferences would severely impair the target detectability of radar systems. In [18], based on the worstcase SNR over the uncertain on steering vector, an iterative algorithm is presented to optimize the waveform covariance matrix under a power constraint. In [19], assuming either the peak sidelobe level (PSL) or the integrated sidelobe level (ISL) as figure of merit, the robust waveform covariance matrixes design has been addressed for the purpose of optimizing the worstcase transmit beampattern over steering mismatches accounting for power constraint and 3 dB mainbeam width constraint.
In this paper, we still focus on the robust waveform design of MIMO radar considering practical constraints for enhancing target detectability in the presence of signaldependent interferences. Precisely, in order to improve the worstcase SINR over the uncertain sets of the steering vectors, we first synthesize waveform covariance matrix accounting for a uniform power constraint by resorting to SDPrelated technique. Then, based on least square (LS) approach, we exploit cyclic algorithm (CA) to design waveform under constant modulus and similarity constraints so as to approximate the obtained waveform covariance matrix. Finally, at the analysis stage, we consider two scenarios of nonuniform pointlike clutter and uniform clutter to evaluate the performance of the proposed devise procedure. Results exhibit that the proposed algorithm has the capability of ensuring an improved worstcase performance.
The remainder of the paper is organized as follows. In Section 2, we show the system model. In Section 3, we consider the robust design of waveform covariance matrix. In Section 4, we consider the waveform synthesis under some practical constraints. In Section 5, we evaluate the performance of the proposed procedure. Finally, in Section 6, we provide concluding remarks and possible future research tracks.
1.1 Notation
We adopt the notation of using boldface for vectors a (lower case) and matrices A (upper case). ∥A∥ denotes the twonorm of A. The transpose, the conjugate, and the conjugate transpose operators are denoted by the symbols (·)^{ T }, (·)^{∗}, and (·)^{ † } respectively. tr(·) denotes the trace of square matric. I _{ N } denotes N×Ndimensional identity matrix. \({\mathbb {C}}^{N}\) and \({\mathbb {H}}^{N}\) are respectively the sets of Ndimensional vectors of complex numbers and N×N Hermitian matrices. The curled inequality symbol ≽ (and its strict form ≻) is used to denote generalized matrix inequality: for any \(\boldsymbol {A}\in {\mathbb {H}}^{N}\), A≽0 means that A is a positive semidefinite matrix (A≻0 for positive definiteness).
The vec (A) denotes the column vector obtained by stacking the columns of A. The letter j represents the imaginary unit (i.e., \(j=\sqrt {1}\)). For any complex number x, we use ℜ(x) respectively the real part of x. In addition, x and arg(x) represent respectively the modulus and the argument of x. \(\mathbb {E}[\cdot ]\) denotes statistical expectation.
2 System model

α _{0} is a complex parameter accounting for the target radar cross section (RCS), channel propagation effects, and other terms involved into the radar range equation.

\(\mathbf{A}(\theta)={\boldsymbol {a}^{*}_{r}(\theta)}{{\boldsymbol {a}_{t}}^{\dag }{(\theta)}}\), in which for the azimuth angle θ, a _{ t }(θ) and a _{ r }(θ) denote, respectively, the transmit spatial steering vector and the receive spatial steering vector. In particular, for the uniform linear arrays (ULAs), they are given by$${\boldsymbol{a}_{t}}(\theta)=\frac{1}{\sqrt{N_{T}}}[1,e^{j2\pi\frac{d_{T}}{\lambda}{\sin{\theta}}},\cdots,e^{j2\pi\frac{d_{T}}{\lambda}{(N_{T}1)\sin{\theta}}}]^{T}, $$with d _{ T } and d _{ R }, respectively, the array interelement spacing of the transmitter and the receiver.$${\boldsymbol{a}_{r}}(\theta)=\frac{1}{\sqrt{N_{R}}}[1,e^{j2\pi\frac{d_{R}}{\lambda}{\sin{\theta}}},\cdots,e^{j2\pi\frac{d_{R}}{\lambda}{(N_{R}1)\sin{\theta}}}]^{T} $$

\(\boldsymbol {d}(n)\in \mathbb {C}^{N_{R}}, n=1,2,\cdots,M\), accounts for K signaldependent uncorrelated pointlike interfering scatterers. Specifically, considering the kth interfering source located at θ _{ k }, k=1,2,⋯,K, the received interfering vector d(n) can be expressed as the superposition of the returns from K interference sources, i.e.,$$ \boldsymbol{d}(n)=\sum\limits_{k=1}^{K}\rho_{k}\mathbf{A}(\theta_{k}){\boldsymbol{s}(n)}, $$(2)
with ρ _{ k } being the complex amplitude of the mth interferences.

\(\boldsymbol {v}(n)\in \mathbb {C}^{N_{R}}, n=1,2,\cdots,M\), denotes additive noise, modeled as independent and identically distributed (i.i.d.) complex circular zeromean Gaussian random vector, i.e., \(\boldsymbol {v}(n)\thicksim \mathcal {CN}(0,{\sigma }_{v}^{2}\mathbf {I}_{N_{R}})\).
3 Robust waveform covariance matrix design
In this section, we formalize the problem of the design of robust waveform covariance matrix in order to maximize the worstcase output SINR criterion under specific practical constraints. Finally, we provide the related SDP technique to solve the considered problem.
3.1 Output SINR
Interestingly, inspection of (3) and (4) exhibits that the receiving steer vectors have no effect on the useful power as well as the interfering power. In addition, we also observe that the useful power functionally depends on waveform covariance matrix R so does the clutter power.
with \(\delta _{k}=\mathbb {E}[\\alpha _{k}\^{2}]\), where σ ^{2} denotes the noise power. We notice that the objective function ρ(R) requires the explicit knowledge of the steer vectors a _{ t }(θ _{ k }),k=0,1,⋯,K. However, from a practical point of view, the exact knowledge of a _{ t }(θ _{ k }) can not be available. Hence, in the next subsection, some practical constraints are considered to overcome this drawback.
3.2 Uncertain sets of steering vectors and power constraint
where \(\boldsymbol {B}_{km}\in {\mathbb {H}}^{N_{T}}\), \(\boldsymbol {b}_{km}\in {\mathbb {C}}^{N_{T}}\), and b _{ km }, u _{ km } both are real values for m=1,2,k=0,1,⋯,K. We remark that generalized similarity and conical and norm constraints are the special cases of (6). In particular, the detailed illumination of these constraints can be obtained in [19].
where c is the transmitted power of each emitter.
3.3 Waveform covariance matrix design problem
where \(\mathcal {K}=\{0,1,\cdots,K\}\). (8) is in general NPhard which has no closedform solution.
3.4 Waveform covariance matrix design algorithm
and \(\bar {t}_{1}={\min _{\boldsymbol {a}_{0}\in \mathcal {A}_{0}}{\delta _{0}{\boldsymbol {a}_{0}}^{\dag }\boldsymbol {R}^{*}{{\boldsymbol {a}_{0}}}}}\). Conversely, assuming that \((\bar {\boldsymbol {R}}^{*},t^{*},t^{*}_{1})\) is an optimal solution to (10), R ^{∗}/t ^{∗} is an optimal solution to (9).
which can be efficiently computed by related SDP technique [20].
4 Constrained waveform design with known R
In this section, we consider the waveform design problem by minimizing approximation error accounting for constant modulus and similarity constraints. Finally, we provide CA [21] to design the transmit waveform by exploiting the obtained R.
4.1 Constant modulus and similarity constraints
In practical applications, the synthesized waveform should be unimodular (i.e., constant modulus) due to the limit of nonlinear radar amplifiers. Hence, we here enforce the modulus of each element of s _{ n },n=1,2,⋯,N _{ T } to be constant, i.e., \(\boldsymbol {s}_{n}(m)=\sqrt {c}, n=1,2,\ldots,N_{T},m=1,2,\cdots,M\).
where \(\boldsymbol {s}_{n0}\in \mathbb {C}^{M}\) is the reference sequence vector at the nth transmission interval and ξ _{ n } is a real parameter ruling the extent of the similarity. Without loss of generality, we assume the same similarity parameter ξ (i.e., \(\phantom {\dot {i}\!}\xi =\xi _{1}=\cdots =\xi _{N_{T}}\)) [20, 22, 23], on the sought transmit waveform.
4.2 Constrained waveform design problem
where \(\boldsymbol {S}=[\boldsymbol {s}_{1},\boldsymbol {s}_{2},\cdots,\boldsymbol {s}_{N_{T}}]^{T}\in \mathbb {C}^{N_{T}\times M}\).
4.3 Waveform synthesis algorithm
where \(\phantom {\dot {i}\!}\boldsymbol {S}=[\boldsymbol {s}_{1},\boldsymbol {s}_{2},\cdots,\boldsymbol {s}_{N_{T}}]^{T}\), \(\boldsymbol {U}\in \mathbb {C}^{N_{T}\times M}\) is an arbitrary unitary matrix demanding \(\boldsymbol {U}\boldsymbol {U}^{\dagger }=\boldsymbol {I}_{N_{T}}\), and γ _{ nm }= args _{0n }(m)− arccos(1−ξ ^{2}/2), δ _{ n }=2 arccos(1−ξ ^{2}/2) with \(\xi =\frac {{\xi }_n}{\sqrt {c}}\). In particular, we can observe that for ξ=0, the designed s _{ n } is identical to known s _{0n }, whereas the similarity constraint boils down to only the constant modulus constraint when ξ=2.
Based on the above discussion, we can perform the same procedure to obtain the remaining variables among S. Finally, the CA procedure involved in designing S and U is summarized as Algorithm 1. It is worth pointing out that the total computational complexity of CA is related with the iteration number, the size of S. In particular, each iteration requires to handle a singular value decomposition (SVD) of N _{ T }×M dimension matrix with corresponding to computational complexity \(O(M{N^{2}_{T}}+{N^{3}_{T}})\)[24].
5 Numerical results
where the numerator and denominator of (23) can be efficiently computed by related SDP technique [19], respectively.
5.1 Nonuniform pointlike clutter
The angles and corresponding powers of seven interfering sources
Angle (deg)  −20  5  35  60  70  −30  −15 
Power (dB)  20  20  30  25  28  30  10 
5.2 Uniform clutter
In this subsection, we consider a uniform clutter scenario where we select [−10°,50°] as clutter region which is uniformly discretized with a grid size 1°. In particular, for each azimuth clutter bin, we consider a cluttertonoise ratio (CNR) of 30 dB.
6 Conclusions

We have optimized the waveform covariance matrix to improve the worstcase SINR against the uncertain sets of steering vector under a uniform power constraint through SDPrelated technique. Based on LS approach, we have designed the MIMO waveform to approximate the desired waveform covariance matrix by using CA accounting for constant modulus and similarity constraints.

We have provided numerical simulations to assess the performance of the proposed procedure. Results that the proposed algorithm enjoys the ability of ensuring an improved worstcase performance. We also observed that the approximation error decreases with the improvement of similarity level and sample length. As a consequence, we should choose reasonably the parameters of similarity level and sample length according to practical requirements.
Possible future research tracks might concern the extension of the proposed framework to account for both uncertain sets of angle and Doppler frequency of targets and interferes [15, 26], as well as the joint design of the transmit signal and receive filter for MIMO radar [20].
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grants 61201276, 61178068, and 61301266, the Fundamental Research Funds of Central Universities under Grants ZYGX2012Z001, ZYGX2013J012, ZYGX2014J013, and ZYGX2014Z005, the Chinese Postdoctoral Science Foundation under Grant 2014M550465, and by the Program for New Century Excellent Talents in University under Grant A1098524023901001063.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 MI Skolnik, Radar Handbook, 2nd Edition, (New York: McGrawHill, 1991).Google Scholar
 SP Sira, Y Li, A PapandreouSuppappola, D Morrell, D Cochran, M Rangaswamy, Waveformagile sensing for tracking. IEEE Signal Process. Mag. 26(1), 53–64 (2009).View ArticleGoogle Scholar
 ZQ Luo, WK Ma, AC So, Y Ye, S Zhang, Semidefinite relaxation of quadratic optimization problems. IEEE Signal Process. Mag. 27(3), 20–34 (2010).View ArticleGoogle Scholar
 A De Maio, Y Huang, M Piezzo, A Doppler robust maxmin approach to radar code design. IEEE Trans. Signal Process. 58(9), 4943–4947 (2010).MathSciNetView ArticleGoogle Scholar
 A De Maio, Y Huang, M Piezzo, S Zhang, F Alfonso, Design of optimized radar codes with a peak to average power ratio constraint. IEEE Trans. Signal Process. 59(6), 2683–2697 (2011).MathSciNetView ArticleGoogle Scholar
 MM Naghsh, M Soltanalian, P Stoica, M ModarresHashemi, A De Maio, A Aubry, A Doppler robust design of transmit sequence and receive filter in the presence of signaldependent interference. IEEE Trans. Signal Process. 62(4), 772–785 (2014).MathSciNetView ArticleGoogle Scholar
 A Aubry, A De Maio, MM Naghsh, Optimizing radar waveform and Doppler filter bank via generalized fractional programming. IEEE J. Sel. Topics Signal Process. 9(8), 1387–1399 (2015).View ArticleGoogle Scholar
 SM Karbasi, A Aubry, A De Maio, HM Bastani, Robust transmit code and receive filter design for extended targets in clutter. IEEE Trans. Signal Process. 63(8), 1965–1976 (2015).MathSciNetView ArticleGoogle Scholar
 B Tang, J Tang, in 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Robust waveform design of wideband cognitive radar for extended target detection (Shanghai, 2016), pp. 3096–3100.Google Scholar
 Y Yang, RS Blum, Minimax robust MIMO radar waveform design. IEEE J. Sel. Topics Signal Process. 1(1), 147–155 (2007).View ArticleGoogle Scholar
 P Stoica, J Li, Y Xie, On probing signal design for MIMO radar. IEEE Trans. Signal Process. 55(8), 4151–4161 (2007).MathSciNetView ArticleGoogle Scholar
 B Jiu, H Liu, D Feng, Z Liu, Minimax robust transmission waveform and receiving filter design for extended target detection with imprecise prior knowledge. Signal Process. 92(1), 210–218 (2012).View ArticleGoogle Scholar
 N Shariati, D Zachariah, M Bengtsson, in 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Minimum sidelobe beampattern design for MIMO radar systems: a robust approach (Florence, 2014), pp. 5312–5316.Google Scholar
 H Wang, B Pei, Y Bai, in Signal Processing, Communications and Computing (ICSPCC) 2014 IEEE International Conference on. Robust waveform design for MIMOSTAP with imperfect clutter prior knowledge (Guilin, 2014), pp. 578–581.Google Scholar
 SM Karbasi, A Aubry, V Carotenuto, MM Naghsh, HM Bastani, Knowledgebased design of spacetime transmit code and receive filter for a multipleinputmultipleoutput radar in signaldependent interference. IET Radar Sonar Navig. 9(8), 1124–1135 (2015).View ArticleGoogle Scholar
 W Zhu, J Tang, Robust design of transmit waveform and receive filter for colocated MIMO radar. IEEE Signal Process. Lett. 22(11), 2112–2116 (2015).MathSciNetView ArticleGoogle Scholar
 A Aubry, A De Maio, A Farina, M Wicks, Knowledgeaided (potentially cognitive) transmit signal and receive filter design in signaldependent clutter. IEEE Trans. Aerosp. Electron. Syst. 49(1), 93–117 (2013).View ArticleGoogle Scholar
 H Wang, G Liao, J Li, W Guo, Robust waveform design for MIMOSTAP to improve the worstcase detection performance. EURASIP J. Adv. Signal Process. 2013(1), 1 (2013).View ArticleGoogle Scholar
 A Aubry, A De Maio, Y Huang, MIMO Radar beampattern design via PSL/ISL optimization. IEEE Trans. Signal Process. 64(15), 3955–3967 (2016).MathSciNetView ArticleGoogle Scholar
 G Cui, H Li, M Rangaswamy, MIMO radar waveform design with constant modulus and similarity constraints. IEEE Trans. Signal Process. 62(2), 343–353 (2014).MathSciNetView ArticleGoogle Scholar
 P Stoica, J Li, X Zhu, Waveform synthesis for diversitybased transmit beampattern design. IEEE Trans. Signal Process. 56(6), 2593–2598 (2008).MathSciNetView ArticleGoogle Scholar
 A De Maio, S De Nicola, Y Huang, Z Luo, S Zhang, Design of phase codes for radar performance optimization with a similarity constraint. IEEE Trans. Signal Process. 57(2), 610–621 (2009).MathSciNetView ArticleGoogle Scholar
 A Aubry, A De Maio, M Piezzo, A Farina, M Wicks, Cognitive design of the receive filter and transmitted phase code in reverberating environment. IET Radar Sonar Navig. 6(9), 822–833 (2012).View ArticleGoogle Scholar
 GH Golub, CFW Loan, Matrix Computations, 4rd Edition (MD: The Johns Hopkins University Press, Baltimore, 2013).MATHGoogle Scholar
 M Grant, S Boyd, CVX package. http://www.cvxr.com/cvx.r. Accessed Feb 2012.
 X Yu, G Cui, L Kong, V Carotenuto, in 2016 IEEE Radar Conference. Spacetime transmit code and receive filter design for colocated MIMO radar (Philadelphia, 2016), pp. 1–6.Google Scholar