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Robust constrained waveform design for MIMO radar with uncertain steering vectors
EURASIP Journal on Advances in Signal Processing volume 2017, Article number: 2 (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.
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.
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.
System model
We consider a colocated narrow band MIMO radar system with N _{ T } transmitting antennas and N _{ R } receiving antennas. Each transmitting antenna emits a distinct waveform s _{ n }(m),n=1,2,⋯,N _{ T }, m=1,2,⋯,M, with M being the sample number of each transmitting pulse. Let us denote by \(\boldsymbol {s}(n)=[\boldsymbol {s}_{1}(n),\boldsymbol {s}_{2}(n),\cdots,\boldsymbol {s}_{N_{T}}(n)]^{T}\in \mathbb {C}^{N_{T}}\), the nth sample of the N _{ T } waveforms. At each receiver, the received waveform is downconverted to baseband, undergoes a pulsematched filtering operation, and then is sampled. Hence, the observations of the nth sample for a farfield target at the azimuth angle θ _{0} can be expressed as
where

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
$${\boldsymbol{a}_{t}}(\theta)=\frac{1}{\sqrt{N_{T}}}[1,e^{j2\pi\frac{d_{T}}{\lambda}{\sin{\theta}}},\cdots,e^{j2\pi\frac{d_{T}}{\lambda}{(N_{T}1)\sin{\theta}}}]^{T}, $$$${\boldsymbol{a}_{r}}(\theta)=\frac{1}{\sqrt{N_{R}}}[1,e^{j2\pi\frac{d_{R}}{\lambda}{\sin{\theta}}},\cdots,e^{j2\pi\frac{d_{R}}{\lambda}{(N_{R}1)\sin{\theta}}}]^{T} $$with d _{ T } and d _{ R }, respectively, the array interelement spacing of the transmitter and the receiver.

\(\boldsymbol {d}(n)\in \mathbb {C}^{N_{R}}, n=1,2,\cdots,M\), 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., \(\boldsymbol {v}(n)\thicksim \mathcal {CN}(0,{\sigma }_{v}^{2}\mathbf {I}_{N_{R}})\).
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.
Output SINR
In this subsection, for signalindependent interferences, we derive the expression of SINR with respect to waveform covariance matrix R. Specifically, according to the signal model (1), the received useful power can be computed as
where \(\boldsymbol {R}=\frac {1}{M}\sum \limits _{m=1}^{M}{\boldsymbol {s}(m)}{\boldsymbol {s}^{\dagger }(m)}\) stands for the waveform covariance matrix. Similarly, since the scatterers are uncorrelated, the received disturbance power is given by
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.
Finally, on the basis of the above equations, the SINR can be defined as
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.
Uncertain sets of steering vectors and power constraint
In practical considerations, due to array nonidealities and imperfect calibration, the actual steering vector can not be exactly known. To this end, we enforce some quadratic constraints into a _{ k },k=0,1,⋯,K, (where we write a _{ t }(θ _{ k }) into a _{ k } for simplifying the notations) i.e.,
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].
Furthermore, to control the amount of transmitted power, we force a uniform power constraint in each transmitting antenna, i.e.,
where c is the transmitted power of each emitter.
Waveform covariance matrix design problem
Based on the aforementioned discussion, the robust design problem of waveform covariance matrix R to optimize the worstcase SINR over mismatching steer vectors can be summarized as follows:
where \(\mathcal {K}=\{0,1,\cdots,K\}\). (8) is in general NPhard which has no closedform solution.
Waveform covariance matrix design algorithm
In this subsection, we focus on studying the solution of (8) by using SDRrelated technique. Before proceeding further, we first transform (8) as follows:
Interestingly, we observe that (9) can be solved by exploiting SDR technique [19]. Specifically, after the same line of reasoning as [19], this problem is tantamount to SDP problem given by
In particular, given an R ^{∗} to problem (9), \((\boldsymbol {R}^{*}\bar {t},\bar {t},\bar {t}_{1})\) is an optimal solution to (10), where
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).
Finally, it is worth pointing out that, for the case of no steering vector mismatches, the optimization problem (8) can be recast as
which can be efficiently computed by related SDP technique [20].
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.
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., \(\boldsymbol {s}_{n}(m)=\sqrt {c}, n=1,2,\ldots,N_{T},m=1,2,\cdots,M\).
Additionally, we enforce N _{ T } different similarity constraints on the emitting waveforms, which employs a known code as a benchmark allowing the designed code to enjoy some good ambiguity characteristics of the known code, namely
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.
Constrained waveform design problem
In order to approximate a desired waveform covariance matrix R computed by (9), based on LS approach, thus the waveform synthesis problem under constant modulus and similarity constraints can be expressed as,
where \(\boldsymbol {S}=[\boldsymbol {s}_{1},\boldsymbol {s}_{2},\cdots,\boldsymbol {s}_{N_{T}}]^{T}\in \mathbb {C}^{N_{T}\times M}\).
Waveform synthesis algorithm
In this subsection, we study the design procedure of transmit signal. Specifically, after some algebraic manipulations about similarity constraint [22] and according to [21], the waveform synthesis problem can be equivalent to
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.
In particular, here we resort to CA to solve (15). Precisely, given a S, the closedform solution to (15) is \(\boldsymbol {U}=\boldsymbol {U}_{l}\boldsymbol {U}^{\dagger }_{r}\), where both \(\boldsymbol {U}_{l}\in \mathbb {C}^{N_{T}\times N_{T}}\) and \(\boldsymbol {U}_{r}\in \mathbb {C}^{M\times N_{T}}\) are unitary matrixes demanding \(\sqrt {M}{\boldsymbol {R}}^{\frac {1}{2}}\boldsymbol {S}=\boldsymbol {U}_{l}\boldsymbol {\Lambda }\boldsymbol {U}^{\dagger }_{r}\) with \(\boldsymbol {\Lambda }\in \mathbb {H}^{N_{T}}\) being a diagonal matrix. The detailed derivation can be found in [21]. Exploiting the estimated U, we can find the nearest matrix to approximate \(\sqrt {M}{\boldsymbol {R}}^{\frac {1}{2}}\boldsymbol {U}\). Specifically, with given a U, (15) associated to the mth variables of s _{ n } is expressed as
with z _{ nm } is the (n,m)th entry of \(\sqrt {M}\boldsymbol {R}^{\frac {1}{2}}\boldsymbol {U}\). After some algebraic manipulations, (16) can be transformed as
Further, the above problem is equivalent to
where φ _{ nm } and φ _{ c } are the phases of s _{ n }(m) and \(z^{*}_{nm}\), respectively. Hence, the optimization solution of (18) can be obtained by
otherwise, the optimal solution \(\varphi ^{*}_{nm}\) is given
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].
Numerical results
In this section, we evaluate the performance of proposed design technique of robust waveform focusing on a uniform linear array (ULA) transmit elements with N _{ T }=16 assuming an interelement spacing d _{ t }=λ/2. In particular, we consider the orthogonal linear frequency modulation (LFM) as the reference waveform s _{0} [20]. Specifically, the (n _{ t },k)th entry of the reference S ^{(0)} is given by
where n _{ t }=1,2,⋯,N _{ T } and m=1,2,⋯,M. We suppose that the target is located at θ _{0}=0° with power \({\sigma _{0}^{2}}=25\) dB. As to the Gaussian white noise, we set the noise variance to \({\sigma _{v}^{2}}=0\) dB. The nominal ULA steering direction is given by
Additionally, we consider c=1, \(\boldsymbol {B}_{k1}=\boldsymbol {I}_{N_{T}}\phantom {\dot {i}\!}\), \(\boldsymbol {b}_{k1}=\bar {\boldsymbol {a}}(\theta _{k})\phantom {\dot {i}\!}\), b _{ k1}=1, u _{ k1}=ε, \(\boldsymbol {B}_{k2}=\boldsymbol {I}_{N_{T}}\phantom {\dot {i}\!}\), b _{ k2}=0, b _{ k2}=−1, u _{ k2}=0, k=0,1,⋯,K. To solve the convex optimisation problem (10), we resort to the CVX toolbox [25]. Finally, we consider the exit condition κ=10^{−4} for Algorithm 1, i.e.,
Hereafter, for any fixed waveform covariance matrix R, the worstcase SINR ρ ^{wr} can be obtained by
where the numerator and denominator of (23) can be efficiently computed by related SDP technique [19], respectively.
Nonuniform pointlike clutter
In this subsection, we consider a nonuniform pointlike clutter scenario where the angles and corresponding powers of seven interfering sources are reported in Table 1.
Figure 1 shows the worstcase SINR behavior versus uncertain parameter ε considering the nonrobust and robust design of waveform covariance matrix, respectively. Note that the curve of nonrobust case is plotted based on the obtained R according to (8) when no steering vector mismatches are foreseen at the design stage. These curves both decrease as the uncertain parameter ε grows up, implying that the steering vector mismatches indeed impair the detection performance. Interestingly, the robust design exhibits an improvement gain of SINR than that of nonrobust design. In particular, the higher ε, the gap of SINR between robust design and nonrobust design becomes larger and larger.
In the following, based on the R computed by SDPrelated technique, we will focus on the waveform synthesis under constant modulus and similarity constraints by exploiting Algorithm 1. Specifically, we fix the uncertainly parameter ε=0.04 for the following simulations. In addition, the approximation error η between the obtained \(\hat {\boldsymbol {R}}\) using Algorithm 1 and R is computed by
Figure 2 depicts the approximation error η behavior versus iteration number considering different similarity levels for M=20. Notice that these obtained curves are averaged results of 100 statistic independent trials. As expected, the η decreases with the increasing iteration number. Besides, the higher similarity level ξ, the lower η is achieved due to the feasible set of optimization problem (15) becomes larger and larger.
Next, we analyze the impact of similarity parameters and sample length M on the approximation error η and SINR. In particular, we plot both the η and SINR (dB) behaviors versus sample length M considering ξ=0.3,.7,1.1,1.7,2 in Fig. 3 a, b, respectively. Notice that the black line in Fig. 3 b represents the theoretical value without the synthetic loss of SINR. Results highlight that both the approximation error and SINR are associated with both similarity parameter ξ and sample length M. Precisely, as ξ and M increases, the better approximation between \(\hat {\boldsymbol {R}}\) and R can be achieved, which leads to higher gains of SINR. This is a reasonable behavior since the more degree of freedom (DOF) of waveform at design stage is obtained. In particular, when M≥220, the curve of SINR at ξ=2 overlaps with the theoretical curve showing that the constant modulus constraint causes no loss of SINR in the correspondence of the analysed parameters. However, it is worth pointing out that a loss of SINR can be observed due to the introduction of the similarity constraint. For example, we have the SINR gain of 2.8 dB from the black line in Fig. 3 b, whereas the SINR becomes 2.2 dB at ξ=1.7. Besides, interestingly, for large M, both an increasing trend for SINR and a reducing trend for η show unremarkable in Fig. 3 b. This performance behavior provides a foundation for how to choose M at waveform design stage. Finally, it is worth highlighting that a tradeoff should be considered between similarity level, sample length, and SINR.
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.
In Fig. 4, the worstcase SINR behavior considering the nonrobust and robust design of waveform covariance matrix is plotted versus uncertain parameter ε. The similar observation as Fig. 1 can be obtained. Indeed, the increasing ε, both the nonrobust design and robust design experience worse and worse SINR values due to the enlargement of mismatches between actual steering vectors and nominal ones. Again, it can be found that the robust design ensures a enhanced worst case performance in comparison with that of nonrobust design. Finally, it is also interesting to notice that the gap between curves of nonrobust design and robust design becomes larger and larger with the improvement of ε.
In Fig. 5, the approximation error η behavior, averaged over 100 independent trials of Algorithm 1, is plotted versus iteration number for ξ=0.3,0.7,1.1,1.7,2, M=20. Again, the smaller η can be obtained as the increasing of iteration number and ξ, revealing the better approximation between \(\hat {\boldsymbol {R}}\) and R.
In Fig. 6, the approximation error η and SINR (dB) behaviors are plotted versus sample length M considering ξ=0.3,0.7,1.1,1.7,2. Notice that the obtained curves are averaged results of 100 independent trials of Algorithm 1 and the black line in Fig. 6 b stands for the theoretical value without the synthetic loss of SINR. Again, we observe that the higher ξ, the lower η and the better SINR can be reached since the enlargement of the feasible sets of (15). In particular, we also find that both constant modulus and similarity constraints result in a loss of SINR. Specifically, we see the theoretical value of SINR is about −8.7 dB from the black line in Fig. 6 b, whereas it becomes about −9.9 and −9.4 dB for ξ=1.7 and 2, respectively. Besides, both an increasing trend for SINR and an opposite trend for η can be obtained as M increases, whereas these trends are not apparent for large M. As a consequence, in practice, we should reasonably consider the selection of sample length as well as the similarity parameter.
Conclusions
In this paper, we have considered the robust design of MIMO radar waveform under practical constraints for enhancing the targets delectability in the presence of signaldependent interferences. Summarizing,

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].
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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.
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Yu, X., Cui, G., Piezzo, M. et al. Robust constrained waveform design for MIMO radar with uncertain steering vectors. EURASIP J. Adv. Signal Process. 2017, 2 (2017). https://doi.org/10.1186/s1363401604379
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Keywords
 Robust waveform design
 Multipleinput multipleoutput (MIMO)
 Signaldependent interferences
 Signaltointerferenceplusnoise ratio (SINR)
 Waveform covariance matrix
 Constant modulus and similarity constraints