Robust Design of Transmit Waveform and Receive Filter For ...

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Robust Design of Transmit Waveform and Receive Filter For Colocated MIMO Radar

arXiv:1505.08069v1 [cs.IT] 29 May 2015

Wei Zhu and Jun Tang

Abstract—We consider the problem of angle-robust joint transmit waveform and receive filter design for colocated MultipleInput Multiple-Output (MIMO) radar, in the presence of signaldependent interferences. The design problem is cast as a maxmin optimization problem to maximize the worst-case output signal-to-interference-plus-noise-ratio (SINR) with respect to the unknown angle of the target of interest. Based on rank-one relaxation and semi-definite programming (SDP) representation of a nonnegative trigonometric polynomial, a cyclic optimization algorithm is proposed to tackle this problem. The effectiveness of the proposed method is illustrated via numerical examples. Index Terms—MIMO radar, optimization, receive filter, robust design, waveform design.

I. I NTRODUCTION Compared with conventional phased-array radar, multipleinput-multiple-output (MIMO) radar uses multiple antennas to transmit independent waveforms and multiple receivers to receive, which provides extra degrees of freedom in radar system [1]. Due to the improved parameter identifiability and interference rejection capability [1] [2], as well as the enhanced detection and estimation performance [3] [4], MIMO radar has been extensively studied over the last decade. According to the configurations of the transmitters and receivers, MIMO radar is classified into two types, i.e., colocated MIMO radar [2] and distributed MIMO radar [5]. For both types of MIMO radar, one of the most important problem is how to design probing signals properly. According to the design criteria adopted, existing design approaches can mainly be classified into five categories: 1) optimizing the radar ambiguity function [6], [7]; 2); matching a desired beam-pattern [8]–[11]; 3) maximizing the mutual information or relative entropy to improve the detection or estimation performance based on information theory [12]–[15]; 4) optimizing an estimation-oriented lower bound (e.g., Cram´er-Rao bound (CRB) [16] and ReuvenMesser bound [17]) to improve estimation performance and 5) joint transmit sequence and receive filter design to maximize the output signal-to-interference-plus-noise-ratio (SINR), in the presence of interferences [18]–[21]. This letter mainly focus on the last design approach for colocated MIMO radar. In this design framework, joint transmit and receive beamforming is investigated in [19] for an active array in the presence of signal-dependent interference. A sequential optimization algorithm is proposed to maximize the output SINR. In [20], joint transmit waveform and receive filter design is considered under the constant modulus and The authors are with the Department of Electronic Engineering, Tsinghua University, Beijing, 100084, China (email: [email protected], tangj [email protected]).

similarity constraint. Both works rely on exact a-prior angular knowledge of target and interferences. Indeed, the angle and INR of interferences can be obtained from knowledge-aided methods [22] or previously estimated in high INR cases [23]. The known target angle assumption can apply to the detection of the presence of target at some angle bin [20]. However, there are situations where the angle of target is partially known or unknown (e.g., weak target case) and the SINR should be averagely optimized over the uncertain angle area. Hence, angular robust design must be considered for this case and the design can also be used as an initial step for cognitive detection. In previous works [24], [25], robust waveform design has been considered for interpulse (or intrapulse) coding in radar by taking into account the unknown Doppler shift of target. Motivated by these works, in this letter, we consider the problem of angular-robust design for colocated MIMO radar in the presence of signal-dependent interferences. The signal-dependent interferences are induced by the interaction of transmitted waveform with unwanted scatters (e.g., clutter or other targets in multiple targets scenario [23]). Based on the SINR criterion, transmit waveform and receive filter are jointly optimized to maximize the worst-case output SINR. Since the resulting problem is non-convex, cyclic optimization [26] and rank relaxation [27] are used to solve this problem. Although the cyclic optimization might converge to a locally optimal solution far from the globally optimal solution, it is not so bad and it still can yield a good solution with high worst-case SINR, as illustrated in section IV. This is different with the parameter estimation problem, where the local convergence may significantly effect the accuracy of estimation. Notations: Matrices are denoted by bold capital letters, and vectors by bold lowercase letters. (·)T , (·)c and (·)H denote the transpose, conjugate and conjugate transpose, respectively. k · k denotes Euclidean norm. ⊗ denotes Kronecker product. IL means L × L identity matrix. R and C denotes the sets of all real numbers and complex numbers, respectively. δ(·) represents Kronecker delta function. vec (·) denotes vectorization operator. Re {·} denotes the real part of the argument. II. P ROBLEM F ORMULATION Consider a colocated MIMO radar system equipped with NT transmitters and NR receivers. Both the transmit and receive arrays are assumed to be uniform linear arrays (ULA) with half-a-wavelength element-separation. Let S ∈ CNT ×N denote the transmitted waveforms matrix, where N is the number of samples in the duration of the transmitted waveform. The received waveform matrix Y ∈ CNR ×N from

2

NR receivers corresponding to the range cell of interest is corrupted by K signal-dependent interferences from adjacent range cells with the additional noise, and is modeled as T

Y = α0 ar (θ0 )at (θ0 ) S +

K X

T

αk ar (θk )at (θk ) SJrk + N

k=1

where • α0 and αk are the complex amplitudes of the target and the k-th interference source, respectively. • θ0 and θk are the direction of arrivals (DOA) of the target and the k-th interference source, respectively. N ×1 • ar (θ) ∈ C R is the receive steering vector defined by  jπ sin(θ) T ar (θ) , 1, e , · · · , ejπ(NR −1) sin(θ) . N ×1 • at (θ) ∈ C T is the transmit steering vector defined  T by at (θ) , 1, ejπ sin(θ) , · · · , ejπ(NT −1) sin(θ) . • Jrk , rk ∈ {−N + 1, · · · , −1, 0, 1, · · · , N − 1} is an N by-N shift matrix with (l1 , l2 )-th element Jr (l1 , l2 ) , δ(l1 − l2 − r). rk is the range cell index of the k-th interference source relative to the range cell of interest. • N is spatially and temporally white circularly symmetric complex Gaussian noise with mean zero and variance σ 2 . Let y = vec (Y ), s = vec (S) and n = vec (N ). The vectorization form of the measurement model is given by y = α0 A(θ0 )s +

K X

αk B(θk )s + n

(1)

k=1

T where A(θ0 ) = IN ⊗ ar (θ0 )at (θ0 ) and B(θk ) = JrTk ⊗   T ar (θk )at (θk ) . The SINR at the output of the receive filter NR N ×1 w∈C is given by 2 SNR wH A(θ0 )s (2) X (s, w, θ0 ) = H w ΣI (s)w + wH w PK H H where ΣI (s) = with the k=1 INRk B(θk )ss B(θk ) signal-to-noise-ratio (SNR) of target and the interference-tonoise-ratio  (INR) of k-th interference  defined as SNR , E |α0 |2 /σ 2 and INRk , E |αk |2 /σ 2 , respectively. As with prior works [19], [20], [22], [28], we assume that the angle and the INR of the interferences are all known. We assume that the angle of target under test is known to lie in an angular sector Ω = [θC − ∆θ, θC + ∆θ] centred around θC , where ∆θ indicates the level of angular uncertainty. The goal is to maximize the worst-case SINR with respect to θ0 under the waveform energy constraint ksk2 = E. Therefore, the robust design of transmit waveform s and receive filter w can be formulated as the following max-min problem: s,w

θ0 ∈Ω

X (s, w, θ0 )

Then, it follows that |wH A(θ0 )s|2 = p(ν)H Gp(ν) where
∗
T G , H H vec SW H vec SW H H (10)     Let S = s1 , s2 , · · · , sN and W = w1 , w2 , · · · , wN . P N Using SW H = n=1 sn wnH , we can write N N
X
X wn∗ ⊗ sn vec sn wnH = vec SW H = n=1



max min

Let p(ν) = [1, ejν , · · · , ejν(L−1) ]T with L = NR + NT − 1 ˜ T,··· ,H ˜ T ]T where ˜ T,H and ν = π sin(θ0 ). Let H = [H 2 1 NR NR NT ×L N ×L ˜ k ∈ R T , k = 1, · · · , NR , and the H ∈ R , H ˜ k is defined by H ˜ k (m, n) , δ(n − (m, n)-th element of H m−k +1). Then, one can easily show that ar (θ0 )⊗at (θ0 ) = Hp(ν). According to the property of Kronecker products that vec (CXD) = (D T ⊗ C)vec (X), we can show that
T wH A(θ0 )s = A(θ0 )T vec (W ∗ ) s (4)
T ∗ T = vec at (θ0 )ar (θ0 ) W s (5)

T T s (6) = (W H ⊗ INT )vec at (θ0 )ar (θ0 )
T H (7) = s (W ⊗ INT ) ar (θ0 ) ⊗ at (θ0 )
T ∗ (8) = (W ⊗ INT )vec (S) Hp(ν)
H T = vec SW Hp(ν) (9)

subject to

ksk2 = E

(3)

Note that for a-prior known target angle θ0 , (3) reduces to the optimization problem considered in [19]. III. M AX -M IN ROBUST D ESIGN A LGORITHM In this section, we shall present our algorithm to solve the problem (3). To begin with, we make some mathematical transformations to the objective function of the optimization problem. Define W ∈ CNR ×N such that w = vec (W ).

(11)

n=1

Define X = ssH and V = wwH with X ∈ CNT N ×NT N and V ∈ CNR N ×NR N . Partition X and V into a N -by-N block matrix with (n1 , n2 )-th block denoted by X[n1 ,n2 ] and V[n1 ,n2 ] where X[n1 ,n2 ] ∈ CNT ×NT and V[n1 ,n2 ] ∈ CNR ×NR , it follows from (10) and (11) that   X ∗ H (12) G(X, V ) = H H V[n1 ,n2 ] ⊗ X[n 1 ,n2 ] 1≤n1 ,n2 ≤N

where we use the notation G(X, V ) to emphasize G as a function of X and V . Moreover, using wH w = tr (V ) tr (X) and wH ΣI (s)w = tr (ΣI (V )X) with P/E K H ΣI (V ) = k=1 INRk B(θk ) V B(θk ), it is easy to find that the denominator of (2) can be re-written as tr ΣI (V ) +

tr(V ) E INT N X . Consequently, problem (3) can be recast as  p(ν)H G(X, V )p(ν)      max min   tr(V ) X,V ν∈I   tr Σ (V ) + I I NT N X  E  subject to tr (X) = E (13)     X  0, V  0     rank (X) = 1, rank (V ) = 1

where I = [νC − ∆ν, νC + ∆ν] is the corresponding uncertain range of ν after parameter transformation. This class of problem is known to be very difficult [24], [25]. For example, if we simply consider the optimization with respect to X (or V ) for fixed V (or X), one can show that it is equivalent to a quadratically constrained quadratic program (QCQP) with infinitely many non-convex constraints, which is known to be NP-hard in general [29, Chap. 4]. Hence, the aim of this work is to provide a technique which gives a suboptimal solution with good SINR performance. The proposed approach is shown in the sequel.

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A. Optimization with respect to X and V Since the rank constraint in (13) is non-convex, we adopt the commonly-used rank relaxation technique [27] to obtain a relaxed problem by dropping the rank-one constraint in (13):    min   max X,V ν∈I 

H

p(ν) G(X, V )p(ν)    ) X I tr ΣI (V ) + tr(V N N T E

 subject to    

tr (X) = E

(14)

X  0, V  0

or equivalently,  max    U ,V ,t,γ     subject to           

t p(ν)H G(U , V )p(ν) ≥ t, for ∀ν ∈ I    ) (15) tr ΣI (V ) + tr(V E INT N U = 1 tr (U ) = Eγ, γ ≥ 0

U  0, V  0

T  with gl = where U = γX. Let g = g0 , g1 , · · · , gL−1 PN −l G(U , V ) , l = 0, 1, · · · , L−1. One can also show l+k,k k=1 that the constraint p(ν)H Gp(ν) ≥ t in (15) is equivalent to  L−1  X gl e−jlν ≥ 0 f (ν) = g0 − t + 2Re

(16)

l=1

The optimization problem (15) is still non-convex and it includes infinitely many quadratic constraints as ν ∈ I. To deal with this problem, we resort to an equivalent semi-definite programming (SDP) representation for the nonnegativity constraint of the trigonometric polynomial in (16) based on [30, Theorem 3.4], which is quoted below as a lemma. Lemma 1: The trigonometric polynomial f˜(ω) = h0 +  PL−1 −jωl 2Re is non-negative over [α−β, α+β] (with l=1 hl e 0 < β < π) iff there exists an L × L Hermitian matrix Z1  0 and an (L − 1) × (L − 1) Hermitian matrix Z2  0 such that


h = F1H diag F1 Z1 F1H + d ⊙ diag F2 Z2 F2H (17)

T T   where h = h0 , h1 , · · · , hL−1 , d = d0 , d1 , · · · , dL−1 = cos(2πl/L − α) − cos(β), F = 1 with dl f0 , f1 , · · · , fL−1 and F2 = f0 , f1 , · · · , fL−2 where T  fl = 1, e−j2πl/Q , · · · , e−j2πl(Q−1)/Q with Q ≥ 2L − 1. Based on Lemma 1, cyclic optimization [26] can then be performed to tackle problem (15) iteratively. To be specific, we perform the optimization with respect to U for some fixed V , and then conduct it with respect to V for fixed U , repeatedly. To this end, let α = νC , β = ∆ν and h = g − te1 in Lemma 1, where e1 is an L × 1 vector with the first component being one and the others zero. For fixed V , the optimization with respect to U for (15) can be represented by the following SDP,

which can be solved by the CVX toolbox [31]:  max t    U ,Z1 ,Z2 ,t 
   subject to g − te1 = F1H diag F1 Z1 F1H  

+ d ⊙ diag F2 Z2 F2H       )   INT N U = 1 tr ΣI (V ) + tr(V  E    U  0, Z1  0, Z2  0

(18)

Let U ⋆ denote the optimal solution of U to (18). Then, the optimal solution of X is X ⋆ = EU ⋆ /tr (U ⋆ ). We note that the denominator of X (s, w, θ0 ) can also be + INR N ) V ) where ΣI (X) = PKexpressed as tr ((ΣI (X) H INR B(θ )XB(θ ) . So, analogously, the optimizak k k k=1 tion with respect to V for fixed X can be cast as a similar SDP as below.   max t   V ,Z1 ,Z2 ,t 
   subject to g − te1 = F1H diag F1 Z1 F1H  

(19) + d ⊙ diag F2 Z2 F2H     tr ((ΣI (X) + INR N ) V ) = 1      V  0, Z1  0, Z2  0

By starting from a random initial point and cyclically solving (18) and (19) until the SINR improvement is negligible, the value of the objective function is non-decreasing and the convergence of the algorithm can be guaranteed [25]. The cyclic optimization yields a solution which is not only the local optimum, but also the global optimum along X dimension and V dimension separately [18]. To obtain a more accurate result, one can perform this procedure with a large number of random initializations and select the best (X, V ). In section IV, numerical examples show that the proposed algorithm is quite insensitive to initial values. B. Synthesis of s and w from X and V Let (X ⋆ , V ⋆ ) denote the resultant solution via circularly solving (18) and (19). If both X ⋆ and V ⋆ are rank-one, the transmit sequence s⋆ and receive filter w⋆ can be obtained by eigen-decomposition of X ⋆ = s⋆ (s⋆ )H and V ⋆ = w⋆ (w⋆ )H . In this case, the rank-one relaxation in (14) is tight and the solution is optimal. Otherwise, a suboptimal procedure can be adopted following a recently proposed algorithm in [25]. The idea is that X (s, w)
is a scaled version of the numerator tr XA(θ0 )H V A(θ0 ) . s and w should be designed to let |wH A(θ0 )s|2 well approximate the shape of numerator while imposing constraints on the denominator. The interested readers can refer to [25] for detailed motivations. To make this letter self-contained, we will present the synthesis algorithm for our problem in the sequel. The synthesis algorithm for our problem works
as follows. Consider the value of the tr X ⋆ A(θ0 )H V ⋆ A(θ0 ) evaluated on a discrete set of DOAs {ϑ1 , ϑ2 , · · · , ϑM } “uniformly distributed” on Ω:
cm = tr X ⋆ A(ϑm )H V ⋆ A(ϑm ) , m = 1, 2, · · · , M (20)

4

Let Tm = A(ϑm )H V ⋆ A(ϑm ), Qm QH m = Tm and define M auxiliary unit-norm vectors q1 , q2 , · · · , qM . The synthesis of s is formulated as  PM √ ¯ − c m qm k 2 min  m=1 kQm s  ¯ s , q ,··· ,q  1 M   tr(V ⋆ ) ⋆ H ¯ Σ (V ) + subject to s s¯ ≤ ζ ⋆ (21) I I N N T E    kqm k = 1, 1 ≤ m ≤ M

⋆ ) where ζ ⋆ , tr ΣI (V ⋆ ) + tr(V INT N X ⋆ . This problem E can be solved using cyclic minimization by optimizing qm for fixed s¯ and vice versa. For a fixed s¯, the solution to ¯ Qm s (21) is given by qm = kQ ¯k , m = 1, · · · , M . For fixed ms qm , m = 1, · · · , M , problem (21) reduces to a QCQP that can be solved by the CVX package [31]. The initial value of s¯ can be chosen as the eigenvector of X ⋆ corresponding to the largest eigenvalue. Let s¯⋆ denote the optimal solution to√ (21), the optimal transmit waveform s⋆ is given by s⋆ k, considering the energy constraint on s. s⋆ = E s¯⋆ /k¯ eH emQ Analogously, let Tem = A(ϑm )X ⋆ A(ϑm )H and Q m = e Tm , the synthesis of w is similar to (21):  PM √ e min cm qem k2  m=1 kQm w −  w, qe1 ,··· ,qeM (22) subject to wH (ΣI (X ⋆ ) + INR N ) w ≤ η ⋆    kqem k = 1, 1 ≤ m ≤ M where η ⋆ , tr ((ΣI (X ⋆ ) + INR N ) V ⋆ ). As with (21), the cycle optimization can be applied to solve this problem. We note that the randomization optimization approach [27] can also be used to obtain an approximate s⋆ and w⋆ in the non-rank-one case, with known good approximation accuracy. Similar applications can be found in [20], [32], [33]. The synthesis algorithm based on randomization optimization method for our problem is shown in Algorithm 1. Remark: Prior results on the tightness of SDP relaxation [27], [34] indicate that for a separable SDP with P semidefinite variables and J constraints, there exists an optimal rank-one solution if J ≤ P + 2. But this can not guarantee the existence of rank-one solution for our problem, since we have 3 semi-definite variables and L effective constraints for (18) and (19). Nevertheless, as with in [25], one can numerically observe that either the rank of X ⋆ or V ⋆ is rarely greater than one for different random initializations as along as Ω∩Ωc = ∅, where Ωc denotes the set of all interferences angles. IV. N UMERICAL E XAMPLES In this section, numerical examples are conducted to examine the performance of the proposed method. Throughout the simulations, a total of 30 interferences are considered with the range and angle pair (rk , θk ) generated from all possible combinations of {−2, −1, 0, 1, 2} × {−60◦, −50◦ , −40◦ , 40◦ , 60◦ , 70◦ }. The INR of all interferences is 30 dB. In Fig. 1, the output SINR as a function of the true target angle θ0 for the non-robust design and the proposed robust design are compared under four different parameters. The target SNR is −15 dB. The waveform energy E is set to be equal to the number of waveform samples N . For the

Algorithm 1 Synthesis algorithm based on randomization optimization Input: X ⋆ and V ⋆ Output: A randomized approximate solution s⋆ and w⋆ 1: if rank(V ⋆ ) = 1 then 2: find w⋆ via eigen-decomposition V ⋆ = w⋆ (w⋆ )H 3: else 4: draw R random vectors wj from the complex Gaussian distribution CN (0, V ⋆ ), j = 1, 2, · · · , R 5: calculate wjH A(θ0 )X ⋆ A(θ0 )H wj , j = 1, · · · , R θ0 ∈Ω wjH ΣI (X ⋆ )wj + wjH wj PK where ΣI (X ⋆ ) = k=1 INRk B(θk )X ⋆ B(θk )H . let w⋆ = wjmax where ξj = min

6:

jmax = arg max ξj . 1≤j≤R

7: 8: 9: 10: 11: 12:

end if if rank(X ⋆ ) = 1 then find s⋆ via eigen-decomposition X ⋆ = s⋆ (s⋆ )H else draw R random vectors si from the complex Gaussian distribution CN√(0, X ⋆ ), i = 1, 2, · · · , R i calculate s¯i = ksEs and ik |(w⋆ )H A(θ0 )¯ si |2 , i = 1, · · · , R + (w⋆ )H w⋆ PK H where ΣI (¯ si ) = k=1 INRk B(θk )¯ si s¯H i B(θk ) . ⋆ let s = s¯imax where

ζi = min

θ0 ∈Ω (w ⋆ )H ΣI (¯ si )w⋆

13:

imax = arg max ζi . 1≤i≤R

14:

end if

non-robust design, the assumed a-prior target angle is set to be θC , and the optimization algorithm is based on the method presented in [19]. For the robust design, the angle uncertainty ∆θ is set to be 10◦ in all cases. In this example, both X ⋆ and V ⋆ obtained from the cyclic optimization are rank-one, and thus the SDP relaxation is tight. It is shown that the robust design improves the worst-case SINR performance significantly at the cost of peak-SINR degradation. For fixed ∆θ and N , the superiority of robust design increases with the number of transmitting or receiving arrays. H A(θ)sk2 In Fig. 2, we depict the beampattern P (θ) = Nkw 2 2 R NT kwk ksk for parameter setting in Fig. 1(c) and Fig. 1(d) as an example. One can observe that both robust and non-robust design can produce nulls near the DOAs of interferences. From Fig. 1 and Fig. 2, we see that when angle uncertainty ∆θ is large enough relative to the beamwidth, the max-min design can form a wide and flat beam over the uncertain space area to bring robustness. In Fig. 3, we plot the worst-case output SINR versus the target angle uncertainty ∆θ for different NR and NT . The target SNR is set to −15 dB and the waveform energy E = N . As expected, a wider range of target angular uncertainty leads

5

to a worse output SINR. Moreover, the impact of ∆θ on the worst-case SINR performance of non-robust design is more prominent, which suffers a sharp decline as ∆θ increases, due to the effect of the first null near the main lobe. In this example, both X ⋆ and V ⋆ obtained from the cyclic optimization are rank-one for all ∆θ. 15

8.8 8.7 −3

−2

−1 5.9

0

5

5.8

6

robust non−robust

5.7 9.9

9.95

robust non−robust 0 −10 −8 −6 −4 −2 0 2 4 θ0 (degree)

10

5.5 −10 −8 −6 −4 −2 0 2 4 θ0 (degree)

6

8

10

(a)

6

(b)

16

20 10 SINR (dB)

SINR (dB)

12

8

0 −10 −20

4 robust non−robust

robust non−robust −40 10 12 14 16 18 20 22 24 26 28 30 θ0 (degree) −30

0 −10 −8 −6 −4 −2 0 2 4 θ0 (degree)

6

8 10

(c)

L,

8 10

15.41

18.4

15.4

18.38

15.39 15.38 15.37

−60

−30 0 30 θ (degree)

60

−100

−150 −90

20

0

0 SINR (dB)

SINR (dB)

20

−60

8

−60

−30 0 30 θ (degree)

60

90

−80 0

4 8 12 ∆θ (degree)

−40

8 4

−60 −80

16

10

20 30 iteration index

40

50

40

50

(b) 21.72

18.6

−100 0

21.68 21.66 21.64

18.55 21.62 10

20 30 iteration index

(c)

40

50

21.6 0

10

20 30 iteration index

(d)

Fig. 4. The effect of initial values on cyclic optimization. (a) NR = NT = 4, N = 10; (b) NR = NT = 4, N = 20; (c) NR = 4, NT = 8, N = 10. (d) NR = 4, NT = 8, N = 20.

−20 12

10 non−robust robust

18.3 0

50

21.7

Fig. 2. Comparisons of the beampattern P (θ). (left) NR = NT = 8, θC = 0◦ ; (right) NR = 8, NT = 16, θC = 20◦ .

−40

40

18.65

18.5 0

10 8 6 4 2 6

20 30 iteration index

−50

robust non−robust

90

18.34

(a)

robust non−robust

−20

10

18.7

SINR (dB)

BeamPattern (dB)

BeamPattern (dB)

−100

18.36

18.32

0

−50

−150 −90

18.42

15.42

15.36 0

0

(23)

to evaluate the variation of T , where max(T ), min(T ) and mean(T ) denote the maximum, minimum and mean value of T , respectively. The values of L for the four cases are equal to 0.016, 0.0177, 0.0263 and 0.016, respectively. One can see that in our problem, the cyclic optimization is quite insensitive to the initialization.

(d)

Fig. 1. Comparisons of the output SINR. ∆θ = 10◦ . E = N = 20. SNR = −15dB. (a) NR = NT = 4, θC = 0◦ ; (b) NR = 4, NT = 8, θC = 0◦ ; (c) NR = NT = 8, θC = 0◦ ; (d) NR = 8, NT = 16, θC = 20◦ .

max(T ) − min(T ) mean(T )

SINR (dB)

7 6.5

10

SINR (dB)

SINR (dB)

SINR (dB)

8.9

8 7.5

SINR (dB)

9 8.5

different random initializations. Four different parameter settings of NT , NR and N are considered. The waveform energy E is equal to NT N . θC = 0◦ and ∆θ = 10◦ . The target SNR is set to be equal to 1/(NR +NT −1). The cyclic optimization is stopped if either the increment of objective function between two iterations is less than 5×10−3 or the maximum number of iterations reaches. The maximum number of iterations of the cyclic optimization is set to 150. We can see that the values of objective function under different initializations are very close. Let T = {t(1) , t(2) , · · · , t(Γ) } denote the value of the objective function from Γ random initializations. We define the following metric

0 6

8

10

non−robust robust

4 8 12 ∆θ (degree)

16

Fig. 3. The worst-case output SINR versus the angle uncertainty. θC = 0◦ . E = N = 20. SNR = −15dB. (left) NR = 4, NT = 8; (right) NR = NT = 8.

In Fig. 4, we investigate the effect of initial values on the cyclic optimization of X and V . We plot the value of the objective function for the relaxed problem (15) under Γ = 50

In Fig. 5, we illustrate the performance of the synthesis algorithm in the non-rank-one case, which seldom happens in our experiments. In this example, the parameter settings are the same as in figure 4(c). Under a certain random initialization, the cyclic optimization provides a solution with rank(X ⋆ ) = 2 and rank(V ⋆ ) = 1. The receive filter w⋆ is obtained based on eigen-decomposition, and the transmit waveform s⋆ is obtained via the synthesis algorithm. The performance of synthesis algorithm based on solving problem (21) (denoted Method 1) and the algorithm based on randomization optimization method (denoted Method 2) are compared. We plot their corresponding SINR as a function of θ0 according to (2). For the Method 1, the number of DOA samples M is set to 41 and the number of iterations to solve (21) is 50. For the Method 2, the number of random samples is set to be

6

1000. We also plot the SINR Xrelax (X ⋆ , V ⋆ , θ0 ) obtained by directly substituting X ⋆ and V ⋆ into the objective function of relaxed problem (14) for comparison. We can see that their SINR performance are very close to each other. We can also observe that the SINR curve of Method 1 matches well with Xrelax (X ⋆ , V ⋆ , θ0 ). Both synthesis algorithms yield a good solution in the non-rank-one case. The worst-case SINRs for Method 1, Method 2 and Xrelax (X ⋆ , V ⋆ , θ0 ) are 18.478 dB, 18.394 dB 18.526 dB, respectively. Method in [25]

SINR (dB)

19.5

Randomization Optimization SINRrelax

19

18.5 18.8

18.6

18.6

18.5

18 18.4

−10 −9.9 −9.8

−10

−5

18.4 9.9

0 θ (degree)

9.95

10

5

10

0

Fig. 5.

The SINR performance of the synthesis algorithm.

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