Multi-Stage Antenna Selection for Adaptive Beamforming in MIMO

Multi-stage Antenna Selection for Adaptive Beamforming in MIMO Arrays Hamed Nosrati, Studnet Member, IEEE, Elias Aboutanios, Senior Member, IEEE, and David Smith, Member, IEEE

Abstract—Increasing the number of transmit and receive ele- the baseband DSP and RF front-ends are a great deal more ments in multiple-input-multiple-output (MIMO) antenna arrays expensive than the antenna elements. One way to reduce the imposes a substantial increase in hardware and computational cost while maintaining the spectral diversity is to employ a costs. We mitigate this problem by employing a reconfigurable MIMO array where large transmit and receive arrays are large antenna array but select a subset of antennas to feed multiplexed in a smaller set of k baseband signals. We consider through the RF switching network [7]–[9]. In this work, we four stages for the MIMO array configuration and propose four focus on antenna selection in the context of MIMO radar. different selection strategies to offer dimensionality reduction in Over the last decade, antenna selection in MIMO arrays post-processing and achieve hardware cost reduction in digital has commanded significant attention both in wireless commu- signal processing (DSP) and radio-frequency (RF) stages. We define the problem as a determinant maximization and develop nications and radar applications. In communications, antenna a unified formulation to decouple the joint problem and select selection is employed to maximize the channel capacity. To antennas/elements in various stages in one integrated problem. this end, near-optimal strategies that assume perfect knowledge We then analyze the performance of the proposed selection of the channel were proposed in [10] and [11]. In [12], a fast approaches and prove that, in terms of the output SINR, a adaptive antenna selection via discrete stochastic optimization joint transmit-receive selection method performs best followed by matched-filter, hybrid and factored selection methods. The in is proposed, where an aggressive stochastic approximation theoretical results are validated numerically, demonstrating that is employed to generate iteratively a sequence of estimates of all methods allow an excellent trade-off between performance the solution. More recently, antenna selection has been em- and cost. ployed to reduce complexity and power consumption in mm- Index Terms—Antenna selection, MIMO radar, adaptive array wave MIMO systems through compressed spatial sampling beamforming, STAP, convex optimization. of the received signal [13]. In radar, both deterministic and optimization-based methods have been developed to select a I.INTRODUCTION subset of antennas and reconfigure the array architecture in order to maximize the output signal-to-interference and noise The spatial diversity and performance improvements offered ratio (SINR) [8], [9], [14], [15] and enhance the direction of by multiple-input multiple-output (MIMO) antenna systems arrival (DoA) estimation [16]. Antenna selection also plays have led to their widespread use in a variety of applications an important role in aperture sharing in dual function radar including wireless communications e.g. massive MIMO [1], communication systems [17], [18]. [2], radar and sonar [3], [4]. In radar, MIMO arrays have In MIMO radars, antenna selection has been studied mostly proven effective at enhancing the radar’s resolution as they from the perspective of target parameter estimation. An opti- offer increased number of degrees of freedom (DOFs) [5]. A mal antenna placement was proposed in [19] to minimize the MIMO phased array comprises an array of antennas, transmit- Cramer-Rao´ lower bound (CRLB) of the velocity estimates. ting a set of noncoherent orthogonal waveforms that can be ex- In [20] a combinatorial optimization approach was used to tracted at the receiver by a corresponding number of matched achieve resource allocation for localization error minimization filters. Improved spatial diversity, parameter identifiability, and in multiple radar systems. The CRLB for target location in detection performance result from the added DoFs compared MIMO radars with collocated antennas was derived in [21], arXiv:1904.08614v1 [eess.SP] 18 Apr 2019 to single-input multiple-output (SIMO) configurations [3]. allowing its determinant to be minimized. Joint antenna subset The advantages of the MIMO configuration are delivered at selection and optimal power allocation were also implemented the expense of a significant increase in the problem dimension- in [22] for localization in MIMO radar sensor networks via ality and hardware cost [6]. The system hardware include the convex optimization. In a similar vein, the idea of minimum antennas, baseband digital signal processing (DSP) and radio redundancy has been successfully applied to the design of frequency (RF) front-ends comprising the low noise amplifiers physical transmit/receive arrays to form MIMO virtual arrays (LNA), phase shifters, and frequency mixers. Among these, with maximum contiguous aperture, i.e. minimum redundancy H. Nosrati is with the School of Electrical Engineering and Telecommuni- virtual arrays (MRVA) [23]. The two-level autocorrelation cation, University of New South Wales, NSW 2052, Australia, and Data61, property of the difference sets (DSs) was then successfully CSIRO (Commonwealth Scientific and Industrial Research Organisation), exploited to maximize the virtual aperture [24]. NSW 2015, Australia (e-mail:[email protected]). E. Aboutanios is with the School of Electrical Engineering and Telecom- In this paper we address the problem of antenna selection munication, University of New South Wales, NSW 2052, Australia (e- for interference cancellation and SINR maximization. Antenna mail:[email protected]) selection can be applied to the transmit and receive arrays D. Smith, is with Data61, CSIRO (Commonwealth Scientific and Industrial Research Organisation), NSW 2015, Australia (e- separately, jointly to the transmit and receive arrays, or to mail:[email protected]) the matched filter bank (virtual array). We study all of these 2 scenarios and propose a comprehensive optimization method to derive their solutions. We first examine the joint transmit/recieve element selection, which reduces the dimensionality and consequently decreases the computational cost. We then consider the factored selection approach in which we separately select subsets of the transmit and receive arrays. We formulate the factored problem as a coupled optimization such that both selections are solved together. The computational Tx1 Rx1 Tx1\Rx1 Matched Tx2\Rx1 cost of the MIMO radar can also be alleviated by reducing Filter Bank Tx3\Rx1 the number of matched filters used to generate a virtual array Txn\Rx1 Tx2 Rx2 Tx1\Rx2 at the receiver, which involves the application of element Matched Tx2\Rx2 Filter Bank Tx3\Rx2 selection to the virtual array. Finally, we bring these scenarios Txn\Rx2 together in a hybrid selection strategy that is capable of Tx3 Rx3 Tx1\Rx3 Matched Tx2\Rx3 reducing the number of transmitters, receivers and matched Filter Bank Tx3\Rx3 filters simultaneously in a unified approach. Txn\Rx3 The main contributions of this paper are as follows.

1) We express the output SINR, denoted as SINRout, as a Txn Rxn Tx1\Rxn function of selected elements of the MIMO array in a Matched Tx2\Rxn scenario comprising a single target, multiple jammers, Filter Bank Tx3\Rxn Txn\Rxn and clutter. 2) We propose four different selection approaches, each 1 2 3 4 achieving a different efﬁciency in terms of hardware (e.g., baseband and RF), computational, and power cost. Fig. 1. Block diagram of the considered MIMO system. Stage 1: transmit 3) Since the SINRout is a joint function of transmitters and antenna selection. Stage 2: receive antenna selection. Stage 3: matched-ﬁlter receivers in MIMO, we propose a new factored problem selection in DSP. Stage 4: output signal selection. formulation that permits us to decouple the transmit and receive sides and allows their designs to be performed the waveform vector φ(t) = [φ (t), φ (t), ..., φ (t)]. The separately. 1 2 M snapshot vector received by the receive array for pulse τ is 4) We formulate the dual problem and study the per-

formance of the proposed selection methods from a x(t, τ) = xs(t, τ) + xc(t, τ) + xj(t, τ) + xn(t, τ), mathematical point of view. 5) We propose a relaxation method and successfully ap- where xc and xj represent the clutter and jammer respectively. proximate the global solution via a set of problem- The signal of interest (SOI), xs, represents the target reflection specific randomized rounding strategies. and xn is zero-mean additive Gaussian noise with variance σ2 . By applying matched-filtering with respect to the M The rest of this paper is organized as follows. In section II n orthogonal waveforms, the extended receive signal becomes we present the formulation of the SINRout maximization using element selection. We then study the selection approaches in Z ! H Section III. The relaxation strategy is detailed in Section IV, x˜(τ) = vec x(t, τ)φ (t)dt and the numerical results are presented in Section V. Finally Tp some conclusions are drawn in section VI. = x˜s(τ) + x˜c(τ) + x˜j(τ) + xñ(τ), where x˜ represents the vectorized version of the matched- Notation filtered snapshot vector. Exploiting the orthogonality assump- We use bold lower-case letters to denote vectors, and upper- tion, we can write the SOI as case letters for matrices. The notation is the expectation E x˜s(τ) = βsas operator, and Tr(M) denotes the trace of M. (•)T and (•)H a = a (θ ) ⊗ a (θ ), are the Hermitian and transpose operations. The operation s t t r r diag(v) constructs a square diagonal with v along the diagonal, where βs is the target reflection coefficient, which we assume whereas diag(M) extracts the diagonal of M. The function obeys the Swirling II model. The transmit steering vector, real(•) takes the real part of its complex argument. We use ⊗ at(θt), corresponds the Direction-of-Departure (DoD) θt, and for Kronecker product. Finally, 1N is a N × 1 vector of all the receive steering vector, ar(θr), is associated with DoA θr. ones, 0N a vector of zeros, and IN the N ×N identity matrix. In the case of a ULA, the steering vectors are given by

j2πdtsinθt j2π2dtsinθt j2πMdtsinθt at(θt) = [1, e , e , ..., e ], II.PROBLEM FORMULATION j2πdrsinθr j2π2drsinθr j2πNdrsinθr ar(θr) = [1, e , e , ..., e ], Let us consider a MIMO radar equipped with M transmitters and N receivers as shown in Fig. 1. Each transmitter with dt, dr denoting the inter-element spacing employed in emits one of the predesigned orthogonal waveforms from transmit and receive arrays. 3

Nc Now assuming a set of angle cells, {θi}i=1, we model the Similarly, using (1), the covariance matrix for the jamming clutter as the reflections from these directions and extract the signal is found to be received clutter signal as H H Rj = E{x˜j(τ)x˜j (τ)} = AjPjAj , N Xc x˜c(τ) = βiac,i where Aj, and Pj are MN ×MNj, and MNj ×MNj matrices i=1 denoted as ac,i = at(θtc,i) ⊗ ar(θrc,i), Aj = [ar(θrj,1), ..., ar(θrj,1), ..., ar(θrj,Nj ), ..., ar(θrj,Nj )], | {z } | {z } where βi is the reflection coefficient of the i-th clutter cell M times M times from directions θtc,i, and θrc,i with respect to transmit, and = diag(σ2 , ..., σ2 , ..., σ2 , ..., σ2 ), Pj j,1 j,1 j,Nj j,Nj receive sides. Suppose that Nj jamming signals are in the field | {z } | {z } of view of the radar. Then the jamming signal is expressed as M times M times and Nj X x˜ (τ) = α a 2 ∗ j i j,i σj,i = E{αiαi }. i=1 Finally, the covariance matrix of the white noise is given by aj,i = x˜j,i(τ) ⊗ ar(θrj,i), R = {x˜ (τ)x˜H (τ)} = σ2 I . with αi, and x˜j,i(τ) denoting the complex amplitude, and the n E n n n MN matched filtered version of the i-th jamming signal. Given a 2 ∗ Now writing σs = E{βsβs }, the SINR at the output of the strong jamming source xj,i(t, τ) with a power of αî , which filter (2) is emulates the radar orthogonal waveforms we have 2 H −1 M SINRout = σs as R as. X xj,i(t, τ) =α î φi(t, τ), (1) The inverse covariance matrix becomes i=1 −1 −1 H H 2 then, this signal passes through the matched filters and by R = AcPcAc + AjPjAj + σnIMN −1 incorporating αî in αi we can write x˜j,i(τ) = 1 [25], [26]. (a) H 2 The received signal is then input to an adaptive filter with = AjcPjcAjc + σnIMN −1 weights vector, w, giving the output (b) −2 2 −1 H H = σn IMN − Ajc σnPjc + Ajc Ajc Ajc , y(τ) = wH x˜(τ). where in (a) we put Ajc = [Ac, Aj], and Pjc = diag (Pc, Pj), The weights vector that preserves the SOI, x˜s, while suppress- ing the clutter, jammers and noise, thus maximizing the output and (b) follows from the matrix inversion lemma. Now, we SINR, is obtained by solving the following optimization can reformulate the output SINR as 2 H σs H 2 −1 H −1 H min w Rw SINRout = a IMN − Ajc(σ P + A Ajc) A as. w 2 s n jc jc jc σn H s.t. w as = 1. Defining the matrices

This yields the solution [27] 2 −1 h 2 −1i Bjc = σnPjc , As = [as, Ajc], Bs = 0, σnPjc , −1 R as w = , (2) and making use of the determinant formula of block matrices, aH R−1a s s we obtain [17], [28] where R is the interference (jamming and clutter) plus noise aH a aH A covariance matrix of size MN × MN. Assuming the clutter, H s s s jc As As + Bs = H H jammers and noise are statistically independent, we have Ajc as Ajc Ajc + Bjc

= AH A + B R = Rc + Rj + Rn. jc jc jc −1 Taking the clutter scattering coefﬁcients, βi to be mutually H H H ×as IMN − Ajc Bjc + Ajc Ajc Ajc as. uncorrelated, we ﬁnd that

Nc Substituting this into the expression of SINRout yields H X 2 H H Rc = E{x˜c(τ)x˜c (τ)} = σc,iac,iac,i = AcPcAc , H 2 As As + Bs i=1 σs SINRout = 2 , where σn H Ajc Ajc + Bjc σ2 = {β β∗}, c,i E i i where |.| denotes the determinant and the signal-to-noise ratio, σ2 Ac = [ac,1, ac,2, ..., ac,Nc ],MN × Nc s SNR = 2 . This reveals that, although the SNR is constant, σn and the set of active transmit and receive elements directly affects the achieved output SINR by varying the interplay among P = diag σ2 , ..., σ2 . c c,1 c,Nc the jamming and clutter steering vectors and their powers. 4

Let us introduce a binary vector c with elements 0 if their output (stage 4) of Fig. 1. In this case, we reshape the selection corresponding elements are inactive and 1 otherwise. Then vector c as a matrix C the element selection can be incorporated into the expression of the output SINR as follows Tx1 Tx2 ··· TxM   σ2 Rx1 c1,1 c1,2 ··· c1,M SINR (c) = s h(c),   out σ2 Rx2  c2,1 c2,2 ··· c2,M  n =  , C .  . . . .  (5) .  . . .. .  where .  . . .  RxN cN,1 cN,2 ··· cN,M H As diag(c)As + Bs h(c) = . such that c = vec(C), and c is the entry that indicates AH diag(c)A + B i,j jc jc jc whether the j-th matched ﬁlter (which extracts the j-th waveform) in the i-th receiver is selected. The joint thinning mode The optimum set of k active elements that maximizes the SINR is obtained by directly selecting k elements in c. The selection is found via the following optimization: problem of (4) can now be written as

max SINRout(c) (3) c max f(c) 2 c s.t. ci − ci = 0 i = 1...MN, s.t. c2 − c = 0 i = 1...MN, cT c = k. i i cT c = k. The optimization in (3) is maximization of the volume of two ellipsoids. Therefore, we may employ log-determinant In this formulation, we place a constraint only on the number function as of output signals and any subset of the output matched filters is a possible solution. The joint selection finds the best subset, f(c) =logdet (h (c)) which decreases the dimensionality of the signal used in the H post processing (e.g. for detection, estimation or other tasks). =logdet As diag(c)As + Bs However, the entire system including transmitters, receivers, H − logdet Ajc diag(c)Ajc + Bjc . and matched filters should be active, and consequently the hardware cost and power consumption remain high. This problem can be effectively solved via a log-determinant relaxation and a sequential convex programming (SCP) procedure accordingly. We will elaborate on the solution approx- B. Factored Tx and Rx selection imation in Section IV. Considering (3), we propose four different methods to apply As all of the transmitters are required to be active in the selection in a MIMO radar. We list all the requirements in joint selection strategy, transmitters that do not contribute to different modes in a quadratic form. Hence, we cast the general the selected set of matched-filters effectively waste the power problem of antenna selection in a MIMO radar as follows alloted to them. This issue can be mitigated by factorizing the selection problem into transmit and receive sub-problems. max f(c) (4) c Suppose that we select kt out of M transmitting antennas T T (stage 1 in Fig. 1) and kr out of the available N receive c Wic + q c + ri ≤ 0, i = 1, ..., `. i antennas (stage 2 in Fig. 1). In terms of the selection matrix MN MN n (5), this strategy selects kr rows and kt columns. Then the where Wi ∈ S , qi ∈ R , and ri ∈ R. Note that S n factored selection involves the optimization of two selection denotes the set of n×n symmetric matrices, and R represents the set of real vectors of size n. vectors jointly, one for the transmitters and the other for the receivers. We now develop a novel way to reformulate this coupled problem in one unified formulation.

III.SELECTION STRATEGIES Let Vt = {vt,1, ..., vt,M }, and Vr = {vr,1, ..., vr,N } be a set of binary vectors each of which denoting a specific transmitter The selection strategy may be applied at each of the four or receiver in the selection matrix stages of a MIMO radar depicted in Fig. 1. In what follows, we detail and compare these selection approaches. vt,i = vec 0,..., 1 ,..., 0 Txi T ! A. Joint Tx-Rx selection vr,i = vec 0,..., 1 ,..., 0 Rxi The first selection strategy involves thinning the MIMO virtual array by selecting the individual matched-filters at the 5

Then, the factored selection problem may be expressed as (7). We revise the factored problem (7), by introducing new variables X, and Y as max f(c) c −1 −1 2 min logdet(X ) − logdet(Y ) s.t. ci − ci = 0 i = 1...MN, (6a) c T s.t. c Pt,ic ∈ {0, kr} i = 1...M, (6b) H T Λ : X = A diag(c)A + B (8a) c Pr,ic ∈ {0, kt} i = 1...N, (6c) s s s T ∆ : Y = AH diag(c)A + B (8b) c c = ktkr, (6d) jc jc jc µ : c (c − 1) = 0 i = 1...MN, (8c) where i i i T ν : c c = ktkr (8d) Pt,i = diag(vt,i), i = 1, ..., M, Pr,i = diag(vr,i), i = 1, ..., N. T λ : c Qc ≤ krkt(kr + kt) (8e) T In (6b) and (6c) we constrain the number of active elements ρi : c Pt,ic ≤ kr i = 1...M, (8f) in each column (row) to be exactly 0 or kt (0 or kr). T ηi : c Pr,ic ≤ kt i = 1...N, (8g) Now let us deﬁne Q as the rectangular matrix n+1 n with Lagrange multipliers Λ ∈ S (n = Nc + Nj), ∆ ∈ S , Q = PPT , where P = [v , ..., v , v , ..., v ] . t,1 t,M r,1 r,N µ ∈ RMN , ρ ∈ RM , η ∈ RN , and ν, λ ∈ R. We then Theorem 1: Let S be the set of selection vectors in conjunc- introduce the Lagrangian k tion with a factored selection problem comprising of t and Lfct(c, X, Y, Λ, ∆, µ, ν, λ, ρ, η) = kr out of M transmitters and N receivers respectively. Then −1 −1 S is given by logdet(X ) − logdet(Y ) + tr(XΛ) + tr(Y∆) M N T X X S = S1 ∩ S2 ∩ S3 ∩ S4, + c (diag(µ) + νI + λQ + ρiPt,i + ηiPr,i)c i=1 i=1 where the sets S1 - S4 are deﬁned as − Λ AH diag(c)A + B − ∆ AH diag(c)A + B T s s s jc jc jc S1 = {c | c Qc = krkt(kr + kt)} T T − µ c − ktkrν − λkrkt(kr + kt) S2 = {c | c Pt,ic ≤ kr i = 1...M, } M N S = {c | cT P c ≤ k i = 1...N} X X 3 r,i t − kr ρi − kt ηi. T S4 = {c | c c = ktkr}. i=1 i=1 Proof: See Appendix A. By rearranging the Lagrangian we get Like the constraint in (6a), the binary constraints involving Lfct(c, X, Y, Λ, ∆, µ, ν, λ, ρ, η) = the quadratic forms in (6b) and (6c) are non-convex. Therefore, logdet(X−1) − logdet(Y−1) + tr(XΛ) + tr(Y∆) we propose relaxing them by employing the following set of quadratic constraints instead − tr(BsΛ) − tr(Bjc∆) MN T X h iT h i h iT h i c Qc = krkt(kr + kt) T H T H + ci As Λ As + Ajc ∆ Ajc + µi i i i i T ≤ k i = 1...M i=1 c Pt,ic r | {z } ω cT P c ≤ k i = 1...N, i r,i t M N T X X Using Theorem 1, the factored selection problem becomes + c (diag(µ) + νI + λQ + ρiPt,i + ηiPr,i)c i=1 i=1 max f(c) c − ktkrν − λkrkt(kr + kt) M N s.t. ci(ci − 1) = 0 i = 1...MN, (7a) X X T − N ρi + M ηi. c c = ktkr (7b) i=1 i=1 cT Qc = k k (k + k ) (7c) r t r t We minimize Lfct with respect to c, X, and Y. Noting that T c Pt,ic ≤ kr i = 1...M, (7d) Lfct is a mixture of two volume covering ellipsoids in terms T c Pr,ic ≤ kt i = 1...N. (7e) of X, and Y (see p 222 in [29], and Appendix in [30]) and given the set of quadratic forms in c, we arrive at the Lagrange We recast the added binary constraints in the factored dual function gfct in (9). problem, into a quadratic form as a special case of (4). Theorem 2: Let f jnt be the optimal value associated with This enables us to compare the performance of the factored the joint selection (3) and f fct be that of the factored selection selection with that of the joint selection. We show that the fct fct (7). Also, let kr , kt be the number of selected transmitters optimum solution (i.e. SINR) obtained by the Lagrange dual and receivers in the factored selection, and kjnt the number of of the joint selection optimization is always greater than or jointly selected elements such that kjnt = kfctkfct. Then equal to that yielded by the factored problem. To this end, we r t derive a dual problem for the factored selection problem in f jnt ≥ f fct. (10) 6

gfct(Λ, ∆, µ, ν, λ, ρ, η) = inf Lfct(c, X, Y, Λ, ∆, µ, ν, λ, ρ, η)} c,X,Y  M N −1  logdet(Λ) + logdet(∆) + 2n + 1 − tr(ΛB ) − tr(∆B ) − 1 ωT diag(µ) + νI + λQ + P ρ P + P η P ω  s jc 4 i t,i i r,i  i=1 i=1  M N  P P  −ktkrν − λkrkt(kr + kt) − kr ρi − kt ηi = i=1 i=1  M N  if diag(µ) + νI + λQ + P ρ P + P η P 0, X 0, Y 0  i t,i i r,i  i=1 i=1  −∞ otherwise. (9) gjnt(Λ, ∆, µ, ν, λ, ρ, η) = inf Ljnt(c, X, Y, Λ, ∆, µ, ν, λ, ρ, η)} c,X,Y  M N −1  logdet(Λ) + logdet(∆) + 2n + 1 − tr(ΛB ) − tr(∆B ) − 1 ωT diag(µ) + νI + λQ + P ρ P + P η P ω  s jc 4 i t,i i r,i  i=1 i=1  M N  jnt jnt P P  −k ν − λk (M + N) − N ρi − M ηi = i=1 i=1  M N  if diag(µ) + νI + λQ + P ρ P + P η P 0, X 0, Y 0  i t,i i r,i  i=1 i=1  −∞ otherwise. (12)

Proof: Let us recast the joint problem as Therefore, by subtracting (9) and (12) we get

fct jnt jnt fct fct max f(c) g − g = λ k (M + N) − (kr + kt ) c M N s.t. X fct X fct + ρi N − kr + ηi M − kt µi : ci(ci − 1) = 0 i = 1...MN, (11a) i=1 i=1 ν : cT c = kjnt (11b) ≥ 0 λ : cT Qc ≤ kjnt(M + N) (11c) T ρi : c Pt,ic ≤ N i = 1...M, (11d) The factored selection operates on a subset of solutions that is included in the joint selection, and hence may not achieve η : cT P c ≤ M i = 1...N, (11e) i r,i the same optimal solution that is guaranteed by joint selection. We can also reformulate (11) in terms of the equivalent min- Nonetheless, selecting a subset of transmitters allows the avail- imization like (8) and derive the Lagrange dual function gjnt able total transmit power to be allocated only to the chosen as in (12). By minimization reformulation and employing f¯jnt, elements. This is in contrast to the joint selection problem and f¯fct as the corresponding optimal values, the expression where all transmitters must be operational to guarantee that in (10) can be transformed into all matched ﬁlters are available for selection. Thus, assuming a total available transmit power Pe = PT , the transmit power Pt Pt jnt fct per element in the factored case is Pe = as opposed to f¯ ≤ f¯ . kt M for the joint selection case. It is important to note, however that Given the upper bounds in (9) and (12), we show (10) by increasing the allocation of transmit power per element may equivalently proving that be restricted by the hardware limitations of the components in the RF chain, such as ampliﬁer linear range. This may limit gjnt ≤ gfct. the gain achievable by the factored approach.

Now we have that C. Matched Filter Constrained Selection We can adjust the transmitter power and SNR by a factored Q 0, Pt,i 0, i = 1, ..., M , Pr,i 0, i = 1, ..., N. selection. Moreover, the number of receivers is decreased, which leads to a considerable hardware reduction. Since the Also, the Karush-Kuhn-Tucker (KKT) conditions [29] imply number of transmitters is reduced in a factored selection, the that spatial diversity is reduced significantly [31]. To preserve the spatial diversity provided by MIMO arrays but still reduce λ ≥ 0, ρ ≥ 0, η ≥ 0. hardware and computation overheads, we propose restricting 7 the number of matched filters in each receiver, as well as the where card(x) denotes the cardinality, the number of non-zero number of receivers, in a matched filter constrained (MFC) elements of vector x. We first relax (14c) via (15d), and (15f) selection strategy [32]. Using this, we decrease the number of in the following optimization RF front-ends on the receive end (stage 2 in Fig.1) as well as the required processing blocks in DSP (stage 3 in Fig.1). max f(c) c We specify the MFC selection to select km matched filters 2 s.t. c − ci = 0 i = 1...MN, (15a) in kr receivers as follows i T c c = kmkr, (15b) max f(c) T c card c Pr ≤ kt, (15c) 2 T 2 s.t. ci − ci = 0 i = 1...MN, (13a) c Qrc = kmkr (15d) T T c c = kmkr, (13b) c Pt,ic ≤ kr i = 1...M, (15e) T 2 T c Qrc = kmkr, (13c) c Pr,ic ≤ km i = 1...N, (15f) T c Pt,ic <= kr i = 1...M, (13d) T where Pt is c Pr,ic <= km i = 1...N, (13e) where Qr is Pt = [vt,1, ..., vt,M ] .

T Qr = PrPr , Pr = [vr,1, ..., vr,N ]. The cardinality constraint (15c), is a nonconvex constraint. The best relaxation strategy for a cardinality or norm-0 constraint Comparing (13) with (7) makes it obvious that the factored is a norm-1 constraint [33], which is already met by (15b). selection is a special case of the MFC selection. Therefore, To tackle this constraint, we employ alternative constraints for concision we state the following theorem without proof. described in the following theorem. mfc Theorem 3: Let f be the optimal value of the MFC Theorem 4: Let c be the selection vector associated with jnt selection (13) and f the optimal value of the joint selection the hybrid selection satisfying (15a) to (15b). Then, we can fct (11), and f the optimal value of the factored selection (7). replace (15c) by the following inequality constraints mfc Moreover, let km be the number of selected matched-filters mfc fct fct and receivers in MFC selection , kr , let kr , kt be the 2 (krkm) T 2 number of selected transmitters and receivers respectively, in ≤ c Qtc ≤ k km, k r factored selection, and kjnt be the number of jointly selected t jnt mfc mfc fct fct elements such that k = km kr = kr kt . Then T where Qt = PtPt . f jnt ≥ f mfc ≥ f fct. Proof: See Appendix B. We relax (15c) through (16c), and ultimately reformulate the hybrid selection as follows D. Hybrid Selection max f(c) By the giving up of spatial diversity described for factored c 2 selection , there is a benefit for transmit power and selecting s.t. ci − ci = 0 i = 1...MN, (16a) the optimum subset of matched filters with a decrease in the T c c = kmkr (16b) number of transmitters and receivers. On the other hand, in 2 (krkm) T 2 MFC selection we maintain spatial diversity at the expense ≤ c Qtc ≤ kr km, (16c) of transmit power. In this subsection, we propose integrating kt T 2 these two methods into a hybrid algorithm by which the c Qrc = kmkr, (16d) T MIMO array is entirely controlled. In a MIMO radar compris- c Pr,ic ≤ km i = 1...N, (16e) ing M transmitters and N receivers, the hybrid selection finds T c Pt,ic ≤ kr i = 1...M. (16f) the optimum subset including k = krkm matched filters such that exactly k out of k active transmitters are used in each m t In the following theorem, we study the performance of the k receivers. By adopting the same methodology introduced r hybrid selection with respect to the rest of the modes. in (6), we define the hybrid selection as Theorem 5: Let f fct be the optimal value of the factored hyb hyb max f(c) selection (7), f the MFC selection (13), and f the optimal c value of the hybrid selection (16). Also, let kfct, kfct be 2 r t s.t. ci − ci = 0 i = 1...MN, (14a) the number of selected transmitters and receivers in factored T mfc mfc c c = kmkr, (14b) selection, km , kr the number of selected matched filters khyb khyb khyb cT P c ∈ {0, k } i = 1...M, (14c) and receivers in MFC selection, and m , r , t the t,i r number of selected matched filters, receivers, and the number T card c Pr ≤ kt, (14d) of allowed transmitters in hybrid selection. Moreover, suppose 8 that k elements are selected in each selection strategy subject to the following conditions Joint selection Hybrid selection fct fct mfc mfc hyb hyb MFC selection A : k = kr kt = km kr = km kr Factored selection fct mfc hyb B : kr = kr = kr fct mfc hyb C : kt = km = km hyb hyb D : km ≤ kt Then, fct hyb mfc Fig. 2. Geometric illustration of feasible regions and relationship of different f ≤ f ≤ f selection methods. Proof: We can rewrite (7) as max f(c) (17a) E. Performance and Complexity Analysis c s.t. (17b) By Theorem 3 , and Theorem 5, we conclude the following, 2 in terms of the output SINR, µi : ci − ci = 0 i = 1...MN, (17c) T ν : c c = ktkr (17d) f fct ≤ f hyb ≤ f mfc ≤ f jnt, T 2 λ : c Qrc = kt kr (17e) Also, based on this conclusion the feasible regions for different κ : cT Q c = k k2 (17f) t t r selection modes given by intersection of positive semidefinite T ρi : c Pt,ic ≤ kr i = 1...M, (17g) cones can be illustrated as in Fig. 2. T ηi : c Pr,ic ≤ kt i = 1...N, (17h) The computational complexity of the minimum volume ellipsoid problem when solving via barrier-generated path- while, we revise (14) as: following interior-point method with n variables and m con- max f(c) straints can be approximated by O m2.5(n2 + m) (see The- c orem 6.5.1 in [34]). We can represent the joint selection with s.t. (18a) at least mjnt = 2 constraints. However, the factored and MFC 2 fct mfc µi : ci − ci = 0 i = 1...MN, (18b) selections need at least m = m = 3+M +N constraints, T fct mfc ν : c c = ktkr (18c) and the hybrid selection requires m = m = 5 + M + N constraints. Hence, the computational complexities can be λ : cT Q c = k2k (18d) r t r summarized as T 2 κ : c Qtc = ktkr (18e) T 2 jnt fct mfc hyb σ : c Qtc < kr kt (18f) O ≤ O = O ≤ O . 2 T (krkm) τ : c Qtc ≥ (18g) kt IV. RELAXATION AND APPROXIMATION T ρi : c Pt,ic ≤ kr i = 1...M, (18h) T In the previous section, we expressed the selection methods ηi : c Pr,ic ≤ km i = 1...N. (18i) as a difference log-determinant maximization with quadratic By computing the Lagrangian dual function of (17a), and (18) constraints. Although we decoupled the factored problem in we get subsection III-B, and relaxed the norm constraints in sub- 2 section III-D, the problem still is not tractable due to the hyb fct 1 T 2 (krkm) g − g = − ω ((σ + τ) Q ) ω − σk kt − τ . nonconvexity of (a) a difference of concave functions, (b) 4 t r k t quadratic equalities, e.g. (7a)- (7c), and (c) the quadratic Due to the KKT conditions, we note that inequality from constructing a nonconvex set, e.g. (18g). σ ≥ 0, τ ≥ 0. To handle these nonconvexities, we propose a variation of sequential convex programming (SCP) with an exact penalty Therefore, approach [35]. Let the primal problem be defined as ghyb−gfct ≤ 0, max f(c) c and consequently s.t. ci ∈ {0, 1} i = 1...MN, f fct ≤ f hyb. g1(c) ≥ 0 hyb mfc Following the same process for comparing f and f gi(c) ≤ 0, i = 1, ..., `, yields ei(c) = 0, i = 1, ..., . f hyb ≤ f mfc. Where f(c) is a concave-concave function, g(c), and e(c) represent quadratic functions. By relaxing the binary constraint 9 and using a convex local approximation we specify the prob- Algorithm 1: Structured binary rounding lem in the `-th iteration as 1 switch Selection method do  2 case Joint selection do ˆ (l) X (l) ∗ max f(c ) + ψ ei(c ) (19) 3 ˆ (l) sort z in descending order c jnt i=1 4 round the first k elements to one and the (i) s.t. 0 ≤ ci ≤ 1 i = 1...MN, remaining elements to zero (l) gˆ1(c ) ≥ 0 5 case Factored selection do ∗ (l) 6 reshape ˆz as a MIMO selection matrix gi(c ) ≤ 0, i = 1, ..., `. 7 calculate sum of the rows, and columns. Then, Where we use a first Taylor expansion to get an affine sort it in descending order fct fct approximation over the trust region of a box around the current 8 round the first kr rows (receivers), and kt point. Employing this for f(c) we will have columns (transmitters) to one and the rest to zero ˆ (l) (l) (l) (l) (l) (l−1) f(c ) = f1(c ) − f2(c ) − Of2(c )(c − c ) 9 case MFC selection do ∗ H (l) 10 reshape ˆz as a MIMO selection matrix = logdet As diag(c )As + Bs 11 calculate sum of the rows, and columns. Then, H (l−1) − logdet Ajc diag(c )Ajc + Bjc sort it in descending order 12 −1 sort rows descending H H (l−1) mfc mfc − diag as Ajc diag c Ajc + Bjc as . 13 round the first km elements in the first kr fct rows (receivers), and kt columns (transmitters) We also use an affine approximation for g1(c) to tackle the to one and the rest to zero nonconvex constraint 14 case Hybrid selection do ∗ 15 reshape ˆz as a MIMO selection matrix (k) (k) (k) (k) (k−1) gˆ1(c ) = g1(c ) + Og1(c )(c − c 16 calculate sum of the columns. Then, sort it in (k)T (k) T (k) (k) (k−1) descending order = c W1c + r1 + W1 + W1 c c − c 17 sort rows descending mfc mfc After solving the relaxed version of the problem by an SCP 18 round the first km elements in the first kr procedure, we find the suboptimal solution of the original rows (receivers), to one and the rest zero problem by an appropriate randomized rounding strategy. Let (c?) be the optimal value obtained by relaxation and SCP. Also, assume z ∈ RMN is a Gaussian variable with This optimization is another form of binary programming. distribution z ∼ N (µ, Σ). Then, the following maximization Heuristically, we approximate this problem in two successive max E (f(c)) steps. We first find the projected sample by applying a relax- c ation as follows s.t. E (ci) ∈ {0, 1} i = 1...MN, min kˆz − zk E (g1(c)) ≥ 0 ˆz E (gi(c)) ≤ 0, i = 1, ..., `, s.t. 0 ≤ ci ≤ 1 i = 1...MN, E (hi(c)) = 0, i = 1, ..., , gi(c) ≤ 0, i = 1, ..., `, h (c) ≤ 0, i = 1, ..., . is solved by z for µ = c?. As for Σ, we use Σ = i ∗ diag ([var(ci)]) where ci is a vector comprising the sequence Then the best feasible projected sample ˆz is obtained through of solutions of (19). We take a sample from z for a sufficent successive evaluations of the cost function. We then employ a number of times and and keep the best sample yielding the structured rounding strategy to round ˆz∗ to the nearest binary maximum value. Nevertheless, this direct sampling does not point, while still meeting the structure of the selection vector immediately provide a feasible point regarding the embedded for different selection methods. The corresponding structured quadratic constraints. Hence, we need to project the direct rounding for each selection method is listed in Algorithm 1, sampled vector onto the constraints feasible set. To do this we while the final algorithm is summarized in Algorithm 2. add one more step after sampling, where we find the projected point ˆz via the following optimization V. SIMULATION min kˆz − zk (20) To evaluate the performance of the proposed selection ˆz methods, we use a uniform linear collocated MIMO phased s.t. ci ∈ {0, 1} i = 1...MN, array comprising 5 transmitters (M = 5), and 5 receivers λ g1(c) ≥ 0 (N = 5). We use the inter-element spacing of dr = 2 for the elements in the receive array. To maintain a non-overlapping g (c) ≤ 0, i = 1, ..., `, i virtual array and maximum aperture we place the transmit hi(c) = 0, i = 1, ..., . antennas dt = Ndr apart. We include two jamming signals 10

Joint Selection with azimuth angles of θj = [20, 50] degrees and powers of 36 MVDR 13dBW. We vary the azimuth angle of the received signal, θs 34 Exhausive search ◦ ◦ Optimal rounding from 0 to 90 , while the power is fixed at 20 dBW. Also, 32 ◦ we assume that clutter is contributed from angles between 0 30 ◦ and 90 , and we use a low rank model with rank 5 and a total (dB) 28 out clutter-to-noise ratio of 13 dB. We define the problem such 26 that 12 elements are selected in total according to different SINR 24 jnt selection methods, such that for joint selection k = 12, and 22 kfct = 3, kfct = 4 for the factored selection t r . The selection 20 criteria for MFC and hybrid selections are kmfc = 3, kmfc = 4 m r 18 hyb hyb hyb 0 10 20 30 40 50 60 70 80 90 and k = 4, k = 3, k = 4. The maximum SINRout t m r Azimuth is first calculated by an exhaustive search for each of the (a) strategies and the suboptimal solution is then obtained from Factored Selection Algorithm 2. For each direction, we used the SDPT3 solver 36 MVDR embedded in CVX [36] to solve (19) for 10 iterations, and then 34 Exhausive search Exhausive search with power adjustment proceeded to take 1000 samples from a random distribution as 32 Optimal rounding Optimal rounding with power adjustment described in Algorithm 2. The achieved values for SINRout 30 resulting from the exhaustive search and proposed method (dB) 28 out are depicted in Figs. 3(a)-3(d) along with the corresponding 26

SINR for the full array achieved by an MVDR beamformer. SINR out 24 The approximated value is close to the exhaustive search, 22 confirming that a good approximation ratio is achieved by 20 the proposed relaxation strategy. For the factored and hybrid 18 selection two sets of curves are shown in Figs. 3(b), 3(d) to 0 10 20 30 40 50 60 70 80 90 Azimuth demonstrate the effect of the power adjustment. The additional (b) transmit power afforded by allocating the total available power MFC Selection to fewer transmit waveforms improves the SINRout signifi- 36 MVDR cantly in these modes of selection. 34 Exhausive search Optimal rounding Next, we show in Fig. 4 the selected elements yielded by 32 each of the selection strategies. In this example, we fix the 30 ◦ impinging signal direction at θs = 18 . We can see from (dB) 28 out Fig. 4(a) that all of the transmitters and receivers must be 26 operational in the joint selection in order to be able to select SINR 24 from the corresponding matched-filters. On the other hand, 22 exactly two transmitters and two receivers were deactivated in 20 factored selection as shown in Fig. 4(b). For MFC selection 18 three matched filters are selected precisely at four receivers. 0 10 20 30 40 50 60 70 80 90 Azimuth While in hybrid selection the three matched filters are selected (c) only from 4 possible options as Tx-1 is deactivated. Hybrid Selection We now compare the performances of the selection meth- 36 MVDR ods. In Fig. 5 we plot the SINRout achieved by the different 34 Exhausive search Exhausive search with power adjustment selection methods via exhaustive search, While in Fig. 6 32 Optimal rounding Optimal rounding with power adjustment the optimal values are shown for the proposed optimization. 30

Ignoring the power adjustment option, we see in both ﬁgures (dB) 28 out that joint selection achieves the highest SINRout followed 26 by MFC, hybrid, and then factored selection, which veriﬁes SINR 24

Algorithm 2: Optimal Randomized Algorithm 18 0 10 20 30 40 50 60 70 80 90 ? 1 Solve SCP relaxed problem (19) to get c Azimuth ∗ ∗ 2 Initialize the best point ˆz := 0 and f := 0 (d) 3 for each iteration n do 4 Random sampling z ∼ N (µ, Σ) Fig. 3. SINRout value estimated by MVDR beamformer in the full array 5 Solve (20) and get ˆz vs. the optimum value achieved by exhaustive search, and the proposed best 6 if f(ˆz) ≥ f then optimization for (a) Joint selection, (b) Factored selection, (c) MFC selection, best ∗ and (d) Hybrid selection. 7 f = f(ˆz) and ˆz = ˆz

8 Employ Algorithm 1 to obtain z 11

(a) Joint selection

Tx-1 Rx-1 Tx-2 Rx-2 Tx-3 Rx-3 Tx-4 Rx-4 Tx-5 Rx-5 (b) Factored selection

Tx-1 Rx-1 Tx-2 Rx-2 Tx-3 Rx-3 Tx-4 Rx-4 Tx-5 Rx-5 (c) MFC selection

Tx-1 Rx-1 Tx-2 Rx-2 Tx-3 Rx-3 Tx-4 Rx-4 Tx-5 Rx-5 (d) Hybrid selection

Tx-1 Rx-1 Tx-2 Rx-2 Tx-3 Rx-3 Tx-4 Rx-4 Tx-5 Rx-5 Active antenna Deactive antenna Active matched-filter Deactive matched-filter

Fig. 4. Illustration of the output selection in different selection modes.

Exhaustive search 36 26 to 5 and the clutter rank to 10. The five jamming signals have MVDR 24 Joint selection 34 azimuth angles θj = [0, 5, 15, 25, 30] degrees and powers of Factored selection 22 Factored selection with power adjustment 13dBW, and the total clutter-to-noise ratio is set to 13 dB. The 32 MFC selection Hybrid selection 20 SOI is assumed to have a power of 20dBW at the reciever 30 Hybrid selection with power adjustment ◦ ◦ 18 and its azimuth is varied between 0 to 30 . We select 54 46 48 50 52 54 56 58 jnt (dB) 28 elements in total such that for joint selection k = 54, and out fct fct 26 for the factored selection kt = 6, kr = 9. The selection SINR mfc mfc 24 criteria for MFC are km = 6, kr = 9 and for the hybrid strategy khyb = 7, khyb = 6, khyb = 9. The optimal values 22 t m r achieved by the proposed optimization are depicted in Fig. 7. 20 Firstly, observe the excellent performance achieved for a larger 18 0 10 20 30 40 50 60 70 80 90 array with a denser interference environment. The proposed Azimuth selection strategies exhibit similar trends to the previous, smaller example, with the joint as well as factored and hybrid Fig. 5. SINRout value estimated by MVDR beamformer in the full array vs. the optimum value achieved by exhaustive search in different selection strategies (with power adjustment) being comparable to the modes. Optimal rounding MVDR with almost half the number of elements. 36 26 MVDR Joint selection In the following example, we again use the 5 × 5 array 34 24 Factored selection ◦ Factored selection with power adjustment of the first example. We fix the signal direction θs = 18 MFC selection 32 22 Hybrid selection and solve the selection problem for an increasing subset of Hybrid selection with power adjustment 30 20 antennas ranging from 2 to 25. The preset values for all the

18 46 48 50 52 54 56 58 selection methods are listed in Table I. As is revealed in Fig. 8,

(dB) 28

out the joint selection outperforms the rest of selection methods 26

SINR when power adjustment is not employed, followed by MFC, 24 hybrid and factored approaches. When the transmit power is 22 adjusted, we see that the factored and hybrid approaches are 20 able to achieve a higher SINRout. Also, notice that the output

18 SINR given by the selection remains comparable with that of 0 10 20 30 40 50 60 70 80 90 Azimuth the full array even when a signiﬁcantly smaller number of pairs are used. For instance selecting 15 out of 25 elements Fig. 6. SINRout value estimated by MVDR beamformer in the full array vs. would substantially reduce the dimensionality and hardware the optimum value achieved by proposed optimization in different selection modes. cost but would result in a SINRout loss with respect to the full array of less than 0.5 dB for joint selection and 2.24 dB for factored selection (when the power is not adjusted). Theorems (2) and (5). However, when the power adjustment Adjusting the transmit power, however, can have a signiﬁcant for factored and hybrid selections is included, the increased effect giving, even showing an improvement over the full array. transmit signal power means that the factored and hybrid selections surpass the joint mode. VI.CONCLUSION The above example employed a 5 × 5 array to permit us to In this paper, we formulated the antenna selection in MIMO compare the performance to the exhaustive search. Now we arrays with beamforming to mitigate interference signals in demonstrate the performance with a larger array such that the the form of multiple jamming signals and clutter. We devised exhaustive search is not possible. To this end, we employ a four different selection approaches to control different aspects 10×10 array. Additionally, we increase the number of jammers of a MIMO system. We cast the problem as a determinant 12

TABLE I relaxation and approximation method to tackle the noncon- THENUMBEROFSELECTEDELEMENTSINEACHSELECTIONMETHODFOR vexity of the primal problem. Finally, we presented extensive THE RESULTS SHOWN IN FIG. 8 simulations that verified the theoretical findings and confirmed fct fct mfc mfc hyb hyb hyb kjnt kt kr km kr kt kr km the effectiveness of the proposed techniques in reducing the 2 1 2 1 2 4 2 1 problem dimensionality while maintaining a performance that 3 1 3 1 3 4 3 1 is comparable to the full array. 4 2 2 2 2 4 2 2 5 1 5 1 5 4 5 1 6 2 3 2 3 4 3 2 APPENDIX A 8 2 4 2 4 4 4 2 PROOFOF THEOREM 1 9 3 3 3 3 4 3 3 10 2 5 2 5 4 5 2 Proof: For an arbitrary selection vector we can extend S1 12 3 4 3 4 4 4 3 under S2 and S3 as 15 3 5 3 5 4 5 3 16 4 4 4 4 4 4 4 M+N kt−1 kr−1 T X T 2 X 2 X 2 20 4 5 4 5 4 5 4 c Qc = c [P]i = i(kt − i) + ζi(kr − i) , 25 5 5 5 5 5 5 5 i=1 i=0 i=0 (21) ∗ Optimal rounding for 10 10 array where [P]i denotes the i-th column of P, and i, ζi ∈ Z , MVDR 29 Joint selection are appropriate integers equal to the number of times the Factored selection Factored selection with power adjustment ∗ MFC selection corresponding squared value appears. Here Z denotes the set 28 Hybrid selection Hybrid selection with power adjustment of non-negative integers. 27 Now, we can establish a set of equations based on S1 to S4

(dB) 26 and apply (21) as follows out kt−1 SINR 25 2 X 2 2 S1 : 0kt + i(kt − i) + ζ0kr (22) 24 i=1

kr−1 23 X 2 2 2 + ζi(kr − i) = krkt + ktkr (23) 22 0 5 10 15 20 25 30 i=1 Azimuth S2 : i ≤ kr i = 0, ..., kt − 1 (24) S : ζ ≤ k i = 0, ..., k − 1 (25) Fig. 7. The optimum SINRout value achieved by the proposed optimization 3 i t r with varying number of selected elements in different selection modes for a kt−1 10 × 10 array X S4 : i(kt − i) = ktkr (26) 22 i=0 21 k −1 Xr 20 S4 : ζi(kr − i) = ktkr. (27)

19 i=0

(dB) 18 Given the above set of equations, we need to show that the out 17 only non-trivial solution is SINR 16 0 = kr and i = 0 for i = 1, ..., kt − 1 Joint selection 15 Factored selection ζ = k and ζ = 0 for i = 1, ..., k − 1. Factored selection with power adjustment 0 t i r 14 MFC selection Hybrid selection Hybrid selection with power adjustment Based on (26) and (27) we can write 13 2 3 4 5 6 8 9 10 11 12 15 16 20 25 kt−1 kr−1 Cardinality X (kt − i) X (kr − i) 0 = kr − i , ζ0 = kt − ζi . kt kr Fig. 8. The optimum SINRout value achieved by the proposed optimization i=1 i=1 with varying number of selected elements in different selection modes. Then, we extend (22) as

kt−1 kt−1 X (kt − i) 2 X 2 (kr − i )kt + i(kt − i) maximization problem with quadratic constraints to decouple kt the problem and optimize three joint vectors separately in i=1 i=1 kr−1 kr−1 a uniﬁed problem. Since the selection strategies return the X (kr − i) X + (k − ζ )k2 + ζ (k − i)2 = k k2 + k k2. optimal subarrays for the given scenario compared to any arbi- t i k r i r r t t r i=1 r i=1 trary subarray, the maximum output SINR is always achieved. We then presented a theoretical study demonstrating that the Hence, we have joint selection gives the optimum solution followed by the kt−1 kr−1 X X suboptimal solutions offered by MFC, hybrid and factored ii (i − kt) + ζii (i − kr) = 0. (28) modes respectively. Moreover, we proposed an appropriate i=1 i=1 13

Noting that [9] X. Wang, M. Amin, and X. Cao, “Analysis and Design of Optimum Sparse Array Conﬁgurations for Adaptive Beamforming,” IEEE Trans. i ≥ 0, ζi ≥ 0, i ≥ 1, (i − kt) < 0, (i − kr) < 0, Signal Process., vol. 66, no. 2, pp. 1–1, 2017. [10] M. Gharavi-Alkhansari and A. B. Gershman, “Fast antenna subset It follows easily that (28) holds if and only if selection in MIMO systems,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 339–347, 2004. i = 0 for i = 1, ..., kt − 1, ζi = 0 for i = 1, ..., kr − 1, [11] A. Gorokhov, “Antenna selection algorithms for MEA transmission systems,” IEEE International Conference on Acoustics Speech and and accordingly Signal Processing, no. 2, pp. III–III, 2002. [12] I. Berenguer and V. Krishnamurthy, “Adaptive MIMO antenna selection = k , ζ = k . via discrete stochastic optimization,” IEEE Trans. Signal Process., 0 r 0 t vol. 53, no. 11, pp. 4315–4329, 2005. [13] R. Mendez-Rial, C. Rusu, N. Gonzalez-Prelcic, A. Alkhateeb, and R. W. Heath, “Hybrid MIMO Architectures for Millimeter Wave Communica- tions: Phase Shifters or Switches?” IEEE Access, vol. 4, pp. 247–267, APPENDIX B 2016. [14] Y. Liu, Q. H. Liu, and Z. Nie, “Reducing the number of elements in PROOFOF THEOREM 4 multiple-pattern linear arrays by the extended matrix pencil methods,” Proof: We introduce the quadratic function e(y), y ∈ Rkt IEEE Trans. Antennas Propag., vol. 62, no. 2, pp. 652–660, feb 2014. [15] W. P. M. N. Keizer, “Linear array thinning using iterative FFT tech- as niques,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2757–2760,

M kt aug 2008. X 2 X e(y) = cT Q c = cT P PT c = cT [P ] = y2. [16] X. Wang, E. Aboutanios, and M. G. Amin, “Adaptive array thinning for t t t t i i enhanced DOA estimation,” IEEE Signal Process. Lett., vol. 22, no. 7, i=1 i=1 pp. 799–803, July 2015. To find the lower-bound, we solve the following minimization [17] A. Deligiannis, M. Amin, S. Lambotharan, and G. Fabrizio, “Optimum sparse subarray design for multitask receivers,” IEEE Trans. Aerosp. Electron. Syst., pp. 1–1, 2018. [18] H. Nosrati, E. Aboutanios, and D. Smith, “Array partitioning for multi- min e(y) task operation in dual function MIMO systems,” Digital Signal Process., y vol. 82, pp. 106–117, nov 2018. T s.t. 1k y = krkm. [19] Q. He, R. S. Blum, H. Godrich, and A. M. Haimovich, “Target t velocity estimation and antenna placement for MIMO radar with widely Solving this problem using the Lagrangian method yields separated antennas,” IEEE J. Sel. Topics Signal Process., vol. 4, no. 1, pp. 79–100, feb 2010. k k [20] H. Godrich, A. P. Petropulu, and H. V. Poor, “Sensor selection in y = r m , i = 1, ..., k i k t distributed multiple-radar architectures for localization: A knapsack t problem formulation,” IEEE Trans. Signal Process., vol. 60, no. 1, pp. and subsequently the lower-bound is given by 247–260, jan 2012. [21] A. A. Gorji, R. Tharmarasa, and T. Kirubarajan, “Optimal antenna 2 (krkm) allocation in MIMO radars with collocated antennas,” IEEE Trans. e(y) = . Aerosp. Electron. Syst., vol. 50, no. 1, pp. 542–558, jan 2014. kt [22] B. Ma, H. Chen, B. Sun, and H. Xiao, “A joint scheme of antenna selection and power allocation for localization in MIMO radar sensor The upper-bound is achieved by letting km = kt yielding networks,” IEEE Commun. Lett., vol. 18, no. 12, pp. 2225–2228, dec 2 2014. e(y) = k km. r [23] C. Y. Chen and P. P. Vaidyanathan, “Minimum redundancy MIMO radars,” in Proceedings - IEEE International Symposium on Circuits and Systems. IEEE, 5 2008, pp. 45–48. [24] J. Dong, Q. Li, and W. Guo, “A combinatorial method for antenna array REFERENCES design in minimum redundancy MIMO radars,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1150–1153, 2009. [1] E. Larsson, O. Edfors, F. Tufvesson, and T. Marzetta, “Massive MIMO [25] Y. Li, S. A. Vorobyov, and A. Hassanien, “MIMO radar capability on for next generation wireless systems,” IEEE Commun. Mag., vol. 52, powerful jammers suppression,” in 2014 IEEE International Conference no. 2, pp. 186–195, 2 2014. on Acoustics, Speech and Signal Processing (ICASSP). IEEE, may [2] J. Mietzner, R. Schober, L. Lampe, W. Gerstacker, and P. Hoeher, 2014. “Multiple-antenna techniques for wireless communications - a compre- [26] P. P. Vaidyanathan and P. Pal, “MIMO radar, SIMO radar, and IFIR hensive literature survey,” IEEE Communications Surveys & Tutorials, radar: a comparison,” in 2009 Conference Record of the Forty-Third vol. 11, no. 2, pp. 87–105, 2009. Asilomar Conference on Signals, Systems and Computers. IEEE, 2009. [3] J. Li and P. Stoica, “MIMO Radar Diversity Means Superiority,” in [27] H. L. Van Trees, Optimum Array Processing, 2002, vol. Part IV, no. S1. MIMO Radar Signal Processing. Hoboken, NJ, USA: John Wiley & [28] X. Wang, E. Aboutanios, and M. G. Amin, “Slow radar target detection Sons, Inc., ch. 1, pp. 1–64. in heterogeneous clutter using thinned space-time adaptive processing,” [4] A. Hassanien and S. A. Vorobyov, “Why the phased-mimo radar IET Radar, Sonar & Navigation, vol. 10, no. 4, pp. 726–734, apr 2016. outperforms the phased-array and MIMO radars,” in European Signal [29] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge Processing Conference, vol. 58, no. 6, 6 2010, pp. 1234–1238. University Press, 2004. [5] D. Bliss and K. Forsythe, “Multiple-input multiple-output (MIMO) radar [30] S. Joshi and S. Boyd, “Sensor selection via convex optimization,” IEEE and imaging: degrees of freedom and resolution,” Conference Record Trans. Signal Process., vol. 57, no. 2, pp. 451–462, 2009. of the Thirty-Seventh Asilomar Conference on Signals, Systems and [31] H. Nosrati, E. Aboutanios, and D. Smith, “Receiver-Transmitter Pair Computers, vol. 1, pp. 54–59, 2003. Selection in MIMO Phased Array Radar,” The 42nd IEEE International [6] A. F. Molisch and M. Z. Win, “MIMO systems with antenna selection,” Conference on Acoustics, Speech and Signal Processing, pp. 3206–3210, IEEE Microw. Mag., vol. 5, no. 1, pp. 46–56, 2004. 2017. [7] R. Heath, S. Sandhu, and A. Paulraj, “Antenna selection for spatial [32] H. Nosrati, E. Aboutanios, and D. B. Smith, “Matched filter constrained multiplexing systems with linear receivers,” IEEE Commun. Lett., vol. 5, MIMO array spatial thinning for interference mitigation,” in 2017 IEEE no. 4, pp. 142–144, apr 2001. Radar Conference (RadarConf). IEEE, may 2017. [8] X. Wang, E. Aboutanios, M. Trinkle, and M. G. Amin, “Reconfigurable [33] S. Boyd, l1-norm methods for convex cardinality problems. Lecture Adaptive Array Beamforming by Antenna Selection,” IEEE Trans. Notes for EE364b, Stanford University. Available at http://www. stan- Signal Process., vol. 62, no. 9, pp. 2385–2396, 5 2014. ford. edu/class/ee364b, 2007. 14

[34] Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in procedure,” Optimization and Engineering, vol. 17, no. 2, pp. 263–287, Convex Programming. Society for Industrial and Applied Mathematics, nov 2015. jan 1994. [36] M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab software for disciplined [35] T. Lipp and S. Boyd, “Variations and extension of the convex–concave convex programming,” 2008.