A RLT relaxation via Semidefinite cut for the detection of QPSK signaling in MIMO channels

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1 A RLT relaxation via Semidefinite cut for the detection of QPSK signaling in MIMO channels Rupaj Kumar Nayak Harish Kumar Sahoo Abstract A detection technique of quadrature phase shift keying (QPSK) signaling in multiple-input, multiple-output (MIMO) channel is considered, based on several convex relaxation of the maximum likelihood (ML) problem. It is because many ML detectors are, in general, intractable optimization problems. The ML problem is formulated as a non-convex nonlinear optimization model and a tight Semidefinite (SDP) based relaxations coupled with a new SDP formulation for boolean quadratic programming (BQP) embedded with reformulation linearization technique (RLT) is presented for solving non-convex programming problems. In essence, this process yields a new BQP formulation along with an RLT relaxation that is combined with valid inequalities called as triangle inequality, which further tightened the relaxation is proposed. Simulation results designate that the proposed cutting plane method provides a paramount tightening of the lower bound obtained by utilizing RLT along with triangle inequality that enhances the detection of QPSK signaling in MIMO channel. Keywords: Multiple-input Multiple-output, Maximum likelihood detection, Semidefinite relaxation, reformulation linearization technique, quadratically constrained quadratic programming. 1 Introduction Multiple input and Multiple output (MIMO) wireless communications systems uses multiple antennas to increase the data rates and improved link reliability without additional bandwidth [4, 27]. Detection of QPSK symbols in presence of inter symbol interference and additive channel noise is addressed by different researchers using different signal processing techniques. Maximum likelihood (ML) detection of transmitted symbols require mathematical formulation of likelihood or log-likelihood functions. But, the performance Corresponding author. Department of Mathematics, IIIT Bhubaneswar, India, ; rupaj@iiit-bh.ac.in Department of Electronics and Telecommunication, VSSUT, Burla, ; harish sahoo@yahoo.co.in of ML technique is limited because it requires a recursive formulation of likelihood function. Despite rich theoretical support, ML detectors are, in general, intractable optimization problems. In many instances these problems are non-convex, often making the computation unmanageable. A common alternative to these bottleneck is the use of convex relaxations. The applied mathematics community are always interested about tight convex relaxation, because in the worst case scenario when the problem is NP-hard, the relaxed problem can be solvable in polynomial time. Although there are several pros and cons in this approach, we focus on semidefinite programming based relaxations, because of the its rich theory and semidefinite relaxation (SDR) based algorithms have gained much attention in MIMO detection due to their computational efficiency and strong empirical performance. Many studies reveal that SDR is capable of providing accurate (and sometimes near-optimal) approximations and the performance of SDR has motivated many researchers which leads to the creation of new research trends [2, 3, 19, 23, 28, 29, 30]. Several instances of signal processing and sensor network localization problem have established that SDR is an effective technique for handling the challenges for the optimization [1, 17, 31]. It is to be noted that SDR approximation accuracies relative to the true ML have been discussed in the research by Kisialiouetal. [19]. The ML detection problem in this paper is a non-convex optimization problem. Sherali [6] presented a reformulation-linearization technique (RLT), which is capable of solving wide class of both discrete and continuous non-convex optimization problems to global optimality. The RLT approach by the author possess tight linear or convex programming relaxations. Also, this relaxation is an automatic reformulation technique for tightening model, and can be used to near optimal solution algorithms. Anstreicher [10] considered relaxations for nonconvex quadratically constrained quadratic programming (QCQP) based on semidefinite programming (SDP) and the reformulation-linearization technique (RLT). The author presented a method that is further strengthened by adding RLT to SDP relaxation for QCQP with box constraints : X ij min{x i,x j } and X ij max{0,x i +x j 1},0 i j n, which are derived from the box constraints. Wang et al. [24] presented a SDP-cut based relax- 1

2 ation which is tighter than spectral relaxation and somehow as tight as SDR. This solution technique of SDP has complexity of O(kn 3 ) where k is the number of gradientdescent steps of L-BFGS-B. However, the authors used the dual form of the relaxed SDP for the optimal solution. As the problem addressed in the paper is based on correct detection of symbols, in spite of the dispersive nature of the channels, proper choice of algorithm plays a very important role. Thus, motivated by the RLT relaxation technique of Sherali [6], SDP+RLT of Anstreicher [10] and SDP-cut based relaxation by Wang et al. [24], here a tight Semidefinite relaxation is presented, that emphasizes on convex relaxations for boolean quadratic programming problem (BQP). It is based on cutting plane technique for the detection of QPSK signaling in MIMO channel. A new formulation of BQP is proposed which is embedded with reformulation-linearization technique (RLT) and a modified triangle inequality (TI) cut is presented to compare the signal to noise ratio (SNR) vs bit-error rate (BER) in the random channel. The paper is organized as follows. Section 2, describes the problem of QPSK signaling detection in MIMO channel where, we describe the least square formulation of ML detection of the said problem. Also the BQP formulation of the least square problem is presented. In Section 3, some convex relaxation of BQP model is presented. In Section 4, the proposed SDP-cut model of BQP formulation and the proposed triangle inequality cut is presented for BQP model. In Section 5, simulation results are discussed which establishes our claim that SDP+RLT+TI reduces the relative gap to the near optimum in ML problem and as the SNR increases the BER reduces substantially. Finally, Section 6 presents the concluding remarks. 2 The QPSK signaling detection problem The problem considered here is QPSK signaling detection in MIMO channel, a RF technology that is being mentioned and used in many new technologies these days. To put it into context, we consider a generic N-input M-output model. That is, a transmitter simultaneously transmits N symbols, s 1,...,s N, from a finite alphabet or constellations C. At the receiver,m signals such asy 1,...,y M are received, each signal being a linear combination of the N input symbols plus an additive noise component. Throughout the paper, N is assumed to less thanm. A linear MIMO channel is considered which is modeled as y C = H C s C +v C, (2.1) where, y C C M are multi-receiver output vector, s S N are transmitted symbols, H C C N M are MIMO channel matrix, and v C M is Gaussian noise with zero mean and covarianceσ 2 I. The channel matrixh C is assumed that to be previously estimated, by the receiver. Thus, the objective of the receiver will be to estimate s C from the pair (y C ;H C ). Also, the channel matrixh C is generally treated as a random quantity and where no explicit distribution is assigned toh C, the channel matrix may be considered a deterministic quantity by simply considering a degenerate distribution. Also, the transmitted symbols assumed to follow a quaternary phase-shift-keying (QPSK) constellation; i.e., s Ci {±1±j} for all i. There are a wide variety of techniques for doing this but we are fascinated with the maximum-likelihood (ML) MIMO detection, which is optimal in yielding the minimum error probability of detecting s C. It can be shown that the ML detection problem of s C is equivalent to the discrete least squares problem. min y C H C s C 2 s C {±1±j} N 2, (2.2) 1.1 Notation In this paper X 0 denotes, a symmetric matrix X which is positive semidefinite and X Y denotes X Y is positive semidefinite. We denote X Y as the matrix inner product X Y = trace(x Y), where, X and Y are n n matrices. Similarly, diag(x) is the vector x where, x i = X ii, i = 1,...,n with X is n n matrix, and we use e to denote a vector with each component equal to one. The rank of a matrix is denoted as rank( ). The 1 and 2 denote thel 1 andl 2 norm of a vector respectively. Also X 2 F = Trace(XX ) = Trace(X X) is denoted as the Frobenius norm. The ML problem (2.2) is NP hard [26]. It is to be noted that for moderate problem sizes N, there are several efficient algorithms available for the solution or approximation of ML problem (2.2). Recent advances in MIMO detection have provided a practically efficient way of finding a globally optimal ML solution. There are two efficient methods for the globally optimal ML solution such as sphere decoding method [18] and SDR method. Although Sphere decoding is computationally fast for small to moderate problem sizes; e.g., N 20, but the complexity of the algorithm is exponential in N even in an average sense [7]. On the other hand, SDR method can be used to produce an approximate solution to the ML MIMO detection problem in O(N 3.5 ) time, which is polynomial in N. 2

3 2.1 An BQP model To relax (2.2) to a SDP we first convert it into a real valued homogenous quadratically constrained quadratic program (QCQP) by letting [ ] [ ] R{yC } R{sC } y =, s = H = I{y C } [ R{HC } I{H C } I{H C } R{H C } I{s C } ]. The optimization problem (2.2) can be converted to min s {±1} 2N y Hs 2 2, (2.3) Since the ML problem (2.3) is not homogenous, we can homogenize as follows: min ty Hs 2 s {±1} 2N 2, (2.4) s.t. t 2 = 1, s 2 i = 1, i = 1,2,...,2N. where t is an augmented variable for the homogenizing of problem (2.3). Thus, problem (2.4) can then be expressed as a homogeneous QCQP as follows: min ty Hs 2 2, s {±1} 2N = min s H Hs 2s H yt+ y 2 s {±1} 2N 2 t2, [ = min s t ] [ ][ H H H y s s {±1} 2N y H y 2 2 t ], (2.5) Thus, homogenous problem (2.4) can further be written as [ min s t ] [ ][ ] H H H y s s R n y H y 2 (2.6) 2 t subject to t 2 = 1, s 2 i = 1, i = 1,2,...,2N. [ H Let x = [s t] and C = H H y y H y 2 2 optimization problem (2.6) can be presented as ]. The BQP : min x R nx Cx (2.7) subject to x 2 i = 1, i = 1,2,...,n. where x i is i-th component of x and n = 2N + 1. The problem (2.7) can be viewed as a boolean quadratic problem (BQP). Even if the BQP (2.7) can be solved in polynomial time, it is NP-hard. It is essential to write the constellation constraint on x i in the above problem as a quadratic form which limits the SDR approach to the case of QPSK. It is to be noted that the constraint given by x n = 1 does not need to be maintained explicitly since, if a solution where x n = 1 is obtained, it is sufficient to multiply x by 1 to recover the correct solution. 3 Convex relaxations for BQP The BQP (2.7), which is an instance of quadratically constrained quadratic programs (QCQP), is a non-convex optimization problem because of the non-convex boolean constraintx 2 i = 1. In this Section, we present two convex relaxations of BQP such as RLT relaxation and rank relaxation. 3.1 RLT relaxation The RLT relaxation of BQP (2.7) is based on LP relaxation proposed by Sherali and Adams [5]. The relaxation consists of two phases such as reformulation phase and linearization phase. In the reformulation phase, additional implied constraints are generated, and in linearization phase, suitable new variables are introduced to replace each distinct nonlinear polynomial term. For two variablesx i andx j constraints x i + 1 0,1 x i 0,x j + 1 0,1 x j 0 are obtained. Multiplying each of the constraints involvingx i by a constraint involvingx j, and replacing the product termx i x j with the new variabley ij, we obtain the constraints: y ij 1+x i x j, y ij 1 x i +x j, y ij x i +x j 1, y ij x i x j 1. (3.1a) (3.1b) for all i,j = 1,2,...,n. It is to be noted that these constraints also hold when i = j in which case the first two constraints are identical. Moreover the first two constraints are identical for all i, j once the symmetric condition on X is imposed. Clearly, the linearization of the modified problem named BQP RLT has n(n+3) variables and 2 relaxed problem is an ordinary linear programming problem. Although it is solvable in polynomial time, the two major drawbacks are: (i) the problem is higher in dimension and (ii) provides weak lower bounds. 3.2 Rank relaxation Another method of SDR is obtained by relaxing the nonconvex rank-one constraint and present a spectral cut formulation of the SDP. LettingX = xx, the equivalent problem of (2.7) is min C X (3.2) subject to X ii = 1, i = 1,2,...,n, rank(x) = 1. Here the original problem is lifted to the space of rank-one p.s.d. matrices of the formx = xx, and thereby the number of variables increases from n to n(n+1) 2. Relaxing the rank constraint, the problem (3.2) can be converted to a SDP, min C X (3.3) subject to diag(x) = e, X 0. 3

4 The above problem conveniently can solved by standard convex optimization toolboxes, e.g., SeDuMi [8] and SDPT3 [15, 16]. The advantage of the SDP formulation is the competency of solving quandaries of more general forms, e.g., quadratically constrained quadratic program (QCQP). Despite the drawbacks of the poor scalability to large problems, the powerful toolboxes e.g., SeDuMi and SDPT3, utilize the interior-point method for solving SDP problems. 4 Main result In the present section, the proposed SDP-cut formulation for BQP, is presented, which in the association with RLT and triangle inequality cut further tightened the SDP relaxation. Wang et al. [24] presented a relaxation which is tighter than spectral relaxation and somehow as tight as SDP relaxation given in (3.3). It has complexity of O(kn 3 ) where k is the number of gradient-descent steps of L-BFGS-B. Let C = {X S n : X 0, diag(x) = e}. It is to be noted thatc is the intersection of the closed convex cone of SDP matrices and the affine subspace diag(x) = e; and thusc is a closed convex subset ofs n. Further, for the set C we have the following theorem: Lemma 1. Let X C. Then we have the inequality X F n in which equality holds if and only if X is a rank-one matrix. Proof. For proof see [9]. The constraints diag(x) = e, implies trace(x) = n. Thus, it can be said that X F n is involved in the SDP formulation of BQPs. Since it imposes on X to lie on the Euclidian sphere of center0and radiusnins n, spherical constraint X F = n is replaced by X 2 F = n2. Thus, the rank relaxed problem (3.2) is: min C X (4.1) subject to diag(x) = e, X 2 F = n 2, X 0, which is a non-convex SDP. Thus, the non-convex spherical constraint X 2 F = n2 is relaxed to the convex inequality constraint X 2 F n2 and the convex relaxed SDP is min C X (4.2) subject to diag(x) = e, X 2 F n2 X 0. Thus, following Wanget al., the following SDP formulation is considered: min C X ρ(n 2 X 2 F ) (4.3) subject to diag(x) = e, X 0. where,ρis a scalar parameter. It can also be written as BQP SDP : min C X ρ(n 2 X X) (4.4) subject to diag(x) = e, X 0. It is observed that the problem (4.4) finds a sub-optimal solution to (3.3). The gap between the solution of them vanishes when ρ approaches 0. The following theorem provides the result regarding the tightness of optimal bound of SDP (4.4) over that of SDP (3.3). Theorem 1. Iff(X) = C X andf ρ (X) = C X ρ(n 2 X 2 F ), then, f(x ) f ρ (X ) ǫ. Proof. f(x ) f ρ (X ) = ρ(n 2 X 2 F) ρn 2 Setting ρ = ǫ, we can say that ǫ > 0, ρ > 0, the n2 f(x ) f ρ (X ) ǫ. The problem BQP SDP is convex and tractable and specially it is an SDP. Although the BQP SDP can be solved by many efficient interior point algorithm despite the fact that it is not considered as a low complexity computation, it is guaranteed to terminate in O(n 1/2 log(1/ǫ)) iterations with a solution accuracyǫ > 0. Also one more disadvantage of this SDR is that it provides weaker lower bound [12]. Problems of smaller size can be solved quite efficiently with a relative gap to the optimality. But, for larger n, the relative gap increases to the optimality. However, the value ofρ plays a vital role in achieving near optimal solution which is discussed in Section Adding RLT to SDP relaxations The SDP relaxation of BQP given in (4.4) can be further be strengthened by requiring X to satisfy some inequalities called as reformulation linearization technique (RLT). Many researchers [5, 10, 12, 6] have proposed the combined SDP+RLT relaxation method which is proved to be very effective over the SDP or RLT relaxations used alone. To continue with, we present a SDP+RLT relaxation by adding RLT condition to the semidefinite relaxation of BQP. The resulting SDP+RLT relaxation can be written as follows: BQP SDP+RLT :min x, X C X ρ(n2 X X) (4.5) s.t. diag(x) = e, y ij 1+x i x j,y ij 1 x i +x j, y ij x i +x j 1,y ij x i x j 1, ( ) 1 x Y = 0. x X 4

5 where,y ij = x i x j. The SDP+RLT relaxation performs well on all types of QCQP problems reducing the relative gap better than using SDP relaxations but, on extremely large problems it has larger relative gap. 4.2 Valid inequalities The SDP+RLT relaxation can be further strengthened by requiring Y to satisfy the triangle inequalities. Padberg [22] showed that the following triangle inequalities are valid inequalities for BQP n whenz {0,1} n. t ij +t ik t jk z i, z i +z j +z k t ij t ik t jk 1. (4.6a) (4.6b) where,t ij = z i z j. The Padberg s triangle inequality further tights the bounds of SDP+RLT. However, the number of constraints increases in the relaxed problem of BQP with the inclusion of TI to SDP+RLT. To over come this bottleneck we have proposed a triangle inequality which is valid and derived by the combination of the above triangle inequalities given in (4.6). Forz {0,1} n, z i +z j +z k 1+t ij +t ik +t jk, 1+2t jk +z i. To make further tight our proposed valid triangle inequality is as follows: z j +z k t jk 1. (4.7) Since in the present BQP problem (2.7), x { 1,1} n, then the above triangle inequalities in the present domain can be obtained by setting z i = x i +1, with the 2 justification that z {0,1} n implies x { 1,1} n. Thus, the proposed triangle inequality (4.7) for the BQP problem (4.5) is represented as: x j +x k y jk 1 (4.8a) where y jk = x j x k. We collect the following RLT and triangle inequalities symbolically as A(X) b, where A is an operator mapping of symmetric matrices. y ij 1+x i x j, y ij 1 x i +x j, y ij x i +x j 1, y ij x i x j 1, x i +x j y ij 1, x j +x k y jk 1, x i +x k y ik 1. Hence adding triangle inequality constraint BQP SDP+RLT, the relaxed BQP SDP+RLT+TI is as follows: BQP SDP+RLT+TI :min C X, (4.9) s.t. diag(x) = e,a(x) b, X 0. ( ) n This is again a semidefinite program, but it has 6 RLT 3 and triangle inequalities in addition to the n equations fixing the main diagonal of X to e. The computational effort to solve this problem is nontrivial, which is seen in the next section. 4.3 Tightness of SDP bounds Thus, the following relaxations for BQP problems with nonconvex quadratic constraints are considered: The basic SDP relaxation: BQP SDP given in (3.3) SDP+RLT relaxation: BQP RLT given in (4.5) SDP+RLT+TI: BQP SDP+RLT+TI given in (4.9). The relationship between the SDP and RLT constraints is discussed in [13]. Burer showed that SDP and RLT constraints together give a full characterization of X [14]. Although the triangle inequalities and RLT constraints fully characterize the BQP, but these constraints combined with the SDP constraint do not give a complete characterization of X [25]. It is seen in research paper of Anstreicher [11] that the following relationship of tightness holds. SDP opt (SDP +RLT) opt (4.10) (SDP +RLT +TI) opt. wheresdp opt, (SDP +RLT) opt, and(sdp +RLT + TI) opt are the optimal solutions to their corresponding problems. Therefore, we have seen that the convex relaxation (SDP+RLT+TI) can produce lower bound for BQP at least as tight as the existing SDP bounds and SDP+RLT bounds. 5 Simulation result In this section, the efficiency of the class of SDR detectors in an MIMO system using computer simulations is demonstrated. The simulation settings are as follows. The channel H is randomly generated following an independent identically distributed complex Gaussian distribution with zero mean and covariance σ 2 I. Some simulation results are shown to illustrate how well our proposed detector SDP+RLT+TI performs in practice. For this, simulations for some classical works like 2 2, 4 4, 6 6, 8 8 5

6 MIMO models are presented. The performance in reducing the BERs with the increase of SNR of all the detectors is appreciable. In Fig. 1 to Fig. 4, four performance curves are presented for MIMO detectors that provide a comparison between the BERs obtained by the detectors SDR, SDP+RLT, and SDP+RLT+TI and a benchmark detector like zero-forcing (ZF) detector. All the SDP s are solved by SDPT3 [15, 16]. While solving, we use the computational engine within the optimization modeling language as CVX [20, 21]. The class of SDR detectors use the SDP- Interior Point Method (SDR-IPM) and are solved by infeasible path following algorithm in CVX. We can see that all the detectors provides near-optimal bit error probability, and gives notably better performance than other MIMO detectors like ZF decoders under test. The number of simulation in each iteration is set at100 and white Gaussian noise with a variable SNR range up to 16dB is considered while evaluating the BERs at the receiving end. The purpose of our computer simulation is to show that the new class of SDP relaxation which is coupled with RLT and triangle inequality cut, generate strictly tighter lower bounds for BQP problems. To provide some benchmark, the zero-forcing (ZF) detector were also tested. As all the three detectors use IPM, therefore they maintain a polynomial time complexity with respect ton. Since all the detectors provides almost the same BERs, we are interested in the performance in terms of number of iterations required of the proposed detector SDP+RLT+TI. For this and to establish our claim of tightness of bound, we make a comparison among average optimal values of various detectors like SDR, SDP+RLT, SDP+RLT+TI. The results are plotted in Fig. 5, where we fix SNR= 12dB andm = N. The key observations in Fig. 5 is that SDP+RLT+TI is more tight in comparison with the other two detectors. But whenn > 12, the gap between the optimal bounds in the case of SDP+RLT and SDP+RLT+TI is nominal. However, the gap between the optimal values of SDP and SDP+RLT+TI is very high making the bound of the later tight. It is largely due to two things: (i) the new SDP cut formulation and (ii) proposed triangle inequality cut. We have also observed from our simulations that the value of ρ plays a vital role in this achievement. With several experiments done with the value of parameterρ, we observed that for ρ = n+1, SDP+RLT+TI performs a tight optimal objective value. It is to be noted that the above setting of ρ is also taken in SDR+RLT and SDR detectors. Table 1 gives the execution times and the objective values of the various detectors. It can be seen that the SDP+RLT+TI detector takes slightly more time than the SDR and SDP+RLT detectors, however, provides a tight bound among the three. BER SDR detector RLT detector RLT TI detector ZF detector SNR (db) Fig 1: Bit error probability performance of various MIMO detectors in a QPSK 2 2 MIMO system. BER RLT TI detector RLT detector SDR detector ZF detector SNR (db) Fig 2: Bit error probability performance of various MIMO detectors in a QPSK 4 4 MIMO system. BER SDR detector SDR RLT detector SDR RLT TI detector ZF detector SNR (db) Fig 3: Bit error probability performance of various MIMO detectors in a QPSK 6 6 MIMO system. 6

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