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

Rupaj Kumar Nayak∗ Harish Kumar Sahoo†

Abstract

A detection technique of quadrature phase shift key-ing (QPSK) signaling in multiple-input, multiple-output(MIMO) channel is considered, based on several convexrelaxation of the maximum likelihood (ML) problem. Itis because many ML detectors are, in general, intractableoptimization problems. The ML problem is formulated asa non-convex nonlinear optimization model and a tightSemidefinite (SDP) based relaxations coupled with a newSDP formulation for boolean quadratic programming (BQP)embedded with reformulation linearization technique (RLT)is presented for solving non-convex programming prob-lems. In essence, this process yields a new BQP formulationalong with an RLT relaxation that is combined with validinequalities called as triangle inequality, which furthertightened the relaxation is proposed. Simulation resultsdesignate that the proposed cutting plane method providesa paramount tightening of the lower bound obtained byutilizing RLT along with triangle inequality that enhancesthe detection of QPSK signaling in MIMO channel.

Keywords: Multiple-input Multiple-output, Maximumlikelihood detection, Semidefinite relaxation,reformulation linearization technique, quadratically con-strained quadratic programming.

1 Introduction

Multiple input and Multiple output (MIMO) wireless com-munications systems uses multiple antennas to increase thedata rates and improved link reliability without additionalbandwidth [4, 27]. Detection of QPSK symbols in pres-ence of inter symbol interference and additive channel noiseis addressed by different researchers using different signalprocessing techniques. Maximum likelihood (ML) detectionof transmitted symbols require mathematical formulation oflikelihood or log-likelihood functions. But, the performance

∗Corresponding author. Department of Mathematics, IIIT Bhubaneswar,India, 751003; E-mail: rupaj@iiit-bh.ac.in

†Department of Electronics and Telecommunication, VSSUT, Burla,768018; E-mail:harishsahoo@yahoo.co.in

of ML technique is limited because it requires a recursiveformulation of likelihood function. Despite rich theoreti-cal support, ML detectors are, in general, intractable opti-mization problems. In many instances these problems arenon-convex, often making the computation unmanageable.A common alternative to these bottleneck is the use of con-vex relaxations. The applied mathematics community are al-ways interested about tight convex relaxation, because in theworst case scenario when the problem is NP-hard, the re-laxed problem can be solvable in polynomial time. Althoughthere are several pros and cons in this approach, we focus onsemidefinite programming based relaxations, because of theits rich theory and semidefinite relaxation (SDR) based algo-rithms have gained much attention in MIMO detection dueto their computational efficiency and strong empirical per-formance. Many studies reveal that SDR is capable of pro-viding accurate (and sometimes near-optimal) approxima-tions and the performance of SDR has motivated many re-searchers which leads to the creation of new research trends[2, 3, 19, 23, 28, 29, 30]. Several instances of signal pro-cessing and sensor network localization problem have es-tablished that SDR is an effective technique for handling thechallenges for the optimization [1, 17, 31]. It is to be notedthat SDR approximation accuracies relative to the true MLhave been discussed in the research by Kisialiouetal. [19].

The ML detection problem in this paper is anon-convex optimization problem. Sherali [6] presented areformulation-linearization technique (RLT), which is capa-ble of solving wide class of both discrete and continuousnon-convex optimization problems to global optimality. TheRLT approach by the author possess tight linear or convexprogramming relaxations. Also, this relaxation is an auto-matic reformulation technique for tightening model, and canbe used to near optimal solution algorithms.

Anstreicher [10] considered relaxations for non-convex quadratically constrained quadratic programming(QCQP) based on semidefinite programming (SDP) andthe reformulation-linearization technique (RLT). The authorpresented a method that is further strengthened by addingRLT to SDP relaxation for QCQP with box constraints :Xij ≤ min{xi, xj} andXij ≥ max{0, xi + xj − 1}, 0 ≤

i ≤ j ≤ n, which are derived from the box constraints.Wanget al. [24] presented a SDP-cut based relax-

1

ation which is tighter than spectral relaxation and some-how as tight as SDR. This solution technique of SDP hascomplexity ofO(kn3) wherek is the number of gradient-descent steps of L-BFGS-B. However, the authors used thedual form of the relaxed SDP for the optimal solution.

As the problem addressed in the paper is based oncorrect detection of symbols, in spite of the dispersive na-ture of the channels, proper choice of algorithm plays avery important role. Thus, motivated by the RLT relax-ation technique of Sherali [6], SDP+RLT of Anstreicher [10]and SDP-cut based relaxation by Wanget al. [24], here atight Semidefinite relaxation is presented, that emphasizeson convex relaxations for boolean quadratic programmingproblem (BQP). It is based on cutting plane technique forthe detection of QPSK signaling in MIMO channel. A newformulation of BQP is proposed which is embedded withreformulation-linearization technique (RLT) and a modifiedtriangle inequality (TI) cut is presented to compare the sig-nal to noise ratio (SNR) vs bit-error rate (BER) in the ran-dom channel.

The paper is organized as follows. Section 2, de-scribes the problem of QPSK signaling detection in MIMOchannel where, we describe the least square formulation ofML detection of the said problem. Also the BQP formula-tion of the least square problem is presented. In Section 3,some convex relaxation of BQP model is presented. In Sec-tion 4, the proposed SDP-cut model of BQP formulation andthe proposed triangle inequality cut is presented for BQPmodel. In Section 5, simulation results are discussed whichestablishes our claim that SDP+RLT+TI reduces the relativegap to the near optimum in ML problem and as the SNRincreases the BER reduces substantially. Finally, Section6presents the concluding remarks.

1.1 Notation

In this paperX � 0 denotes, a symmetric matrixX whichis positive semidefinite andX � Y denotesX − Y ispositive semidefinite. We denoteX • Y as the matrix in-ner productX • Y = trace(X⊤Y ), where,X andY aren × n matrices. Similarly, diag(X) is the vectorx where,xi = Xii, i = 1, ..., n with X is n × n matrix, and weusee to denote a vector with each component equal to one.The rank of a matrix is denoted asrank(·). The‖·‖1 and‖·‖2 denote thel1 andl2 norm of a vector respectively. Also‖X‖2F = Trace(XX⊤) = Trace(X⊤X) is denoted as theFrobenius norm.

2 The QPSK signaling detectionproblem

The problem considered here is QPSK signaling detectionin MIMO channel, a RF technology that is being mentionedand used in many new technologies these days. To put it intocontext, we consider a genericN -input M -output model.That is, a transmitter simultaneously transmitsN symbols,s1, ..., sN , from a finite alphabet or constellationS ⊆ C. Atthe receiver,M signals such asy1, ..., yM are received, eachsignal being a linear combination of theN input symbolsplus an additive noise component. Throughout the paper,N

is assumed to less thanM .A linear MIMO channel is considered which is mod-

eled as

yC = HCsC + vC , (2.1)

where,yC ∈ CM are multi-receiver output vector,s ∈ SN

are transmitted symbols,HC ∈ CN×M are MIMO chan-nel matrix, andv ∈ C

M is Gaussian noise with zero meanand covarianceσ2I. The channel matrixHC is assumed thatto be previously estimated, by the receiver. Thus, the ob-jective of the receiver will be to estimatesC from the pair(yC ;HC). Also, the channel matrixHC is generally treatedas a random quantity and where no explicit distribution isassigned toHC , the channel matrix may be considered a de-terministic quantity by simply considering a degenerate dis-tribution. Also, the transmitted symbols assumed to followa quaternary phase-shift-keying (QPSK) constellation; i.e.,sCi

∈ {±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 minimumerror probability of detectingsC . It can be shown that theML detection problem ofsC is equivalent to the discreteleast squares problem.

minsC∈{±1±j}N

‖yC −HCsC‖22, (2.2)

The ML problem (2.2) is NP hard [26]. It is to be notedthat for moderate problem sizesN , there are several effi-cient algorithms available for the solution or approximationof ML problem (2.2). Recent advances in MIMO detectionhave provided a practically efficient way of finding a glob-ally optimal ML solution. There are two efficient methodsfor the globally optimal ML solution such as sphere decod-ing method [18] and SDR method. Although Sphere decod-ing is computationally fast for small to moderate problemsizes; e.g.,N ≤ 20, but the complexity of the algorithm isexponential inN even in an average sense [7]. On the otherhand, SDR method can be used to produce an approximatesolution to the ML MIMO detection problem inO(N3.5)

time, which is polynomial inN .

2

2.1 An BQP model

To relax (2.2) to a SDP we first convert it into a real val-ued homogenous quadratically constrained quadratic pro-gram (QCQP) by letting

y =

[

R{yC}

I{yC}

]

, s =

[

R{sC}

I{sC}

]

H =

[

R{HC} −I{HC}

I{HC} R{HC}

]

.

The optimization problem (2.2) can be converted to

mins∈{±1}2N

‖y −Hs‖22, (2.3)

Since the ML problem (2.3) is not homogenous, wecan homogenize as follows:

mins∈{±1}2N

‖ty −Hs‖22, (2.4)

s.t. t2 = 1, s2i = 1, i = 1, 2, ..., 2N.

wheret is an augmented variable for the homogenizing ofproblem (2.3). Thus, problem (2.4) can then be expressed asa homogeneous QCQP as follows:

mins∈{±1}2N

‖ty −Hs‖22,

= mins∈{±1}2N

s⊤H⊤Hs− 2s⊤H⊤yt+ ‖y‖22t2,

= mins∈{±1}2N

[

s⊤ t]

[

H⊤H −H⊤y

−y⊤H ‖y‖22

] [

s

t

]

,

(2.5)

Thus, homogenous problem (2.4) can further be written as

mins∈Rn

[

s⊤ t]

[

H⊤H −H⊤y

−y⊤H ‖y‖22

] [

s

t

]

(2.6)

subject tot2 = 1, s2i = 1, i = 1, 2, ..., 2N.

Let x = [s⊤ t]⊤ andC =

[

H⊤H −H⊤y

−y⊤H ‖y‖22

]

. The

optimization problem (2.6) can be presented as

BQP : minx∈Rn

x⊤Cx (2.7)

subject tox2i = 1, i = 1, 2, ..., n.

wherexi is i-th component ofx andn = 2N + 1. Theproblem (2.7) can be viewed as a boolean quadratic problem(BQP). Even if the BQP (2.7) can be solved in polynomialtime, it is NP-hard. It is essential to write the constellationconstraint onxi in the above problem as a quadratic formwhich limits the SDR approach to the case of QPSK. It isto be noted that the constraint given byxn = 1 does notneed to be maintained explicitly since, if a solution wherexn = −1 is obtained, it is sufficient to multiplyx by −1 torecover the correct solution.

3 Convex relaxations for BQP

The BQP (2.7), which is an instance of quadratically con-strained quadratic programs (QCQP), is a non-convex opti-mization problem because of the non-convex boolean con-straintx2

i = 1. In this Section, we present two convex relax-ations of BQP such as RLT relaxation and rank relaxation.

3.1 RLT relaxation

The RLT relaxation of BQP (2.7) is based on LP relaxationproposed by Sherali and Adams [5]. The relaxation consistsof two phases such as reformulation phase and linearizationphase. In the reformulation phase, additional implied con-straints are generated, and in linearization phase, suitablenew variables are introduced to replace each distinct nonlin-ear polynomial term. For two variablesxi andxj constraintsxi + 1 ≥ 0, 1 − xi ≥ 0, xj + 1 ≥ 0, 1 − xj ≥ 0 are ob-tained. Multiplying each of the constraints involvingxi by aconstraint involvingxj , and replacing the product termxixj

with the new variableyij , we obtain the constraints:

yij ≤ 1 + xi − xj , yij ≤ 1− xi + xj , (3.1a)

yij ≥ xi + xj − 1, yij ≥ −xi − xj − 1. (3.1b)

for all i, j = 1, 2, ..., n. It is to be noted that theseconstraints also hold wheni = j in which case the firsttwo constraints are identical. Moreover the first two con-straints are identical for alli, j once the symmetric condi-tion onX is imposed. Clearly, the linearization of the mod-

ified problem named BQPRLT hasn(n+ 3)

2variables and

relaxed problem is an ordinary linear programming prob-lem. Although it is solvable in polynomial time, the two ma-jor drawbacks are: (i) the problem is higher in dimensionand (ii) provides weak lower bounds.

3.2 Rank relaxation

Another method of SDR is obtained by relaxing the non-convex rank-one constraint and present a spectral cut formu-lation of the SDP. LettingX = xx⊤, the equivalent problemof (2.7) is

min C •X (3.2)

subject toXii = 1, i = 1, 2, ..., n, rank(X) = 1.

Here the original problem is lifted to the space of rank-onep.s.d. matrices of the formX = xx⊤, and thereby the num-ber of variables increases fromn to n(n+1)

2 . Relaxing therank constraint, the problem (3.2) can be converted to a SDP,

min C •X (3.3)

subject todiag(X) = e, X � 0.

3

The above problem conveniently can solved by stan-dard convex optimization toolboxes,e.g., SeDuMi [8] andSDPT3 [15, 16]. The advantage of the SDP formulationis the competency of solving quandaries of more generalforms, e.g., quadratically constrained quadratic program(QCQP). Despite the drawbacks of the poor scalability tolarge problems, the powerful toolboxese.g., SeDuMi andSDPT3, utilize the interior-point method for solving SDPproblems.

4 Main result

In the present section, the proposed SDP-cut formulation forBQP, is presented, which in the association with RLT andtriangle inequality cut further tightened the SDP relaxation.Wanget al. [24] presented a relaxation which is tighter thanspectral relaxation and somehow as tight as SDP relaxationgiven in (3.3). It has complexity ofO(kn3) wherek is thenumber of gradient-descent steps of L-BFGS-B.

Let C = {X ∈ Sn : X � 0, diag(X) = e}. It is tobe noted thatC is the intersection of the closed convex coneof SDP matrices and the affine subspacediag(X) = e; andthusC is a closed convex subset ofSn.

Further, for the setC 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 ifX is arank-one matrix.

Proof. For proof see [9].

The constraintsdiag(X) = e, impliestrace(X) =

n. Thus, it can be said that‖X‖F ≤ n is involved in theSDP formulation of BQPs.

Since it imposes onX to lie on the Euclidian sphereof center0 and radiusn in Sn, spherical constraint‖X‖F =

n is replaced by‖X‖2F = n2. Thus, the rank relaxed prob-lem (3.2) is:

min C •X (4.1)

subject todiag(X) = e, ‖X‖2F = n2, X � 0,

which is a non-convex SDP. Thus, the non-convexspherical constraint‖X‖2F = n2 is relaxed to the convexinequality constraint‖X‖2F ≤ n2 and the convex relaxedSDP is

min C •X (4.2)

subject todiag(X) = e, ‖X‖2F ≤ n2 X � 0.

Thus, following Wanget al., the following SDP for-mulation is considered:

min C •X − ρ(n2 − ‖X‖2F ) (4.3)

subject todiag(X) = e, X � 0.

where,ρ is a scalar parameter. It can also be written as

BQPSDP : min C •X − ρ(n2 −X⊤ •X) (4.4)

subject todiag(X) = e, X � 0.

It is observed that the problem (4.4) finds a sub-optimal so-lution to (3.3). The gap between the solution of them van-ishes whenρ approaches0. The following theorem providesthe result regarding the tightness of optimal bound of SDP(4.4) over that of SDP (3.3).

Theorem 1. If f(X) = C•X andfρ(X) = C•X−ρ(n2−

‖X‖2F ), then,|f(X∗)− fρ(X∗)| ≤ ǫ.

Proof.

|f(X∗)− fρ(X∗)| = ρ(n2 − ‖X∗‖2F ) ≤ ρn2

Settingρ =ǫ

n2, we can say that∀ ǫ > 0, ∃ ρ > 0, the

|f(X∗)− fρ(X∗)| ≤ ǫ.

The problem BQPSDP is convex and tractable andspecially it is an SDP. Although the BQPSDPcan be solvedby many efficient interior point algorithm despite the factthat it is not considered as a low complexity computation,it is guaranteed to terminate inO(n1/2 log(1/ǫ)) iterationswith a solution accuracyǫ > 0. Also one more disadvantageof this SDR is that it provides weaker lower bound [12].Problems of smaller size can be solved quite efficiently witha relative gap to the optimality. But, for largern, the rela-tive gap increases to the optimality. However, the value ofρ

plays a vital role in achieving near optimal solution which isdiscussed in Section 5.

4.1 Adding RLT to SDP relaxations

The SDP relaxation of BQP given in (4.4) can be furtherbe strengthened by requiringX to satisfy some inequali-ties called as reformulation linearization technique (RLT).Many researchers [5, 10, 12, 6] have proposed the combinedSDP+RLT relaxation method which is proved to be veryeffective over the SDP or RLT relaxations used alone. Tocontinue with, we present a SDP+RLT relaxation by addingRLT condition to the semidefinite relaxation of BQP. Theresulting SDP+RLT relaxation can be written as follows:

BQPSDP+RLT : minx, X

C •X − ρ(n2 −X⊤ •X) (4.5)

s.t. diag(X) = e,

yij ≤ 1 + xi − xj , yij ≤ 1− xi + xj ,

yij ≥ xi + xj − 1, yij ≥ −xi − xj − 1,

Y =

(

1 x⊤

x X

)

� 0.

4

where,yij = xixj . The SDP+RLT relaxation performs wellon all types of QCQP problems reducing the relative gapbetter than using SDP relaxations but, on extremely largeproblems it has larger relative gap.

4.2 Valid inequalities

The SDP+RLT relaxation can be further strengthened by re-quiring Y to satisfy the triangle inequalities. Padberg [22]showed that the following triangle inequalities are valid in-equalities for BQPn whenz ∈ {0, 1}n.

tij + tik − tjk ≤ zi, (4.6a)

zi + zj + zk − tij − tik − tjk ≤ 1. (4.6b)

where,tij = zizj .The Padberg’s triangle inequality further tights the

bounds of SDP+RLT. However, the number of constraintsincreases in the relaxed problem of BQP with the inclusionof TI to SDP+RLT. To over come this bottleneck we haveproposed a triangle inequality which is valid and derived bythe combination of the above triangle inequalities given in(4.6). Forz ∈ {0, 1}n,

zi + zj + zk ≤ 1 + tij + tik + tjk,≤ 1 + 2tjk + zi.

To make further tight our proposed valid triangle inequalityis as follows:

zj + zk − tjk ≤ 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 settingzi =xi + 1

2, with the

justification thatz ∈ {0, 1}n implies x ∈ {−1, 1}n. Thus,the proposed triangle inequality (4.7) for the BQP problem(4.5) is represented as:

xj + xk − yjk ≤ 1 (4.8a)

whereyjk = xjxk. We collect the following RLTand triangle inequalities symbolically asA(X) ≤ b, whereA is an operator mapping of symmetric matrices.

yij ≤ 1 + xi − xj , yij ≤ 1− xi + xj ,

yij ≥ xi + xj − 1, yij ≥ −xi − xj − 1,

xi + xj − yij ≤ 1, xj + xk − yjk ≤ 1,

xi + xk − yik ≤ 1.

Hence adding triangle inequality constraintBQPSDP+RLT , the relaxed BQPSDP+RLT+TI is as

follows:

BQPSDP+RLT+TI : min C •X, (4.9)

s.t. diag(X) = e, A(X) ≤ b,

X � 0.

This is again a semidefinite program, but it has6

(

n

3

)

RLT

and triangle inequalities in addition to then equations fixingthe main diagonal ofX to e. The computational effort tosolve this problem is nontrivial, which is seen in the nextsection.

4.3 Tightness of SDP bounds

Thus, the following relaxations for BQP problems with non-convex quadratic constraints are considered:

• The basic SDP relaxation: BQPSDP given in (3.3)

• SDP+RLT relaxation: BQPRLT given in (4.5)

• SDP+RLT+TI: BQPSDP+RLT+TI given in (4.9).

The relationship between the SDP and RLT con-straints is discussed in [13]. Burer showed that SDP andRLT constraints together give a full characterization ofX

[14]. Although the triangle inequalities and RLT constraintsfully characterize the BQP, but these constraints combinedwith the SDP constraint do not give a complete characteri-zation ofX [25]. It is seen in research paper of Anstreicher[11] that the following relationship of tightness holds.

SDPopt ≤ (SDP +RLT )opt (4.10)

≤ (SDP +RLT + TI)opt.

whereSDPopt, (SDP + RLT )opt, and(SDP +RLT +

TI)opt are the optimal solutions to their correspondingproblems. Therefore, we have seen that the convex relax-ation (SDP+RLT+TI) can produce lower bound for BQP atleast as tight as the existing SDP bounds and SDP+RLTbounds.

5 Simulation result

In this section, the efficiency of the class of SDR detec-tors in an MIMO system using computer simulations isdemonstrated. The simulation settings are as follows. ThechannelH is randomly generated following an indepen-dent identically distributed complex Gaussian distributionwith zero mean and covarianceσ2I. Some simulation re-sults are shown to illustrate how well our proposed detec-tor SDP+RLT+TI performs in practice. For this, simulationsfor some classical works like2 × 2, 4 × 4, 6 × 6, 8 × 8

5

MIMO models are presented. The performance in reduc-ing the BERs with the increase of SNR of all the detec-tors is appreciable. In Fig. 1 to Fig. 4, four performancecurves are presented for MIMO detectors that provide acomparison between the BERs obtained by the detectorsSDR, SDP+RLT, and SDP+RLT+TI and a benchmark detec-tor like zero-forcing (ZF) detector. All the SDP’s are solvedby SDPT3 [15, 16]. While solving, we use the computa-tional engine within the optimization modeling language asCVX [20, 21]. The class of SDR detectors use the SDP-Interior Point Method (SDR-IPM) and are solved by infea-sible path following algorithm in CVX. We can see that allthe detectors provides near-optimal bit error probability, andgives notably better performance than other MIMO detec-tors like ZF decoders under test. The number of simulationin each iteration is set at100 and white Gaussian noise witha variable SNR range up to16dB is considered while eval-uating the BERs at the receiving end. The purpose of ourcomputer simulation is to show that the new class of SDPrelaxation which is coupled with RLT and triangle inequal-ity cut, generate strictly tighter lower bounds for BQP prob-lems. To provide some benchmark, the zero-forcing (ZF) de-tector were also tested. As all the three detectors use IPM,therefore they maintain a polynomial time complexity withrespect toN .

Since all the detectors provides almost the sameBERs, we are interested in the performance in terms ofnumber of iterations required of the proposed detectorSDP+RLT+TI. For this and to establish our claim of tight-ness of bound, we make a comparison among averageoptimal values of various detectors like SDR, SDP+RLT,SDP+RLT+TI. The results are plotted in Fig. 5, where wefix SNR= 12dB andM = N . The key observations in Fig.5 is that SDP+RLT+TI is more tight in comparison with theother two detectors. But whenN > 12, the gap between theoptimal bounds in the case of SDP+RLT and SDP+RLT+TIis nominal. However, the gap between the optimal valuesof SDP and SDP+RLT+TI is very high making the bound ofthe later tight. It is largely due to two things: (i) the new SDPcut formulation and (ii) proposed triangle inequality cut.Wehave also observed from our simulations that the value ofρ

plays a vital role in this achievement. With several experi-ments done with the value of parameterρ, we observed thatfor

ρ =10−2

2n+ 1,

SDP+RLT+TI performs a tight optimal objective value. Itis to be noted that the above setting ofρ is also taken inSDR+RLT and SDR detectors. Table 1 gives the executiontimes and the objective values of the various detectors. Itcan be seen that the SDP+RLT+TI detector takes slightlymore time than the SDR and SDP+RLT detectors, however,provides a tight bound among the three.

0 2 4 6 8 10 12 14 1610

−3

10−2

10−1

100

SNR (dB)

BE

R

SDR detectorRLT detectorRLT−TI detectorZF detector

Fig 1: Bit error probability performance of variousMIMO detectors in a QPSK2× 2 MIMO system.

0 2 4 6 8 10 12 14 1610

−3

10−2

10−1

100

SNR (dB)

BE

R

RLT−TI detectorRLT detectorSDR detectorZF detector

Fig 2: Bit error probability performance of variousMIMO detectors in a QPSK4× 4 MIMO system.

0 2 4 6 8 10 12 14 1610

−3

10−2

10−1

100

SNR (dB)

BE

R

SDR detectorSDR−RLT detectorSDR−RLT−TI detectorZF detector

Fig 3: Bit error probability performance of variousMIMO detectors in a QPSK6× 6 MIMO system.

6

Method Times(s) Lower boundSDR 1.823 32.514

SDR+RLT 13.109 32.326SDR+RLT+TI 14.884 30.8

Table 1: Execution time and objective values of the com-puted lower bounds of SDP detectors

0 2 4 6 8 10 12 14 1610

−3

10−2

10−1

100

SNR (dB)

BE

R

RLT−TI detectorRLT detectorSDR detectorZF detector

Fig 4: Bit error probability performance of variousMIMO detectors in a QPSK8× 8 MIMO system.

0 5 10 15 2010

0

101

102

103

104

No. of transmitted symbols (N)

Avg

. F V

alue

SDPSDP+RLT detectorRLT+TI

Fig 5: Average optimal values various SDP-class detectors.

6 Conclusion

In this paper, we presented a class of semidefinite relax-ation technique of the ML detector for QPSK signaling overMIMO channels. We find that the proposed detector is notonly a tight ML detector but also a near optimal ML detec-tor in comparison to other suboptimal methods. The compu-tational results show that the performance of the proposedSDP+RLT+TI detector for QPSK signal in MIMO channelis comparable to other tight ML detectors. The theoreticaland computational success of the present work motivates usto explore the effectiveness of the SDP cut formulation cou-pled with the proposed RLT+TI cut towards the detection ofQAM that arising from communication world.

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