490 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 4, APRIL 2012

Ellipse Expanding Method for Minimum Distance-Based MIMO PrecoderXiaodong Xu, Qiang Gu, Yi Jiang, and Xuchu Dai

Abstract—This letter proposes a new MIMO precoding methodfor dual-stream data transmission based on the criterion ofminimum Euclidean distance. The problem is optimally solved byexpanding an ellipse within a 2-dimensional lattice to the max-imum extent. A closed-form expression of minimum Euclideandistance is derived for square quadrature amplitude modulation(QAM) constellations. The ellipse expanding method (EEM)allows for very efficient implementation by checking a smalllookup table with respect to the condition number of channelmatrix. Illustrative examples validate the promising performanceof the proposed method in comparison with existing schemes.

Index Terms—MIMO, precoder, ellipse expanding, minimumEuclidean distance.

I. INTRODUCTION

ASSUMING full channel state information (CSI) at boththe transmitter and receiver, multiple-input multiple-

output (MIMO) precoding is an efficient way to reap the greatgain of MIMO in spectral efficiency and link reliability [1].Large amount of research work has been done on optimizingthe MIMO precoder according to various criteria, such asmaximizing the channel mutual information [2], minimizingthe mean square error (MMSE) [3], maximizing the receivedsignal-to-noise ratio (SNR) of the weakest subchannel [4], andmaximizing the minimum Euclidean distance for constellationinputs under average transmit power constraint [5].

The minimum Euclidean distance criterion is known tobe similar to the bit error rate (BER) criterion given amaximum likelihood (ML) receiver. Unfortunately, a universalsolution remains elusive due to the combinatorial nature of theproblem, despite the recent interesting efforts [6]–[9].

The authors of [5] parameterize the channel matrix andthe precoder matrix via three different angular parametersand then provide a numerical search scheme to obtain thecomplex- valued optimal precoder. However, the method onlyapplies to the scenario of two independent streams of binaryphase shift keying (BPSK) and quadrature phase shift keying(QPSK) input. A suboptimal extension to even-dimensionalindependent data streams transmission is introduced in [6] toobtain a cross-form precoder matrix. Additional extensions canbe found in [7] and [8], where the precoders are implementedfor the special cases of 3-dimensional transmission and 16-

Manuscript received November 17, 2011. The associate editor coordinatingthe review of this letter and approving it for publication was K. K. Wong.

This work was supported in part by the National Natural Science Founda-tion of China under grant number 61071094, and the National Science andTechnology Special Projects of China under grant number 2012ZX03001007.

X. Xu, Q. Gu, and X. Dai are with the University of Science andTechnology of China, Hefei, Anhui, 230027, P.R. China (e-mail: {xdxu,daixc}@ustc.edu.cn, guqiang@mail.ustc.edu.cn).

Y. Jiang is with Qualcomm Inc., 5775 Morehouse Dr., San Diego, CA92121 USA (e-mail: yjiang.ee@gmail.com).

Digital Object Identifier 10.1109/LCOMM.2012.020712.112349

quadrature amplitude modulation (QAM), respectively. Morerecently, paper [9] extends the work of [6] and constrains theprecoder to be real-valued for reduced implementation com-plexity. However, all these approaches involves an exhaustivenumerical search over a multi-dimensional parameter spaceand hence is computationally very expensive especially whenthe number of data streams increases and the QAM size islarge.

In this letter, we propose the ellipse expanding method(EEM) to obtain the optimum real-valued MIMO precoder butwith much smaller computational complexity than the existingalgorithms. We also derive a closed-form expression of theminimum distance for square QAM constellations. We focuson the two-stream case in this letter. However, the underlyingidea of the EEM can also be applied to the general case ofmore than two data streams.

II. PROBLEM FORMULATION

Assuming that perfect CSI is available at both the transmit-ter and the receiver, we consider transmitting two independentdata streams through a quasi-static MIMO flat-fading channelwith Mt transmit antennas and Mr receive antennas. Thereceived signal can be expressed as

y = HFs+ n, (1)

where H ∈ CMr×Mt is the channel matrix; F ∈ C

Mt×2

denotes the precoder matrix; s ∈ C2×1 represents the two-stream data input with entries taken from a square QAMconstellation, and n ∈ CMr×1 is the circularly symmetricGaussian channel noise. Assume that both s and n are zeromean with covariance E[ssH ] = σ2

sI2, E[nnH ] = σ2nIMr ,

respectively, where (·)H stands for conjugate transpose andIk is a k × k identity matrix.

Denote S as the set of all possible data vectors. Then thesquared minimum Euclidean distance between points in thereceived signal space is given by

d2min = minsi,sj∈S,si �=sj

‖HF(si − sj)‖2. (2)

In this paper, we study optimization of F by the criterion ofmaximizing the minimum distance under the transmit powerconstraint:

Fopt = arg maxF∈CMt×2

d2min

subject to trace(FFH) = P0.(3)

Given a maximum likelihood (ML) equalizer, the minimumEuclidean distance criterion is known to be asymptoticallyequivalent to the bit error rate (BER) criterion as the inputSNR goes to infinity [5].

Denoting the difference vector x = si − sj with i �= j, wedefine X as the set of all x’s. Hence

d2min = minx∈X

‖HFx‖2. (4)1089-7798/12$31.00 c© 2012 IEEE

XU et al.: ELLIPSE EXPANDING METHOD FOR MINIMUM DISTANCE-BASED MIMO PRECODER 491

Denote H = UΛHVH as the singular value decomposition(SVD), where U, V are unitary matrices and the diagonalmatrix ΛH contains nonzero singular values λH,1, · · · , λH,Kin decreasing order with K being the rank of H. It is claimedin [10](proposition 1) that the optimal precoder matrix hasSVD F = VΣB, i.e., its left singular matrix is the rightsingular matrix of H. As we constrain our discussion tothe two-stream case, F = VΣB where V is the submatrixconsisting of the first two columns of V. Denoting Λ ∈ R2×2

as the leading principal submatrix of ΛH, we reformulate (4)as

d2min = minx∈X

‖ΛΣBx‖2. (5)

In [9], the rotation matrix B is considered to be a real-valued orthogonal one. Without loss of generality (w.l.o.g.),ψ ∈ [0, π4 ] and θ ∈ [0, π2 ] can be used to parameterize thematrices Σ and B as [9]

Σ =√P0

[ cosψ 00 sinψ

],B =

[ cos θ − sin θsin θ cos θ

]. (6)

Exhaustive numerical search is used in [9] to optimize (ψ, θ).The similar approach of exhaustive search is also adopted in[5]–[8], although the rotation matrix therein is not constrainedto be real-valued.

In this letter, we propose an ellipse expanding method(EEM) to solve (3), assuming that B is real-valued. Theproposed algorithm yields a closed-form solution of the opti-mum precoder, which is computationally more efficient thanthe existing algorithms especially for the higher-order QAMinputs. Moreover, we obtain a closed-form expression of d2minas a by-product of the EEM.

III. ELLIPSE EXPANDING METHOD

A. Finite Lattice Points in 2-dimensional Plane

Note that although d2min of (5) is computed over the set ofcomplex-valued vectors X , it can actually be calculated overa real-valued set as claimed in the following lemma.

Lemma 1: When B is constrained to be real-valued, forsquare M -QAM inputs, the expression of d2min in (5) isequivalent to

d2min = minw∈W

‖ΛΣBw‖2 (7)

where W = Q2×1\{0} with Q = {−√M+1, · · · ,√M−1}.

Proof: From x ∈ X we know that Re{x} ∈ Q2×1 andIm{x} ∈ Q2×1, where Re{·} and Im{·} stand for the realpart and imaginary part, respectively. Under the real-valuedassumption of B, d2min can be decomposed as

d2min = min(Re{x}Im{x}

)∈ Q4×1\{0}

{‖ΛΣBRe{x}‖2+‖ΛΣBIm{x}‖2

}.

(8)As a QAM constellation is symmetric between the real andimaginary (or I and Q) parts, the minimization of d2min canbe achieved if and only if the first part of (8) is minimizedwhile the second part equals zero or vice versa. W.l.o.g., weonly need to perform the real part minimization on the setW = Q2×1\{0} which contains finite lattice points in 2-Dplane and (7) follows from (8).

Use Lemma 1, we can rewrite (3) as

ψopt, θopt, dmin =argmaxψ,θ,d

d

subject to ‖ΛΣBw‖ ≥ d, for ∀w ∈ W .(9)

B. Ellipse Expanding Method

We propose a closed-form solution to (9) from a geometricperspective. Note that the set {x ∈ R

2×1 : ‖ΛΣBx‖ = d} isan ellipse centered at the origin of the R2 plane, where thediagonal entries of ΛΣ determine the shape of this ellipsewhile the orientation of the ellipse is regulated by the rotationmatrix B. The constraint of (9) means that no points in shouldfall within the ellipse. Geometrically speaking, we want tocontrol the shape (by choosing Σ) and orientation of theellipse (by choosing B) such that the ellipse is expanded tothe max extent (to maximize d) while containing no points ofW in its interior. The optimum solution only occurs whenthe ellipse touches two or more points of W . (Note thatsuch points always occur in antipodal pairs.) Based on theobservation, we develop the EEM to solve (9) efficiently.

We first introduce the concept of feasible ellipse.Definition: An ellipse is referred to as feasible if it touches

three pairs of antipodal points in W and there are no otherlattice points which fall within the ellipse. The points touchedby the ellipse are called critical points.

An origin-centered ellipse can be represented in matrix formxTAx = 1, where x = (x1, x2)

T ∈ W and A = [a, b; b, c] orequivalently

ax21 + 2bx1x2 + cx22 = 1, (10)

Given three independent critical points, the triple (a, b, c) canbe solved based on three linear equations.

According to (7), A = BTΣΛ2ΣB can be rewrittenexplicitly as

A =

⎡⎢⎢⎢⎢⎣

λ2H,1 cos2 θ cos2 ψ+ λ2H,2 cos θ sin θ sin

2 ψ−λ2H,2 sin

2 θ sin2 ψ λ2H,1 cos θ sin θ cos2 ψ

λ2H,2 cos θ sin θ sin2 ψ− λ2H,1 sin

2 θ cos2 ψ+

λ2H,1 cos θ sin θ cos2 ψ λ2H,2 cos

2 θ sin2 ψ

⎤⎥⎥⎥⎥⎦ .

Hence one can derive the angular parameters (ψ, θ) withrespect to the triple (a, b, c) as follows

θ =1

2arctan

2b

a− c(11)

ψ = arctan

(√(a+ c)−√(a− c)2 + 4b2

(a+ c) +√(a− c)2 + 4b2

κ

), (12)

where κ � λH,1

λH,2stands for the condition number of H. Hence

if the ellipse is chosen, then the precoder F = VΣB isuniquely determined.

The EEM aims to find out the optimum ellipse efficiently.It consists of two steps. The first step is to compute offlineall the feasible ellipses. Given a combination of critical pointsxi,xj ,xk ∈ W , one can readily obtain A by solving thethree linear equations of the form (10). The solution A shouldmeet two constraints: i) wTAw ≥ 1 for ∀w ∈ W , and

492 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 4, APRIL 2012

−3 −2 −1 0 1 2 3−4

−3

−2

−1

0

1

2

3

4

x

y

Lattice pointE1E2E3E4

Fig. 1. Feasible ellipses in 2-D plane for 16-QAM constellation.

ii) A is positive definite. Searching over the combinationsof the critical points, we can obtain all the feasible ellipsesrepresented by the matrices Ai, i = 1, 2, · · · , N .

As an illustrative example, Fig. 1 presents all the N = 4feasible ellipses obtained by the EEM for 16-QAM constel-lation, labeled as E1 to E4. Note that the number of feasibleellipses increases with QAM size. For 4-QAM, there is onlyone feasible ellipse, while N = 18 and 32 for 64-QAM and256-QAM, respectively. It is important to note that the offlinepart of the EEM only needs to be performed once for all andtherefore does not add computational complexity to its realimplementation.

In the second step, i.e. the online step, the EEM chooses theellipse that yields the optimum dmin [see (7)] for a specificchannel realization. It turns out that we can determine whichellipse to choose based on κ.

Denote λ(i)A,1 ≥ λ(i)A,2 ≥ 0 as the singular values of Ai. It

follows from (10) that in this case the transmit power

pi =λ(i)A,1

λ2H,1+λ(i)A,2

λ2H,2. (13)

Note that xTAix = 1 for some critical point x ∈ W . Underthe power constraint trace(Σ2) = P0, it follows that theminimum distance is

d2min,i =P0

piminx∈W

xTAix =P0(λ

2H,1 + λ2H,2)

λ(i)A,1(1 + κ−2) + λ

(i)A,2(1 + κ2)

.

(14)Hence in the online step, the EEM chooses the ellipse withindex

i = argmaxi

1

λ(i)A,1(1 + κ−2) + λ

(i)A,2(1 + κ2)

. (15)

Fig. 2 shows d2min,i of (14) as a function of κ for the caseof 16-QAM input, where w.l.o.g P0(λ

2H,1 +λ2H,2) is assumed

to be one. Note that the κ-axis is segmented into multipleintervals and we can label each interval by the index ofoptimum ellipse. For instance, corresponding to the intervalκ1 ≤ κ ≤ κ2, the ellipse E2 is the optimum one. The

2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

κ

d2 min

E1E2E3E4L1L2L3L4

κ1

κ2

κ3

Fig. 2. d2min,i versus κ among feasible ellipses for 16-QAM constellation.

TABLE IPARAMETER LOOKUP TABLE FOR 4-, 16-, AND 64-QAM

CONSTELLATIONS

M κ θ(rad) A

41 ∼ 2.6455 π

4[1,-0.5;-0.5,1]

2.6455 ∼ ∞ 0.4636 [1,-2;-2,4]

16

1 ∼ 2.7575 π4

[1,-0.5;-0.5,1]2.7575 ∼ 6.3293 0.4914 [1,-1.5;-1.5,3]6.3293 ∼ 9.7892 0.3474 [1,-2.5;-2.5,7]

9.7892 ∼ ∞ 0.2450 [1,-4;-4,16]

64

1 ∼ 2.7575 π4

[1,-0.5;-0.5,1]2.7575 ∼ 6.3293 0.4914 [1,-1.5;-1.5,3]

6.3293 ∼ 10.2239 0.3474 [1,-2.5;-2.5,7]10.2239 ∼ 13.5809 0.5763 [3,-4.5;-4.5,7]13.5809 ∼ 19.2505 0.2640 [1,-3.5;-3.5,13]19.2505 ∼ 24.1993 0.6235 [7,-9.5;-9.5,13]24.1993 ∼ 27.6122 0.3766 [3,-7.5;-7.5,19]27.6122 ∼ 33.3233 0.5450 [7,-11.5;-11.5,19]33.3233 ∼ 37.5851 0.1757 [1,-5.5;-5.5,31]

37.5851 ∼ ∞ 0.1244 [1,-8;-8,64]

segmentation κi’s can be computed offline and stored ina lookup table (LUT) along with the corresponding ellipsematrices Ai, i = 1, 2, · · · , N . Hence in the online step ofEEM, instead of solving (15), one only needs to check whichinterval κ falls into.

In the above discussion, we have assumed a finite κ. Foran ill-conditioned channel with κ→ ∞, however, the feasibleellipse degenerates to a pair of parallel lines and A becomesrank deficient. In this letter, we still refer to the degradationcurve as feasible ellipse for simplicity’s sake. In this case, thefeasible ellipse only has two pairs of critical points in W .

With the assumption of rank deficient A, Fig. 3 depictsfour pairs of parallel lines obtained by the EEM for 16-QAM constellation, labeled as L1 to L4. And the closed-formexpression of d2min,i in (14) still holds for this case except that

λ(i)A,2 = 0. A comparison of d2min,i among all feasible ellipses

including the ill-conditioned case is shown in Fig. 2. It canbe seen from this diagram that some of the feasible ellipsesare never to be chosen, such as E3, L2, L3 and L4, since theyhave smaller minimum distance at all time.

To summarize, the EEM consists of two steps, i.e., theoffline computation step and the online step. In the offline

XU et al.: ELLIPSE EXPANDING METHOD FOR MINIMUM DISTANCE-BASED MIMO PRECODER 493

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

x

y

Lattice pointL1L2L3L4

Fig. 3. Special case of feasible ellipses in 2-D plane for 16-QAMconstellation.

step, EEM searches and computes all feasible ellipses Ai

and obtains the interval segmentation of κ using (14). Theoptimum ellipse is assigned to each interval according to (15).The pre-computed information is stored in a LUT. During theonline step, the EEM computes κ according to the specificchannel realization and then checks the LUT to determinewhich Ai is to use and obtain θ and ψ from (11) and (12)to form the precoder F = VΣB. In fact, since (11) isindependent of channel realization, θ can also be pre-computedand stored in the LUT. Hence the computational complexityof the online step is extremely low, which is a remarkableadvantage over the existing methods [5]–[9].

Table-I is the LUT generated by the EEM for 4-QAM,16-QAM and 64-QAM, which is approximately identical tothat in [9]. However, using the closed-form expression ofdmin obtained in (14), the EEM achieves better segmentationaccuracy than [9] for high-order QAM. For instance, additionalinterval of κ ∈ [33.3233, 37.5851] is obtained by EEM for64-QAM, which means that we can achieve larger minimumdistance than [9] at this interval. The LUT for 256-QAM

is available online: http://www.sal.ufl.edu/yjiang/papers/EEM-LUT-256QAM.pdf and is omitted here due to space limit.

IV. CONCLUSION

In this letter, we exploit the geometric feature of theMIMO precoder problem and propose an ellipse expandingmethod (EEM) to optimize MIMO precoder for dual-streamdata transmission. Using EEM, an optimum precoder can beachieved with very low implementation complexity. A closed-form expression of the minimum distance for square QAMconstellations is also obtained as a by-product. The extensionto the case of more than two data streams transmission is underinvestigation and may be presented in upcoming literature.

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