Transmitter Design for Uplink MIMO Systems With Antenna ...

1772 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 4, APRIL 2015

Transmitter Design for Uplink MIMO SystemsWith Antenna Correlation

Chongbin Xu, Peng Wang, Member, IEEE, Zhonghao Zhang, and Li Ping, Fellow, IEEE

Abstract—We study the uplink transmission in multiple-inputmultiple-output (MIMO) systems with antenna correlation. Wefocus on schemes that require only channel covariance informationat the transmitter (CCIT), which involves lower cost than fullchannel state information at the transmitter (CSIT). We start frommutual information analysis and show that a simple CCIT-basedscheme, referred to as statistical water-filling (SWF), can per-form close to the optimal full CSIT-based one in MIMO systemswith more receive antennas than transmit ones. We then focuson the implementation of SWF in practically coded systems. Aniterative linear minimum mean squared error (LMMSE) receiveris assumed and an extrinsic information transfer (EXIT) charttype curve matching technique is developed based on Hadamardprecoding techniques. Simulation results show that the proposedscheme can obtain significant performance improvement com-pared to the conventional equal power transmission. Finally, weshow that the proposed scheme is also very efficient in multi-useruplink MIMO systems with distributed channel information.

Index Terms—Uplink MIMO, antenna correlation, channelcovariance information.

I. INTRODUCTION

R ECENTLY, massive multiple-input multiple-output(MIMO) systems [1]–[3] have attracted extensive

research attention. Let us focus on the uplink. Denote byNBS and NMT, respectively, the numbers of antennas at thebase station (BS) and at the mobile terminal (MT). The basicassumptions of massive MIMO are NBS → ∞ and NBS � NMT.These assumptions lead to the so-called favorable propagationeffect. Then equal power and rate allocation over all antennasat an MT is nearly optimal provided that these antennas areuncorrelated [2], [3]. This does not require channel stateinformation (CSI) at the transmitter (CSIT) and so avoidsthe cost related to CSIT acquisition. It is also known thatsimple maximum ratio combining (MRC) is nearly optimal for

Manuscript received March 16, 2014; revised July 14, 2014 and November 4,2014; accepted November 4, 2014. Date of publication November 20, 2014;date of current version April 7, 2015. This work was jointly supported bya grant from the University Grants Committee of the Hong Kong SpecialAdministrative Region, China (Project No. AoE/E-02/08), a strategic researchgrant from City University of Hong Kong, China (Project No. 7002869), and agrant from the Research Grant Council of the Hong Kong SAR, China (ProjectNo. CityU 118013). This paper was presented in part at the 8th InternationalSymposium on Turbo Codes & Iterative Information Processing, Bremen,Germany, August 2014. The associate editor coordinating the review of thispaper and approving it for publication was M. Elkashlan.

C. Xu, Z. Zhang, and L. Ping are with the Department of ElectronicEngineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail:[email protected]; [email protected]; [email protected]).

P. Wang is with the School of Electrical and Information Engineering,University of Sydney, Sydney, NSW 2006, Australia (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2014.2372768

massive MIMO [1]–[3]. Therefore, transmission and detectioncan be greatly simplified when MIMO is “massive”.

However, in practice, the antennas on a MT are typicallycorrelated due to the limited physical size. It is also reasonableto expect that the MIMO size will become large but may not bemassive in the near future due to technical difficulties. There areworks considering antenna correlation or “not so large” MIMO,but they are mostly on mutual information analysis [4]–[6]. Inthis paper we focus on practical transmitter design for suchsystems.

In general, CSIT can be exploited to improve MIMO per-formance [7]–[18]. However, CSIT acquisition becomes costlywhen NBS ×NMT, the number of channel coefficients, is evenjust modestly large. A conventional low-cost option is to utilizechannel covariance information (CCI) at the transmitter (CCIT)[7], [8], since CCI typically changes more slowly than CSIitself, implying that CCIT requires less updates and so lowercost than CSI.

Mutual information analysis for CCIT channels has beenextensively studied [19]–[30]. Optimization techniques havealso been developed for precoder design for uncoded systemswith CCIT [31]–[34]. Coded systems, however, represent adifficulty. Joint forward-error-control (FEC) coding, adaptivemodulation and linear precoding are generally required inMIMO systems to realize the performance promised by capac-ity analysis. Iterative detection can be applied to improve theperformance of such joint schemes. The extrinsic informationtransfer (EXIT) chart technique [35], [36] is a common designtool for systems with iterative detection. However, analysis foriterative system is highly complicated under CCIT, especiallywhen MIMO size is large. This makes it difficult to generatethe EXIT chart functions analytically. Monte Carlo method canbe used instead, but it is costly and so not suitable for real timeapplications.

Detection complexity is another problem. In a MIMO chan-nel with antenna correlation, information rates should be care-fully allocated to different channel eigen-directions. Adaptivemodulation is a standard approach for this purpose [37], [38].Denote the number of bits carried by the symbol on the n-theigen-direction by Qn, and its average over all eigen-directionsby Q̄. In general, Qn varies in the range of [0,NMTQ̄], whichincreases with the MIMO size. Then the demodulation com-plexity O(2Qn) for some large Qn values can be a problem evenif the MIMO size is only modestly large.

This paper is concerned with uplink MIMO systems withantenna correlation at the MT side. We start with a statisticalwater-filling (SWF) technique for channels with CCIT. SWFwas mentioned briefly in [23], [29], only as an alternative

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XU et al.: TRANSMITTER DESIGN FOR UPLINK MIMO SYSTEMS WITH ANTENNA CORRELATION 1773

concept. We show that SWF using simple maximum ratiocombing (MRC) is actually nearly optimal when NBS is suffi-ciently large, but it performs poorly when NBS is “not-so-large”.This motivates the joint FEC coding and linear precodingscheme developed below. The main contributions of this paperare as follows.

• We develop a fast optimization technique for joint FECencoding and linear precoding in MIMO systems withCCIT. This technique is efficient for three reasons. First,simple SWF is used to facilitate precoder design. Second,based on Hadamard precoding [11], [39]–[41], a closedform expression of the EXIT chart function is derivedfor iterative detection in a channel with CCIT and asufficiently large NBS. It facilitates fast EXIT chart curvematching for system optimization. Excellent performanceis demonstrated by simulation results even if NBS is “not-so-large”. Third, the proposed scheme employs a uniformmodulation constellation over all signal streams, but canstill realize the same water-filling effect of adaptive mod-ulation. This greatly eases the demodulation complexityproblem mentioned earlier.

• We extend the discussions to uplink systems in whichmultiple users transmit in the same time slot and sub-carrier simultaneously. To achieve optimal performance,the transmitters of all users should be designed jointlywith global CSIT, which is very costly. We present a verysimple solution in which each transmitter is optimizedbased on its own correlation matrix. We show that thissimple scheme provides a low-cost but nearly optimalsolution when the ratio of NBS/(KNMT) is only moderatelylarge.

The joint FEC coding and linear precoding scheme proposedin this paper represents a good trade-off between cost andperformance for modestly large MIMO systems, as confirmedby both analysis and simulation. This property is attractive forpractical applications.

II. TRANSMISSION STRATEGY WITH CCIT

A. System Model

Consider an uplink MIMO system. For simplicity, we firstassume that the transmissions of multiple users are orthog-onal. (We will discuss non-orthogonal multi-user concurrenttransmissions in Section V.) Then we can write the receivedsignal as

r = Hy+ηηη (1)

where r is an NBS × 1 signal vector received at BS antennas,H an NBS ×NMT channel transfer matrix, y an NMT × 1 signalvector transmitted from MT antennas, and ηηη an NBS ×1 vectorof complex additive white Gaussian noise (AWGN) with mean0 and variance σ2. We will assume that y has a zero meanE[y] = 0 and a total power constraint Pt , i.e.,

tr{Q} ≤ Pt (2)

where Q = E[yyH] is the transmission covariance matrix andtr{·} denotes the trace operation. Throughout this paper, we willalways assume E[H] = 0.

B. Kronecker Model

The discussions in the rest of this paper are based on theKronecker model that has been widely discussed and validatedunder certain practical channel conditions [7], [8], [19]–[30].This model is given by [19], [30]

H = C1/2BS HwC1/2

MT (3)

where Hw is an NBS ×NMT matrix whose elements are inde-pendent and identically distributed (i.i.d.) complex Gaussianrandom variables with distribution CN(0,1), and CBS andCMT are, respectively, NBS ×NBS and NMT ×NMT Hermitianmatrices characterizing the antenna correlation at the receiverand transmitter. We adopt the following normalization

tr{CBS}= NBS and tr{CMT}= NMT. (4)

In this paper, we will always assume that the receiver knowsfull channel state information (given by H) via proper channelestimation. We will also assume that the transmitter only knowschannel covariance information (given by CBS and CMT),which leads to significantly low cost. We will return to this inSection II-E.

C. Statistical Water-Filling (SWF)

Denote by

R(H,Q) = log2 det(INBS +HQHH/σ2) (5)

the achievable rate of the system in (1) with channel H andtransmission covariance matrix Q. With full CSIT, the channelergodic capacity under the power constraint (2) is given by

CFCSIT = EH

[max

tr{Q(H)}≤Pt

R(H,Q(H))

](6)

where Q(H) represents that the transmission covariance matrixQ is optimized based on H and the expectation is over thedistribution of H conditioned on the known CBS and CMT.

Under CCIT assumption, Q is independent of particularchannel realization. The related ergodic capacity is given by

CCCIT = maxtr{Q}≤Pt

EH [R(H,Q)] . (7)

The key here is to find Q that maximizes the average mutual in-formation EH[R(H,Q)]. In general, this is a highly complicatedproblem.

The following is a sub-optimal solution. Let the eigenvaluedecompositions of Q and CMT be

Q =UQDQUHQ, (8)

CMT =UMTDMTUHMT. (9)


It is known that the optimal UQ in (8) under the CCIT assump-tion is [26]–[28]

UQ = UMT. (10)

Following [23], [29], we generate a suboptimal DQ by max-imizing the Jensen bound of the average mutual informationEH[R(H,Q)]. Note that

det(INBS +HQHH/σ2) = det(INMT +QHHH/σ2),

EH[HHH] =NBSCMT.

Then we solve the following problem:

max{DDDQ}

log2 det(INMT +NBSDQDMT/σ2) (11a)

s.t. tr{DQ} ≤ Pt . (11b)

Solving (11) is straightforward; it is equivalent to water-fillingover a parallel channel with channel gains given by the diagonalelements of NBSDMT. This method is referred to as statisticalwater-filling (SWF) in [23]. The resultant Q, denoted by QSWF,serves as a low-cost and sub-optimal solution to (7). In thesequel, we will assess the performance of SWF and developits implementation techniques.

D. Linear Precoding

The SWF scheme can be realized using a precoding matrix

P = UQD1/2Q V (12)

where UQ and DQ are defined in (8) and V is an arbitraryNMT ×NMT unitary matrix that does not affect capacity. (How-ever, V may affect the performance of a practical system. SeeSection IV.) Assume that x is a length NMT ×1 coded sequencewith elements independently drawn from CN(0,1) and y = Px.Then the required Q is realized as

Q = E[yyH] = PPH. (13)

Substituting y = Px into (1), we have

r = HPx+ηηη. (14)

In general, HP is not diagonal and so there is inter-streaminterference. This is inevitable since the transmitter does notknow H perfectly. We will discuss the related issues in detail inSection IV.

E. Cost Related to Estimating CMT

With SWF, P in (12) can be generated solely based onCMT. In a time varying channel, CMT typically changes moreslowly than H, so it requires less updates. With frequencydivision duplex (FDD), this clearly reduces the cost related toobtaining CMT.

With time division duplex (TDD), CMT can be obtainedby sending pilot signals periodically from the BS. As CMT

typically changes slowly, the related overhead is relatively low

(compared with estimating the full channel matrix H involvingfast-changing Rayleigh fading).

Note that an MT needs to estimate channel for data detectionbut this does not, even with TDD, necessarily provide the fullinformation about H. To see this, assume that a linear precoderis used in the downlink. The received signal at an MT can bewritten as

rDL = HDLPDLxDL +ηηηDL (15)

where HDL = HH assuming channel reciprocity, rDL, PDL, xDL

and ηηηDL are defined similarly to their counterparts in the uplinkmodel (14). To detect xDL, it suffices to estimate HDLPDL,instead of HDL. We cannot directly obtain HDL from HDLPDL ifHDLPDL has a smaller size than HDL. (The number of columnsof PDL is given by the number of signal streams in xDL. Thelatter is limited by min(NMT,NBS) that is typically less than thenumber of columns of HDL (i.e., NBS) when NMT < NBS.)

III. MUTUAL INFORMATION ANALYSIS

The low-cost SWF scheme outlined in Section II is sub-optimal. In this section, we will show by mutual informationanalysis that the potential performance loss is actually marginalfor the uplink with NBS > NMT.

A. Asymptotic Analysis

In this subsection, we consider the asymptotic situation ofNBS → ∞. We will assume that CBS does not contain dominanteigenvalues when NBS → ∞, i.e.,

limNBS→∞

λn(CBS)/tr{CBS}= 0,∀n (16)

where λn(CBS) is the nth eigenvalue of matrix CBS. It can beverified that the above assumption holds provided that there issufficient spacing among BS antennas [42], [43].

Denote by CFCSIT the capacity of the system in (1) with fullCSIT (see (6)). Also denote by RSWF the average achievablerate of SWF:

RSWF = EH [R(H,QSWF)] (17)

with QSWF given by (8), (10), and (11). Clearly, CFCSIT ≥RSWF.Proposition 1: The SWF scheme is asymptotically op-

timal, i.e.,

limNBS→∞

|CFCSIT −RSWF|= 0, (18)

if(i) NBS → ∞ with NMT fixed and CBS meets (16); or

(ii) NBS → ∞ with N3MT/NBS → 0, and CBS = I.

Proof: This can be proved using large random matrixanalysis [44], [45]. For details, see Appendix A.

Proposition 1 reveals that the full CSIT capacity can beapproached by simple SWF under the given two conditions.Recall that SWF only requires the knowledge of CMT at thetransmitter. Full CSIT provides no further performance im-provement. This indicates that SWF is an attractive option for


Fig. 1. Performance of systems with different receive antenna configurations.NMT = 8, and ρBS = 0.5. (a) ρMT = 0.2 (b) ρMT = 0.8.

uplink massive MIMO systems. For practical considerations, itis also interesting to examine the implication of Proposition 1in systems with limited size. The numerical results below showthat SWF is still attractive in the latter case.

B. Numerical Results

We now present numerical results to verify the efficiencyof the SWF scheme. For simplicity, we adopt the exponentialmodel [45], [46] to characterize the antenna correlation, i.e.,

CBS(m,n) =ρ|m−n|BS ei(m−n)θBS (19a)

CMT(m,n) =ρ|m−n|MT ei(m−n)θMT (19b)

where iΔ=

√−1, ρBS and ρMT ∈ [0,1] are respectively receive

and transmit correlation factors, and θBS and θMT are uniformlydistributed over [0,2π).

Fig. 1 compares the SWF performance with the full CSIT(FCSIT) capacity. The latter serves as a performance upperbound. We can see that SWF is nearly optimal even formodestly large NBS. This verifies the asymptotic analysis inProposition 1. The gap between the two schemes reduces whenthe transmit correlation factor ρMT increases. Intuitively, this isbecause a larger ρMT implies a lower degree of freedom, whichhas similar effect as a smaller NMT (and so a larger NBS/NMT

ratio).

Fig. 2. Performance of systems with a fixed NBS/NMT ratio. NBS/NMT = 4,and ρBS = 0.5. (a) ρMT = 0.5 (b) ρMT = 0.8.

Fig. 2 provides the performance when NMT varies whilethe ratio NBS/NMT is fixed. For reference, we also includethe performance of systems of no precoding (NP), in whichwe select Q = Pt/NMTI. We can see that SWF can obtainperformance close to the FCSIT upper bound. Its gain over thesystem of no precoding is noticeable, especially when NMT islarge. This indicates the necessity to introduce proper precodingin the uplink when MTs are equipped with multiple antennas.Comparing Fig. 2(a) and 2(b), we can see that the advantageof SWF over NP increases when the transmit correlation factorρMT increases. This is expected, since the related gain is 0 in theextreme case of ρMT = 0 (and so CMT = I) where equal powerallocation is optimal.

Condition (ii) in Proposition 1 requires N3MT/NBS → 0, which

is a relatively strong requirement. In practice, this may nothold and then the terms NMT log2(1 ± 2

√NMT/NBS) in (46)

and (47) (see Appendix A) is not negligible. This effect canbe seen in Fig. 2: the gap between the FCSIT bound and SWFperformance increases slightly as NMT increases.

IV. TRANSMITTER AND RECEIVER DESIGN

We now discuss the implementation of SWF. The mutualinformation analysis in Section III shows that SWF can obtaina good performance even if NBS is “not so large”. However,careful system design is still required for practical systems. Inthis section, we will show the following.


Fig. 3. Performance of MRC receiver with equal power allocation. NMT = 8,ρMT = 0.8, and ρBS = 0.5.

• A simple maximum ratio combining (MRC) receiver per-forms well if NBS → ∞, which is consistent with mutualinformation analysis. However, it may perform poorly fora “not so large” NBS value due to the residual interferenceproblem when the channel is not perfectly diagonalized.

• Iterative detection can be used to suppress interference andprovides a possible way to implement the potential benefitof SWF for a modest large NBS value.

• The transmitter can be optimized by a curve matchingtechnique based on the EXIT chart principle. This match-ing requires only CMT at the transmitter, which suits theSWF scheme well.

• The demodulation complexity of the proposed scheme isconsiderably lower than the conventional adaptive modu-lation techniques for SWF.

A. MRC Detection

Recall from Section III that, with SWF, multiple signalstreams are transmitted along eigen-directions of CMT. Theorthogonality among these signal streams is not guaranteedsince CMT does not provide perfect CSI. When NBS → ∞, thesystem converges to a parallel channel model in which differentsignal streams are orthogonal to each other. (See Appendix A.)MRC is optimal for such a parallel model. This is equiva-lent to the favorable propagation property in massive MIMO[1]–[3]. However, such convergence is quite slow. For practicalNBS values, the interference among different signal streams maynot be negligible, which results in performance loss for MRCreceiver.

Fig. 3 shows the convergence property for a regular (3, 4)LDPC coded system with MRC receiver. We focus on theimpact of the interference among different eigen-directionsof CMT (assuming UQ = UMT). Equal power allocation anduniform QPSK modulation are assumed for simplicity. (Water-filling implies different rates on different eigen-directions ofCMT and so different constellations, which is generally com-plicated.) For reference, an artificial parallel channel model isconsidered in Fig. 3 using the eigenmodes of CMT. We can seethat the performance gaps are quite significant for “not so large”

NBS values. This indicates the necessity to design detectiontechnique carefully for “not so large” MIMO.

B. Hadamard Precoding

In what follows, we develop an improved precoding schemebased on iterative detection. First, let V = VHad in (12), whereVHad is a Hadamard matrix with a proper size. (See Fig. 4.) Forconvenience, we will call

P = UQD1/2Q VHad (20)

a Hadamard precoder. Similar precoder structures have beenpreviously discussed for channels without CSIT in [39] and forchannels with full CSIT in [11], [40]. In this paper, our focus isits application in SWF. As we will see below, the use of VHad in(20) facilitates the EXIT chart type curve matching for SWF.

C. Iterative LMMSE Detection

Substituting (20) into (14), the received signal is given by(see Fig. 4)

r = HUQD1/2Q VHadx+ηηη. (21)

Applying the standard LMMSE detection to r in (21), wehave [47]

x̂ = E[x]+ v(H̃VHad)HR−1 (r− H̃VHadE[x]

)(22a)

where H̃ = HUQD1/2Q , E[x] is the a priori mean of x, vI is the

a priori covariance of x and

R Δ= cov(r,rH) = v(H̃VHad)(H̃VHad)

H +σ2I. (22b)

Initially, if we do not have any information on x, we set E[x] = 0and vI = I (assuming that the symbols in x have average powerof 1). We rewrite (22a) into a symbol-wise form as

x̂(i) = vΩΩΩ(i, i)x(i)+ ξ(i) (23a)

where ΩΩΩ(i, i) is the ith diagonal element of the following matrix

ΩΩΩ Δ=((H̃VHad)

HR−1(H̃VHad))

diag (23b)

and ξ(i) is an interference-plus-noise term. Following the treat-ments in [48], we approximate ξ(i) by an AWGN sample withvariance (c.f. (18) in [49])

Var(ξ(i)) = vΩΩΩ(i, i)(1− vΩΩΩ(i, i))v. (23c)

We can then estimate x(i) using the symbol-wise modelin (23a).

The estimate of x is then processed by a forward-error-control (FEC) decoder (denoted by DEC in Fig. 4) usinga posteriori probability (APP) decoding. The decoding outputcan be used to update the values of E[x] and v for the sequencex. Then LMMSE detection can be performed again. This pro-cess continues iteratively.


Fig. 4. Illustration of the precoded system. ENC stands for encoder and DEC for decoder.

D. Demodulation Complexity

We now briefly discuss the complexity issue. Let x be dividedinto sections; each section contains NMT symbols transmittedover each time use of the channel. Assume that these NMT

symbols carry a total of NMT × Q̄ bits. For simplicity, let Q̄ bean integer.

For the Hadamard precoding scheme in (21), each symbolin x carries Q̄ bits based on a constellation of size 2Q̄. Thedemodulation complexity involved in (23) is fixed at O(2Q̄).

As a comparison, consider an alternative method based onadaptive modulation, in which NMTQ̄ bits are allocated amongNMT symbols in a section of x. Denote by Qn the number ofbits carried by the symbol on the nth eigen-direction, whichcan vary in the range of [0,NMTQ̄]. Note that NMTQ̄ can bequite large for a relatively large MIMO. The demodulationcomplexity O(2Qn) for some large Qn values then becomes aserious problem.

The key here is that the worst-case complexity can be veryhigh if variable constellation sizes are involved. The precod-ing scheme in (21) employs a uniform constellation size forall symbols, which avoids the problem. Note that here theHadamard precoding scheme is still adaptive to channel con-dition, which is achieved by controlling the power allocationmatrix DQ in (21).

E. Transfer Function for LMMSE Detection

Note that in the symbol-wise demodulation in (23), the inter-ference among different symbols in x is treated as additive noise(included in ξ(i)) for simplicity. The residual interference maystill affect performance noticeably if not treated properly. In thefollowing subsections, we outline an optimization procedureto minimize the interference effect, which is equivalent tomaximize the signal to interference-plus-noise ratio (SINR),during the iterative process.

Recall that ΩΩΩ in (23) characterizes the SINR achieved byLMMSE estimation. In general, such SINR can fluctuate fordifferent symbols with different index i. We can smoothenout such fluctuation by a transmission technique involving anaugmented channel model. This is detailed in Appendix Bwhere the following approximation is derived:

ΩΩΩ ≈ ωI (24)

with ω = N−1MTtr{ΩΩΩ}. The use of the Hadamard matrix VHad

in (21) is the key to this approximation. Then, using (24) and(23c), we can compute the SINR in x̂(i) in (23a) as

ρ(i) = |vΩΩΩ(i, i)|2 /Var(ξ(i)) = ω/(1− vω),∀ i. (25a)

Then the LMMSE module can be characterized by a function

ρ= φ(v)Δ= ω/(1− vω). (25b)

Fig. 5. The curves of φ(v) with SWF for NMT = 8, ρMT = 0.8, ρBS = 0.5, andEb/N0 =−10log10(NBS).

Proposition 2: Assume the Kronecker model (3). For UQ =UMT and NBS → ∞, we have

φ(v) = ω∞(v)/(1− vω∞(v)) (26a)

where

ω∞(v) =1

NMT∑n

DMT(n,n)DQ(n,n)vDMT(n,n)DQ(n,n)+σ2/NBS

(26b)

if the two conditions in Proposition 1 hold.Proof: See Appendix C.

Proposition 2 shows that φ(v) is asymptotically determinedby CMT, since both DMT and DQ are functions of CMT. (See(9) and (11) in Section II.) This is consistent with Proposition 1that the system performance is determined by CMT, except herea practical iterative LMMSE receiver is considered.

Recall from Section IV-A that MRC performance requires arelatively large NBS value to converge. On the contrary, the con-vergence speed for LMMSE detection related to Proposition 2is much faster. This is illustrated in Fig. 5 by a numericalexample. The φ(v) curves for the actual channel H and theartificial parallel channel NBSDMT are provided. Noting theterm σ2/NBS in the denominator in (26b), we set Eb/N0 =−10log10(NBS) in Fig. 5 for different NBS values for a faircomparison. We can see that the curves are quite close forall the NBS values considered. This property is crucial for ourdiscussions in the following subsections.

F. Transmitter Optimization

The performance of the FEC decoder is determined by itsinput SINR (denoted by ρ) and the quality of its output canbe measured by the variance (denoted by v). This can becharacterized by a function v = ψ(ρ) that can be produced bypre-simulation [50]. This function is the counterpart of ρ=φ(v)discussed earlier.


Similar to [35], [36], we can show that the performance of theiterative receiver in Fig. 4 is characterized by a fixed point of thetwo transfer functions ρ= φ(v) and v = ψ(ρ). Given ρ= φ(v),we can carefully design an irregular LDPC code such that ψ(ρ)matches φ(v). This EXIT chart technique has been discussed in[51], [52] and details will be omitted.

The above design strategy requires knowing CMT, but not H,at the transmitter. This leads to a key conclusion of this paperthat for an uplink MIMO system with NBS > NMT, an efficienttransmitter can be designed without the explicit knowledge ofthe complete channel coefficients. Knowing CMT is sufficientfor this purpose. This fact greatly relieves the burden of channelstate information acquiring at the transmitter.

G. Variable and Fixed FEC Code Structures

We now consider two situations that may arrive in practice.We first assume FEC code optimization is allowed. In this case,the SWF based design can be used as initial values for theprecoder (see Section II-C for details) that provides φ(v). Thenan LDPC code (characterized by ψ(ρ)) can be optimized usingthe EXIT chart technique as just described. After obtaining theLDPC code, we may optimize the precoder again. However,only marginal improvement is observed in this way.

In the above, when CMT changes, a different LDPC code isoptimized. The transmitter needs to inform the receiver aboutthe structure of this code. This incurs considerable complexityoverhead. Alternatively, we may fix the FEC code and optimizethe precoder. We assume the receiver can acquire the overallequivalent channel including the physical channel and the newprecoder through proper channel estimation. See (15) and therelated discussions. This avoids the overhead mentioned above.An example for such situation can be found in Fig. 8.

H. Numerical Results

We now demonstrate the effectiveness of transmitter opti-mization using numerical results in Fig. 6. For reference, weconsider equal power (EP) allocation (in which the precoderP = (Pt/NMT)

1/2VHad) and a regular LDPC code for the casewithout CSIT. Specifically, in Fig. 6(a), a rate-0.25 (3, 4)LDPC code with length 32768 is adopted, followed by QPSKmodulation. In Fig. 6(a), we can observe an early crossing pointbetween φ(v) for EP precoder and ψ(ρ) for the (3,4) LDPCcode. This leads to considerable performance deterioration, aswill be shown in Fig. 7.

Note that EP in the above still involves a precoder P =(Pt/NMT)

1/2VHad, which is different from NP (i.e., no pre-coding) in Fig. 2 in which P = (Pt/NMT)

1/2I. For mutualinformation analysis, EP and NP are equivalent since they leadto the same transmission covariance matrix. However, for apractically coded system, the use of a Hadamard matrix in EPcan provide a better diversity gain as discussed in [39].

In Fig. 6(b), we first generate φ(v) using the SWFtechnique described in Section II. We then generate a rate-0.25irregular LDPC code with a ψ(ρ) function that best matchesφ(v). (Variable and check nodes degree distributions aregiven, respectively, by λ(x) = 0.570146x1 + 0.047378x2 +0.193282x7+0.035182x8 + 0.154016x34 and ρ(x) = x3.) Note

Fig. 6. SINR-variance transfer curves. NMT = 8, NBS = 64, ρMT = 0.8, andρBS = 0.5.(a) Eb/N0 =−14.6 dB (b) Eb/N0 =−20.4 dB.

Fig. 7. Simulation performance of the proposed scheme. NMT = 8, ρMT = 0.8,ρBS = 0.5, and rate = 4 bits/symbol.

that we can further match the two functions by alternatingbetween optimizing φ(v) for a given ψ(ρ) and optimizing ψ(ρ)for a given φ(v). However, we observed only limited gain inthis way.

Fig. 7 shows the simulated performance for the design ex-amples in Fig. 6. We can see that though the LDPC code isdesigned based on the asymptotic analysis, it works well insystems even for NBS = 8. The proposed scheme obtains a sig-nificant improvement from the EP performance (about 5.0 dBfor NBS = 64 and 5.4 dB for NBS = 8). The threshold given bythe EXIT chart type analysis is less than 0.5 dB away from theFCSIT capacity limit given in Section III.


Fig. 8. Simulation performance of the proposed scheme with varying ρMT.NMT = 8, NBS = 64, ρBS = 0.5.

An alternative is to fix coding for all realizations of CMT

and employ adaptive precoding (by optimizing DQ in (21)) fordifferent CMT. In this way, the transmitter does not need toinform the receiver for the following reason. For data detection,the receiver can estimate HP (instead of H), where HP is theequivalent channel formed by the physical channel H and theprecoder P. This avoids the additional cost.

Fig. 8 shows an example of a fixed LDPC code and anadaptively optimized precoder, as discussed in Section IV-G.We assume that CMT varies with ρMT ∈ [0.7,0.9] with thecode optimized at ρMT = 0.8. We can again see significantperformance gain in this case.

V. EXTENSION TO MULTI-USER SYSTEMS

In a multi-user MIMO system, global CSIT can lead to sig-nificant performance gain [53]. However, this requires feedingback CSI for all users together with centralized optimization[54], [55]. It is very costly in terms of computation and feed-back bandwidth. In this section, we consider an individualCCIT scheme, in which each user individually designs trans-mitter based on its own CCIT. We will show that this low-costoption can still achieve near-optimal performance.

A. System Model

The system model in (1) is rewritten as follows

r =K

∑k=1

√γkHkyk +ηηη (27)

where γk is the large scale fading factor of user k, Hk is theRayleigh-fading channel matrix modeled as (3), and yk thetransmitted signal vector of user k with zero mean and a powerconstraint Pk, i.e.,

E[yk] = 0 and tr{Qk} ≤ Pk (28)

with Qk=E[ykyHk ] the transmission covariance matrix of user k.

Following (3) and (19), we denote by CBS,k and CMT,k the

Fig. 9. Mutual information performance in a K-user system. K = 4, NMT = 8,ρBS = 0.5, Pk = P/K,k = 1, . . . ,K.

receive and transmit correlation matrices of user k modeled by

CBS,k(m,n) =ρ|m−n|BS,k ei(m−n)θBS,k , (29a)

CMT,k(m,n) =ρ|m−n|MT,k ei(m−n)θMT,k . (29b)

And we also assume that the channels of different users areindependent.

B. Mutual Information Performance

Denote by Rsum(S) the achievable sum rate of user subsetS⊆ {1, . . . ,K}, i.e.,

Rsum(S) = log2 det

(INBS + ∑

k∈SγkHkQkHH

k /σ2

). (30a)

The achievable rate region of the system in (27) is given by

R({Hk},{Qk})={(r1, . . . ,rK)| ∑

k∈Srk ≤ Rsum(S),∀S

}.

(30b)

Denote by CFCSIT the full CSIT capacity region that canbe obtained by optimizing {Qk} in (30). Denote by RSWF theachievable rate region of individual SWF (I-SWF) scheme inwhich each user individually performs SWF precoding andignores the presence of other users. We now show that RSWF

approaches CFCSIT asymptotically when NBS → ∞.Proposition 3: With K and NMT fixed, when NBS → ∞ and

(16) holds for each CBS,k, we have ∀(r1, . . . ,rK) ∈ CFCSIT,∃(r′1, . . . ,r′K) ∈ RSWF,

limNBS→∞

maxk

∣∣rk − r′k∣∣= 0. (31)

Proof: See Appendix D.Fig. 9 shows the I-SWF performance. We assume that K MTs

are uniformly distributed in a hexagon region with sides of 1.The large scale fading factor γk consists of path loss with decayorder 4 and lognormal fading with standard derivation 8 dB,and ρMT,k is uniformly distributed over [0.6, 0.9]. The channelcapacity based on global CSIT and no CSIT are included forreference. We can see from Fig. 9 that I-SWF can obtain


performance close to the global CSIT upper bound. It offerssignificant performance improvement compared to systems ofno precoding (NP).

C. Implementation in Practical Systems

To implement I-SWF, we consider the similar scheme asin Section IV except that multi-user detection is adopted. Weoutline the main detection and analysis process as follows.

Substituting (20) into (27), the received signal is given by

r =K

∑k=1

√γkHkUQ,kD1/2

Q,kVHadxk +ηηη. (32)

With LMMSE detection, we have

x̂k = E[xk]+ vk(H̃kVHad)HR−1

(r−

K

∑k=1

H̃kVHadE[xk]

)(33)

where H̃k =√γkHkUQ,kD1/2

Q,k , and R = ∑Kk=1 vkH̃kDQ,kH̃H

k +

σ2I.Similarly as in the single-user case, we can use the following

matrix to characterize the SINR of LMMSE estimation in (33)

ΩΩΩkΔ=((H̃kVHad)

HR−1(H̃kVHad))

diag . (34)

With an augmented channel model, we can show that ΩΩΩk ≈ ωkIwith ωk = N−1

MTtr{ΩΩΩk}. Hence, the LMMSE module of user kcan be characterized by a function

ρk = φk(vk) = ωk/(1− vkωk). (35)

When NBS → ∞, the transfer function in (35) can be simplifiedas given below.

Proposition 4: With K and NMT fixed, and UQ,k = UMT,k,when NBS → ∞ and (16) holds for each CBS,k, we have

φk(vk) = ω∞,k(v)/(1− vkω∞,k(vk)) (36a)

where

ω∞,k(v) =1

NMT∑n

DMT,k(n,n)DQ,k(n,n)

vkDMT,k(n,n)DQ,k(n,n)+σ2/NBS. (36b)

Proof: See Appendix E.Proposition 4 shows that φk(vk) is asymptotically determined

by CMT,k ignoring the presence of other users when NBS issufficiently large. Later, we will show by numerical results thatgood performance can be obtained even if NBS/(KNMT) is onlymoderately large.

Fig. 10 provides the simulation results. Again EP and regularLDPC code are used for systems without CSIT. The irregularLDPC code is fixed to that used in Fig. 7 (i.e., optimizedaccording to ρMT = 0.8). The precoder for each user is opti-mized according to its own transmit correlation matrix. We cansee that the simple individual CCIT based scheme can offersignificant performance improvement compared to EP even ifNBS/(KNMT) is only moderately large (NBS/(KNMT) = 1 and2, respectively, for NBS = 32 and 64 in Fig. 10).

Fig. 10. Simulation performance in a K-user system. K = 4, NMT = 8, ρBS =0.5; γ1 = γ2 = 0.5, γ3 = γ4 = 2; ρMT,1 = ρMT,3 = 0.6, ρMT,2 = ρMT,4 = 0.8.The target BER is set at 10−4 in precoder design.

VI. CONCLUSION

We have studied a simple SWF scheme that requires onlythe correlation matrix CMT at the transmitter in uplink MIMOsystems. This scheme can efficiently relieve the high costrelated to CSIT acquisition in MIMO systems. We show bymutual information analysis that the performance of SWF ispotentially close to that with full CSIT when NBS > NMT.We also show that iterative LMMSE detection can be used tohandle the interference problem related to SWF and transmitteroptimization in this system can also be accomplished using onlyCMT. Significant performance improvement is demonstrated bysimulation results.

APPENDIX

A. Proof of Proposition 1

To prove Proposition 1(i), we first show that the achievablerate R(H,Q) can be asymptotically characterized by CMT andQ regardless of channel realization H.

Remark 1: When NBS → ∞ with NMT fixed, and CBS meetsthe condition in (16), we have

limNBS→∞

R(H,Q)= log2 det(

INMT+C1/2MTQC1/2

MT ·NBS/σ2). (37)

To show (37), we first rewrite R(H,Q) in (5) as follows

R(H,Q)= log2 det(

INMT+C1/2MTQC1/2

MT ·HHwCBSHw/σ

2).

(38)Recall that the elements of Hw are i.i.d. complex Gaussianrandom variables CN(0,1). From the law of large numbers andthe assumption in (16), we can show that

limNBS→∞

(HH

w

)i,: CBS(Hw):, j =

{tr{CBS} i = j;0 i = j.

where (A)i,: denotes the ith row of matrix A, and (A):, j the jthcolumn of matrix A. This means

limNBS→∞

HHwCBSHw = tr{CBS} · INMT = NBSINMT . (39)

Combining (38) and (39), we obtain (37).


With Remark 1, we can obtain an upper bound for the fullCSIT capacity as detailed below.


limNBS→∞

CFCSIT≤ log2det(

INMT+C1/2MTQSWFC1/2

MT·NBS/σ2). (40)

Recall that the full CSIT capacity is defined in (6) by

CFCSIT = EH

[max

tr{Q(H)}≤Pt

R(H,Q(H))

]. (41)

Denote the optimized Q for the channel realization H by QH.From (37), we have

CFCSIT = E[log2 det(INBS +HQHHH/σ2]

→ E[log2 det

(INMT +C1/2

MTQHC1/2MT ·NBS/σ

2)]

. (42)

Note that for the channel characterized by C1/2MT, QSWF is

optimal. Then we have

E[log2 det

(INMT +C1/2

MTQHC1/2MT ·NBS/σ

2)]

≤ log2 det(

INMT +C1/2MTQSWFC1/2

MT ·NBS/σ2). (43)

Combining (42) and (43), we obtain (40).On the other hand, by setting Q = QSWF in (37) and taking

expectation over H, we can show that the upper bound inRemark 2 can be asymptotically approached by SWF as de-tailed below.


limNBS→∞

RSWF= log2 det(

INMT+C1/2MTQSWFC1/2

MT·NBS/σ2). (44)

Combining Remarks 2 and 3 and noting that RSWF ≤ CFCSIT,we obtain Proposition 1(i).

Proof of Proposition 1(ii): Similar to Remark 1, the achiev-able rate R(H,Q) in case (ii) can be characterized by CMT andQ regardless of the channel realization H.

Remark 4: When NBS,NMT → ∞ with N3MT/NBS → 0, and

CBS = I, we have

limNBS→∞

R(H,Q)= log2 det(

INMT+C1/2MTQC1/2

MT·NBS/σ2). (45)

To prove (45), we first introduce the following result.Proposition 5: (c.f. Example 2.50 in [44]) Let H be an

NMT ×NBS random matrix whose elements are zero-mean i.i.d.Gaussian random variables with variance 1/

√NBSNMT and

denote by√

NBS/NMT = ς . It can be shown that as NMT,NBS →∞ with NMT/NBS → 0, the asymptotic spectrum of the matrixHHH − ς

√NMTI is the semicircle law, i.e.,

w(λ) =

{1

2π

√4−λ2 |λ| ≤ 2;

0 |λ|> 2.

Now we consider the NMT × NBS random matrix HHw in

(3) whose elements are i.i.d. random variables CN(0,1).From Proposition 5, as NMT,NBS → ∞ with NMT/NBS → 0,

the asymptotic spectrum of the matrix (HHwHw − NBSINMT)/√

NBSNMT obeys the semicircle law. Hence the eigenvalue λof HH

wHw meets the following condition with probability 1

|(λ−NBS)/√

NBSNMT| ≤ 2.

This means

NBS

(1−2

√NMT/NBS

)≤λ≤NBS

(1+2

√NMT/NBS

).

Then we can obtain

R(H,Q)

= log2 det

(INMT +HH

wHwC1/2

MTQC1/2MT

σ2

)

≥ log2 det

(INMT+NBS

(1−2

√NMT

NBS

)C1/2

MTQC1/2MT

σ2

)

≥ log2 det

(INMT +

NBSC1/2MTQC1/2

MT

σ2

)

+NMT log2

(1−2

√NMT

NBS

). (46)

Similarly, we can show that

R(H,Q)≤ log2 det

(INMT +

NBSC1/2MTQC1/2

MT

σ2

)

+ NMT log2

(1+2

√NMT

NBS

). (47)

Note that NMT/NBS → 0 and ln(1 + x) → x as x → 0, weobtain (45).

With (45) available, Proposition 1(ii) can be proved similarlyas Proposition 1(i).

B. Justification of Equation (24)

In the discussions in Section IV-E, we assume ΩΩΩ ≈ ωI. Thiscan be justified for the case of full CSIT, as discussed in [40]. Inmore general cases of partial CSIT (as discussed in Section II),an augmented channel model can be adopted to ensure (24). Weoutline the main results as follows.

Denote by J the number of samples of H in a coded frame.They can be distributed over different OFDM subcarriers ordifferent time slots. Let the received signals be (see (1))

r( j) = H( j)y( j)+ηηη( j), j = 1, . . . ,J. (48)

Denote by raug = [r(1)T, . . . ,r(J)T]T

, Haug = diag{H(1), . . . ,

H(J)}, yaug = [y(1)T, . . . ,y(J)T]T

, and ηηηaug = [ηηη(1)T, . . . ,ηηη(J)T]T with superscript “aug” for augmentation. The system(48) can be rewritten as

raug = Haugyaug +ηηηaug. (49)


Assume that x is a coded sequence. Following [41], [56],[57], we adopt the precoding technique as follows

yaug = Uaug(Daug)1/2VHadx (50)

where VHad is a JNMT × JNMT Hadamard matrix, Uaug =diag{UQ(1), . . . ,UQ(J)} and Daug = diag{DQ(1), . . . ,DQ(J)}are, respectively, augmentations of the matrices UQ and DQ

in (20).Similar to (21), from (49) and (50), we can write the precoded

system as

raug = H̃augVHadx+ηηηaug (51)

where H̃aug = diag{H̃(1), . . . ,H̃(J)}. Note that H̃aug is block-diagonal and so is R. Denote by R= diag{R(1), . . . ,R(J)}. Thematrix ΩΩΩ can be calculated in a block-diagonal form as follows

ΩΩΩ Δ=(

VHHad(H̃

aug)HR−1H̃augVHad

)diag

=IJ ⊗(

VNMTHad ΩΩΩNMT VNMT

Had

)diag

(52)

where ⊗ denotes the Kronecker product, and

ΩΩΩNMT =J−1J

∑j=1

(H̃( j)

)H(R( j))−1 (H̃( j)

)

Δ=J−1

J

∑j=1

ΩΩΩ( j). (53)

Note that ΩΩΩ( j) in (53) is a Hermitian matrix. Its diagonal ele-ments are real and non-negative while its off-diagonal elementsare complex with phase determined by H̃( j). When {H̃( j)} in(53) are randomly generated, the off-diagonal elements of ΩΩΩ( j)will have phases varying with j, and the summary over j can beconstructive or deconstructive. Note that the diagonal elementsalways sum constructively, we have

ΩΩΩNMT ≈ (ΩΩΩNMT)diag for a large J. (54)

When {H̃( j)} in (53) are quasi-static, additional i.i.d. permuta-tion matrices {ΠΠΠ( j)} can be applied to form equivalent channel{H̃( j)ΠΠΠ( j)} to introduce random phase and obtain (54) [41].

Combining (52) and (54), and using the conclusion in the fullCSIT case, we obtain (24).

C. Proof of Proposition 2

We first rewrite (24) as follows

ω =N−1MTtr

{H̃H(vH̃H̃H +σ2I)

−1H̃}

=N−1MT

NBS

∑n=1

λn(H̃H̃H)/(vλn(H̃H̃H)+σ2)

=N−1MT

NMT

∑n=1

λn(H̃HH̃)/(vλn(H̃HH̃)+σ2) (55)

where λn(A) denotes the nth eigenvalue of A.

From (3), (9), (10) and (22), we have

H̃HH̃ = D1/2Q D1/2

MTUHMT ·HH

wCBSHw ·UMTD1/2MTD1/2

Q .

On the other hand, when the two conditions in Proposition 1is satisfied, from the proof of Proposition 1, we can show thatlimNBS→∞ HH

wCBSHw = NBSINMT. Thus

limNBS→∞

H̃HH̃ = DQDMT ·NBSINMT . (56)

Substituting (56) into (55), we complete the proof.

D. Proof of Proposition 3

To prove Proposition 3, we first show that the achievable rateregion R({Hk},{Qk}) can be asymptotically characterized by{CMT,k} and {Qk} regardless of the channel realizations {Hk}.

Remark 5: When NBS → ∞ with NMT and K fixed, and (16)holds for each CBS,k, ∀S= {i1, . . . , ik} ⊆ {1, . . . ,K}, we have,

limNBS→∞

log2 det(

INBS +∑i∈S γiHiQiHHi /σ

2)

= log2 det(IkNMT +NBSΓΓΓSQSCMT,S/σ

2) (57)

where

ΓΓΓS =diag{γi1 , . . . ,γik}⊗ INMT ,

QS =diag{Qi1 , . . . ,Qik},CMT,S =diag{CMT,i1 , . . . ,CMT,ik}.

To show (57), we rewrite the expression in (57) as follows

log2 det(

INBS +∑i∈S γiHiQiHHi /σ

2)

= log2 det(INBS +HSΓΓΓSQSHH

S/σ2)

= log2 det(IkNMT +ΓΓΓSQSHH

SHS/σ2) . (58)

Recall that the elements of Hw,i are i.i.d. complex Gaussianrandom variables CN(0,1). From the law of large numbers andthe assumption in (16), we can show that

limNBS→∞

HHSHS = NBSCMT,S. (59)

Combining (58) and (59), we obtain (57).Based on Remark 5, Proposition 3 can be proved similarly as

Proposition 1.

E. Proof of Proposition 4

From (34), we have

ωk =N−1MTtr

⎧⎨⎩H̃H

k

(σ2I+

K

∑i=1

viH̃iH̃Hi

)−1

H̃k

⎫⎬⎭

=N−1MTtr

⎧⎨⎩σ−2

(I+

K

∑i=1

σ−2viH̃iH̃Hi

)−1

H̃kH̃Hk

⎫⎬⎭

=N−1MTtr

{σ−2(I+ R̃)−1H̃kH̃H

k

}(60)


where R̃ Δ= ∑K

i=1σ−2viH̃iH̃H

i . From (59), we have

limNBS→∞

(R̃)nH̃kH̃Hk

= limNBS→∞

(K

∑i=1

σ−2viH̃iH̃Hi

)n

H̃kH̃Hk

=(σ−2vkH̃kH̃H

k

)nH̃kH̃H

k . (61)

Hence

limNBS→∞

σ−2(I+ R̃)−1H̃kH̃Hk

= limNBS→∞

σ−2 (I− R̃+ R̃2 − R̃3 + . . .)

H̃kH̃Hk

= limNBS→∞

σ−2 (I−(σ−2vkH̃kH̃H

k

)+ . . .

)H̃kH̃H

k

=(σ2I+ vkH̃kH̃H

k

)−1H̃kH̃H

k (62)

Then Proposition 4 can be proved similarly as Proposition 2.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir valuable suggestions and comments.

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Chongbin Xu received the B.S. degree in informa-tion engineering from Xi’an Jiaotong University in2005 and the Ph.D. degree in information and com-munication engineering from Tsinghua University in2012. He is now a Research Fellow in the Depart-ment of Electronic Engineering, City University ofHong Kong. His research interests are in the areas ofsignal processing and communication theory, includ-ing linear precoding, iterative detection, and randomaccess techniques.

Peng Wang (S’05–M’10) received the B. Eng.degree in telecommunication engineering and theM. Eng. degree in information engineering, fromXidian University, Xi’an, China, in 2001 and 2004,respectively, and the Ph.D. degree in electronic en-gineering from the City University of Hong Kong,Hong Kong SAR, in 2010. He was a Research Fellowwith the City University of Hong Kong and a visit-ing Post-Doctor Research Fellow with the ChineseUniversity of Hong Kong, Hong Kong SAR, bothfrom 2010 to 2012. Since 2012, he has been with

the Centre of Excellence in Telecommunications, School of Electrical andInformation Engineering, the University of Sydney, Australia, where he iscurrently a Research Fellow. His research interests include channel and networkcoding, information theory, iterative multi-user detection, MIMO techniquesand millimetre-wave communications. He has published over 40 peer-reviewedresearch papers in the leading international journals and conferences, andreceived the Best Paper Award at the IEEE International Conference onCommunications (ICC) in 2014. He has also served on a number of technicalprograms for international conferences such as ICC and WCNC.

Zhonghao Zhang received the B.S. degree and M.S.degree in electronic and information engineeringfrom University of Electronic Science and Technol-ogy of China in 2005 and 2008, respectively. Hereceived the Ph.D. degree in electronic engineeringfrom City University of Hong Kong in 2014. Heis now with the information technology group inthe Department of Electronic Engineering at CityUniversity of Hong Kong. His research interests arein the areas of signal processing in communicationsystems, including spatial coupling transmission and

iterative detection.

Li Ping (S’87–M’91–SM’06–F’10) received thePh.D. degree at Glasgow University in 1990. He lec-tured in the Department of Electronic Engineering,Melbourne University, from 1990 to 1992, andworked as a member of research staff at TelecomAustralia Research Laboratories from 1993 to 1995.He has been with the Department of ElectronicEngineering, City University of Hong Kong, sinceJanuary 1996, where he is now a Chair Professor.He received a British Telecom-Royal Society Fel-lowship in 1986, the IEE J J Thomson premium in

1993, the Croucher Foundation Award in 2005, and a British Royal Academy ofEngineering Distinguished Visiting Fellowship in 2010. He served as a memberof Board of Governors of IEEE Information Theory Society from 2010 to 2012.

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