RESEARCH Open Access Interference-aware receiver ......tiveness of the low-resolution LTE precoders for the multi-user MIMO mode and show that multi-user MIMO can bring significant

RESEARCH Open Access

Interference-aware receiver structure for multi-user MIMO and LTERizwan Ghaffar* and Raymond Knopp

Abstract

In this paper, we propose a novel low-complexity interference-aware receiver structure for multi-user MIMO that isbased on the exploitation of the structure of residual interference. We show that multi-user MIMO can deliver itspromised gains in modern wireless systems in spite of the limited channel state information at the transmitter(CSIT) only if users resort to intelligent interference-aware detection rather than the conventional single-userdetection. As an example, we focus on the long term evolution (LTE) system and look at the two importantcharacteristics of the LTE precoders, i.e., their low resolution and their applying equal gain transmission (EGT). Weshow that EGT is characterized by full diversity in the single-user MIMO transmission but it loses diversity in thecase of multi-user MIMO transmission. Reflecting on these results, we propose a LTE codebook design based ontwo additional feedback bits of CSIT and show that this new codebook significantly outperforms the currentlystandardized LTE codebooks for multi-user MIMO transmission.

1. IntroductionThe spatial dimension surfacing from the usage of mul-tiple antennas promises improved reliability, higherspectral efficiency [1], and the spatial separation of users[2]. This spatial dimension (MIMO) is particularly bene-ficial for precoding in the downlink of multi-user cellu-lar systems (broadcast channel), where these spatialdegrees of freedom at the transmitter can be used totransmit data to multiple users simultaneously. This isachieved by creating independent parallel channels tothe users (canceling multi-user interference) and theusers subsequently employ simplified single-user recei-ver structures. However, the transformation of cross-coupled channels into parallel non-interacting channelsnecessitates perfect channel state information at thetransmitter (CSIT) whose acquisition in a practical sys-tem, in particular frequency division duplex (FDD) sys-tem, is far from realizable. This leads to the precodingstrategies based on the partial or quantized CSIT [3],which limit the gains of multi-user MIMO.Ongoing standardizations of modern cellular systems

are investigating different precoding strategies based onlow-level quantized CSIT to transmit spatial streams tomultiple users sharing the same time-frequency

resources. In third-generation partnership project long-term evolution (3GPP LTE) system [4], the CSIT acqui-sition is based on the precoder codebook approach.These LTE precoders are characterized by low resolu-tion and are further based on the principle of equal gaintransmission (EGT). These precoders when employedfor the multi-user MIMO mode of transmission areunable to cancel the multi-user interference therebyincreasing the sub-optimality of conventional single-userdetection. This has led to the common perception thatmulti-user MIMO mode is not workable in LTE [[5],p. 244].Considering multi-user detection, we propose in this

paper a low-complexity interference-aware receiver [6]for the multi-user MIMO in LTE. Though multi-userdetection has been extensively investigated in the litera-ture for the uplink (multiple access channel), its relatedcomplexity has so far prohibited its employment in thedownlink (broadcast channel). For the multiple accesschannel, several multi-user detection techniques exist inthe literature starting from the optimal multi-user recei-vers [7] to their near-optimal reduced complexity coun-terparts (sphere decoders [8]). The complexityassociated with these techniques led to the investigationof low-complexity solutions as sub-optimal linear multi-user receivers [9], iterative multi-user receivers [10,11],and decision-feedback receivers [12,13]. Since in

* Correspondence: [email protected], 2229 route des Crêtes, B.P.193, Sophia Antipolis Cedex, 06904,France

Ghaffar and Knopp EURASIP Journal on Wireless Communications and Networking 2011, 2011:40http://jwcn.eurasipjournals.com/content/2011/1/40

© 2011 Ghaffar and Knopp; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

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practice, most wireless systems employ error controlcoding combined with the interleaving, recent work inthis area has addressed multi-user detection for codedsystems based on soft decisions [14,15].Our proposed low-complexity interference-aware

receiver structure not only reduces one complex dimen-sion of the system but is also characterized by exploitingthe interference structure in the detection process. Con-sidering this receiver structure, we investigate the effec-tiveness of the low-resolution LTE precoders for themulti-user MIMO mode and show that multi-userMIMO can bring significant gains in future wireless sys-tems if the users resort to intelligent interference-awaredetection as compared to the sub-optimal single-userdetection. We further look at the second characteristicof the LTE precoders, i.e., EGT both for the single-userand multi-user MIMO modes. We show that the EGThas full diversity in the single-user MIMO mode (aresult earlier derived for equal gain combining for BPSKin [16] and for EGT in MIMO systems in [17]); how-ever, it suffers from a loss of diversity in multi-userMIMO mode [18]. Based on this analysis, we propose adesign criteria for the precoder codebooks and showthat the additional feedback of two bits for CSIT canlead to significant improvement in the performance ofthe multi-user MIMO.Regarding notations, we will use lowercase or upper-

case letters for scalars, lowercase boldface letters forvectors and uppercase boldface letters for matrices. Thematrix In is the n × n identity matrix. |.| and ||.|| indi-cate norm of scalar and vector while (.)T, (.)*, and (.)†

indicate transpose, conjugate, and conjugate transpose,respectively. (.)R indicates the real part and (.)I indicatesthe imaginary part of a complex number. The notationE (.) denotes the mathematical expectation while

Q(y) =1√2π

∫∞y e−x2/2dx denotes the Gaussian Q-func-

tion. All logarithms are to the base 2.The paper is divided into eight sections. In Sec. II, we

give a brief overview of LTE and define the systemmodel. In Sec. III, we consider a geometric schedulingstrategy for the multi-user MIMO mode in LTE andpropose a low-complexity interference-aware receiverstructure. In Sec. IV, we look at the information theore-tic perspective of the proposed receiver structure. Sec. Vis dedicated to the performance analysis of the EGTthat is followed by the simulation results. Before con-cluding the paper, we propose a design criteria for theprecoder codebooks of the forthcoming standardizationsof LTE. The proof details in the paper have been rele-gated to appendices to keep the subject material simpleand clear.

2. LTE system modelA. LTE–A brief overviewIn 3GPP LTE, a 2 × 2 configuration for MIMO isassumed as the baseline configuration; however, config-urations with four transmit or receive antennas are alsoforeseen and reflected in the specifications [19]. LTErestricts the transmission of maximum of two code-words in the downlink that can be mapped onto differ-ent layers where one codeword represents an outputfrom the channel encoder. Number of layers availablefor the transmission is equal to the rank of the channelmatrix (maximum 4). In this paper, we restrict ourselvesto the baseline configuration with the eNodeB (LTEnotation for the base station) equipped with two anten-nas while we consider single and dual-antenna userequipments (UEs). Physical layer technology employedfor the downlink in LTE is OFDMA combined with bitinterleaved coded modulation (BICM) [20]. Several dif-ferent transmission bandwidths are possible, rangingfrom 1.08 to 19.8 MHz with the constraint of being amultiple of 180 kHz. Resource blocks (RBs) are definedas groups of 12 consecutive resource elements (REs -LTE notation for the subcarriers) with a bandwidth of180 kHz thereby leading to the constant RE spacing of15 kHz. Approximately, 4 RBs form a subband and thefeedback is generally done on subband basis. Sevenoperation modes are specified in the downlink of LTE;however, we shall focus on the following four modes:• Transmission mode 2. Fall-back transmit diversity.

Transmission rank is 1, i.e., one codeword is transmittedby the eNodeB. Employs Alamouti space-time or space-frequency codes [21].• Transmission mode 4. Closed-loop spatial multiplex-

ing. Transmission rank is 2, i.e., two codewords aretransmitted by the eNodeB to the UE in the single-userMIMO mode. UEs need to have minimum of twoantennas.• Transmission mode 5. Multi-user MIMO mode. Sup-

ports only rank-1 transmission, i.e., one codeword foreach UE.• Transmission mode 6. Closed-loop precoding for

rank-1 transmission, i.e., one codeword for the UE inthe single-user MIMO mode.In the case of transmit diversity and closed-loop pre-

coding, one codeword (data stream) is transmitted toeach UE using Alamouti code in the former case andLTE precoders in the latter case. Time-frequencyresources are orthogonal to the different UEs in thesemodes thereby avoiding interference in the system.However, in the multi-user MIMO mode, parallel code-words are transmitted simultaneously, one for each UE,sharing the same time-frequency resources. Note that


Page 2 of 17

LTE restricts the transmission of one codeword to eachUE in the multi-user MIMO mode.For closed-loop transmission modes (mode 4, 5 and

6), precoding mechanisms are employed at the transmitside with the objective of maximizing throughput. Theprecoding is selected and applied by the eNodeB to thedata transmission to a target UE based on the channelfeedback received from that UE. This feedback includesa precoding matrix indicator (PMI), a channel rank indi-cator (RI), and a channel quality indicator (CQI). PMI isan index in the codebook for the preferred precoder tobe used by the eNodeB. The granularity for the compu-tation and signaling of the precoding index can rangefrom a couple of RBs to the full bandwidth. For trans-mission mode 5, the eNodeB selects the precodingmatrix to induce high orthogonality between the code-words so that the interference between UEs is mini-mized. In transmission modes 4 and 6, the eNodeBselects the precoding vector/matrix such that codewordsare transmitted to the corresponding UEs with maxi-mum throughput.In order to avoid excessive downlink signaling, trans-

mission mode for each UE is configured semi-staticallyvia higher layer signaling, i.e., it is not allowed for a UEto be scheduled in one subframe in the multi-userMIMO mode and in the next subframe in the single-user MIMO mode. For the case of eNodeB with twoantennas, LTE proposes the use of following four preco-ders for transmission modes 5 and 6:

p ={

1√4

[11

],

1√4

[1

−1

],

1√4

[1j

],

1√4

[1−j

] }(1)

The number of precoders increases to sixteen in thecase of four transmit antennas; however, in this paper,we restrict to the case of two transmit antennas. Fortransmission mode 4, LTE proposes the use of followingtwo precoder matrices on subband basis.

P ={

1√4

[1 11 −1

],1√4

[1 1j −j

] }(2)

Note that there is a possibility of swapping the columnsin P but the swap must occur over the entire band.

B. System modelWe first consider the system model for transmissionmode 5, i.e., the multi-user MIMO mode in which theeNodeB transmits one codeword each to two single-antenna UEs using the same time-frequency resources.Transmitter block diagram is shown in Figure 1. Duringthe transmission for UE-1, the code sequence c1 is inter-leaved by π1 and is then mapped onto the signalsequence x1. x1 is the symbol of x1 over a signal setχ1 ⊆ C with a Gray-labeling map where |c1| = M1 and

x2 is the symbol of x2 over signal set c2 where |c2| =M2. The bit interleaver for UE-1 can be modeled as π1:k’ ® (k, i) where k’ denotes the original ordering of thecoded bits ck’, k denotes the RE of the symbol x1,k, and iindicates the position of the bit ck’ in the symbol x1,k.Note that each RE corresponds to a symbol from a con-stellation map c1 for UE-1 and c2 for UE-2. Selection ofthe normal or extended cyclic prefix (CP) for eachOFDM symbol converts the downlink frequency-selec-tive channel into parallel flat fading channels.Cascading IFFT at the eNodeB and FFT at the UE

with the cyclic prefix extension, the transmission at thek-th RE for UE-1 in transmission mode 5 can beexpressed as

y1,k = h†1,kp1,kx1,k + h†

1,kp2,kx2,k + z1,k (3)

where y1,k is the received symbol at UE-1 and z1,k iszero mean circularly symmetric complex white Gaussiannoise of variance N0. x1,k is the complex symbol for UE-1 with the variance σ 2

1 and x2,k is the complex symbol

for UE-2 with the variance σ 22 . h

†n,k ∈ C1×2 symbolizes

the spatially uncorrelated flat Rayleigh fading MISOchannel from eNodeB to the n-th UE (n = 1, 2) at thek-th RE. Its elements can therefore be modeled as inde-pendent and identically distributed (iid) zero mean cir-cularly symmetric complex Gaussian random variableswith a variance of 0.5 per dimension. Note that ℂ1 × 2

denotes a 2-dimensional complex space. pn,k denotes theprecoding vector for the n-th UE at the k-th RE and isgiven by (1). For the dual-antenna UEs, the systemequation for transmission mode 5 is modified as

y1,k = H1,k[p1,kx1,k + p2,kx2,k] + z1,k (4)

where y1,k, z1,k Î ℂ2 × 1 are the vectors of the receivedsymbols and circularly symmetric complex white Gaus-sian noise of double-sided power spectral density N0/2at the 2 receive antennas of UE-1, respectively. H1,k Îℂ

2

× 2 is the channel matrix from eNodeB to UE-1.In transmission mode 6, only one UE will be served in

one time-frequency resource. Therefore, the systemequation for single-antenna UEs at the k-th RE is given as

yk = h†kpkxk + zk (5)

where pk is given by (1). For the dual-antenna UEs,the system equation for mode 6 is modified as

yk = Hkpkxk + zk (6)

3. Multi-user MIMO modeWe now look at the effectiveness of the low-resolutionLTE precoders for the multi-user MIMO mode. We


Page 3 of 17

first consider a geometric scheduling strategy [22] basedon the selection of UEs with orthogonal precoders.

A. Scheduling strategyAs the processing at the UE is performed on a RE basisfor each received OFDM symbol, the dependency on REindex can be ignored for notational convenience. Thesystem equation for the case of single-antenna UEs forthe multi-user mode is

y1 = h†1p1x1 + h†

1p2x2 + z1 (7)

The scheduling strategy is based on the principle ofmaximizing the desired signal strength while minimizingthe interference strength. As the decision to schedule aUE in the single-user MIMO, multi-user MIMO ortransmit diversity mode will be made by the eNodeB,each UE would feedback the precoder that maximizesits received signal strength. So this selected precoder bythe UE would be the one closest to its matched filter(MF) precoder in terms of the Euclidean distance.For the multi-user MIMO mode, the eNodeB needs to

ensure good channel separation between the co-sched-uled UEs. Therefore, the eNodeB schedules two UEs onthe same RBs that have requested opposite (orthogonal)precoders, i.e., the eNodeB selects as the second UE tobe served in each group of allocatable RBs, one of theUEs whose requested precoder p2 is 180° out of phasefrom the precoder p1 of the first UE to be served on the

same RBs. So if UE-1 has requested p1 =1√4

[1q

], q Î

{±1, ±j}, then eNodeB selects the second UE that has

requested p2 =1√4

[1

−q

]. This transmission strategy

also remains valid also for the case of dual-antenna UEswhere the UEs feedback the indices of the precodingvectors that maximize the strength of their desired sig-nals, i.e., ||Hp||2. For the multi-user MIMO mode, theeNodeB schedules two UEs on the same RE, which have

requested 180° out of phase precoders. The details ofthis geometric scheduling strategy can be found in [22].Though this precoding and scheduling strategy would

ensure minimization of the interference under the con-straint of low-resolution LTE precoders, the residualinterference would still be significant. Single-user detec-tion, i.e., Gaussian assumption of the residual interfer-ence and its subsequent absorption in noise, would leadto significant degradation in the performance. On theother hand, this residual interference is actually discretebelonging to a finite alphabet and its structure can beexploited in the detection process. However, intelligentdetection based on its exploitation comes at the cost ofenhanced complexity. Here, we propose a low-complex-ity interference-aware receiver structure that on onehand reduces one complex dimension of the systemwhile on the other hand, it exploits the interferencestructure in the detection process.

B. Low-complexity interference-aware receiverFirst, we consider the case of single-antenna UEs. Softdecision of the bit ck’ of x1, also known as log-likelihoodratio (LLR), is given as

LLRi1

(ck′ |y1,h†

1,P)= log

p(ck′ = 1|y1,h†1,P)

p(ck′ = 0|y1,h†1,P)

(8)

We introduce the notation �i1(y1, ck′) for the bit

metric that is developed on the lines similar to the (7)and 9 in [20], i.e.,

�i1(y1, ck′) = log p

(ck′ |y1,h†

1,P)

≈ log p(y1|ck′ ,h†

1,P)

= log∑

x1∈χ i1,ck′

∑x2∈χ2

p(y1|x1, x2,h†1,P)

≈ minx1∈χ i

1,ck′ ,x2∈χ2

1N0

∣∣∣y1 − h†1p1x1 − h†

1p2x2∣∣∣2

(9)

Source

Encoder-1

π1

π2

μ1, χ1

μ2, χ2

OFDM

OFDM

(IFFT + CP

insertion)

(IFFT + CP

insertion)

(Bits)

Turbo

Encoder-2

Turbo

1

2

x1

x2

c1

c2

Source

(Bits)

P

Figure 1 eNodeB in multi-user MIMO mode. π1 denotes the random interleaver, μ1 the labeling map and c1 the signal set for the codewordof UE-1. P indicates the precoding matrix.


Page 4 of 17

where χ i1,ck′ denotes the subset of the signal set x1 Î c1

whose labels have the value ck’ Î {0, 1} in the position i.Here, we have used the log-sum approximation, i.e.,log

∑j zj = maxj log zj and this bit metric is therefore

termed as max-log MAP bit metric. As LLR is the dif-ference of two bit metrics and these will be decoded

using a conventional soft-decision Viterbi algorithm,1N0

(a common scaling factor to all LLRs) can be ignoredthereby leading to

�i1(y1, ck′) ≈ min

x1∈χ i1,ck′ ,x2∈χ2

∣∣∣y1 − h†1p1x1 − h†

1p2x2∣∣∣2

= minx1∈χ i

1,ck′ ,x2∈χ2

{|y1|2 +

∣∣∣h†1p1x1

∣∣∣2 + ∣∣∣h†1p2x2

∣∣∣2 − 2(h†1p1x1y

∗1)R + 2(ρ12x∗

1x2)R − 2(h†1p2x2y

∗1)R

}

]

(10)

where ρ12 =(h†1p1

)∗h†1p2 indicates the cross-correla-

tion between the two effective channels. Here, we haveused the relation |a - b|2 = |a|2 + |b|2 - 2 (a*b)R wherethe subscript (.)R indicates the real part. Note that thecomplexity of the calculation of bit metric (10) isO (|χ1| |χ2|).In (10), we now introduce two terms as the outputs of

MF, i.e., y1 =(h†1p1

)∗y1 and y2 =

(h†1p2

)∗y1. Ignoring |

y1|2 (independent of the minimization operation), the

bit metric is written as

�i1(y1, ck′) ≈ min

x1∈χ i1,ck′ ,x2∈χ2

{∣∣∣h†1p1x1

∣∣∣ + ∣∣∣h†1p2x2

∣∣∣2 − 2(y∗1x1)R + 2ψAx2,R + 2ψBx2,I

}(11)

where

ψA = ρ12,Rx1,R + ρ12,Ix1,I − y2,RψB = ρ12,Rx1,I − ρ12,Ix1,R − y2,I

Note that the subscript (.)I indicates the imaginarypart.For x1 and x2 belonging to equal energy alphabets,∣∣∣h†1p1x1

∣∣∣2 and ∣∣∣h†1p2x2

∣∣∣2 can be ignored as they are inde-

pendent of the minimization operation. The values of x2,R and x2,I that minimize Eq. (11) need to be in theopposite directions of ψA and ψB, respectively, therebyavoiding search on the alphabets of x2 and reducing onecomplex dimension in the detection, i.e.,

�i1(y1, ck′) ≈ min

x1∈χ i1,ck′

{−2y1,Rx1,R − 2y1,Ix1,I − 2|ψA||x2,R| − 2|ψB|x2,I|}(12)

As an example, we consider the case of QPSK for

which the values of x2,R and x2,I are

[± σ2√

2

], so the bit

metric is written as

�i1(y1, ck′) ≈ min

x1∈χ i1,ck′

{−2y1,Rx1,R − 2y1,Ix1,I −

√2σ2|ψA| −

√2σ2|ψB|

}(13)

For x1 and x2 belonging to non-equal energy alpha-

bets, the bit metric is same as (13) but∣∣∣h†

1p1x1∣∣∣2 and∣∣∣h†

1p2x2∣∣∣2 can no longer be ignored thereby leading to

�i1(y1, ck′) ≈ min

x1∈χ i1,ck′

{∣∣∣h†1p1

∣∣∣2|x1,R|2 + ∣∣∣h†1p1

∣∣∣2|x1,I|2 + ∣∣∣h†1p2

∣∣∣2|x2,R|2 + ∣∣∣h†1p2

∣∣∣2|x2,I|2−2y1,Rx1,R − 2y1,Ix1,I − 2|ψA||x2,R| − 2|ψB||x2,I|

} (14)

Note that the minimization is independent of c2though x2 appears in the bit metric. The reason of thisindependence is as follows. The decision regarding thesigns of x2,R and x2,I in (14) will be taken in the samemanner as for the case of equal energy alphabets. Forfinding their magnitudes that minimize the bit metric(14), it is the minimization problem of a quadratic func-tion, i.e., differentiating (14) w.r.t |x2,R| and |x2,I| to findthe global minima that are given as

|x2,R| → |ψA|∣∣∣h†1p2

∣∣∣2 , |x2,I| → |ψB|∣∣∣h†1p2

∣∣∣2 (15)

where ® indicates the discretization process in whichamong the finite available points of x2,R and x2,I, thepoint closest to the calculated continuous value isselected. So if x2 belongs to QAM256, then instead ofsearching 256 constellation points for the minimizationof (14), the metric (15) reduces it to merely two opera-tions thereby trimming down one complex dimension inthe detection, i.e., the detection complexity is indepen-dent of |c2| and reduces to O(|χ1|).As a particular example of the discretization of contin-

uous values in (15), we consider the case of x2 belongingto QAM16. The values of x2,R and x2,I for the case of

QAM16 are

[± σ2√

10,± 3σ2√

10

]so their magnitudes in

(14) are given as

|x2,R| = σ21√10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝2 + (−1)

I

⎛⎜⎜⎜⎜⎝|ψA|<σ2

2∣∣∣h†

1p2

∣∣∣2√10

⎞⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

|x2,I| = σ21√10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝2 + (−1)

I

⎛⎜⎜⎜⎜⎝|ψB|<σ2

2∣∣∣h†

1p2

∣∣∣2√10

⎞⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(16)


Page 5 of 17

and I (.) is the indicator function defined as

I(a < b) ={1 if a < b0 otherwise

Now we look at the receiver structure for the case ofdual-antenna UEs. The system equation for UE-1 (ignor-ing the RE index) is

y1 = H1[p1x1 + p2x2] + z1 (17)

The receiver structure would remain same with h†1

being replaced by H1, i.e., the channel from eNodeB tothe two antennas of UE-1. Subsequently y1 = (H1p1)

†y1and y2 = (H1p2)

†y1 are the MF outputs while r12 =(H1p1)

† H1p2 is the cross-correlation between two effec-tive channels.For comparison purposes, we also consider the case of

single-user receiver, for which the bit metric is given as

�i1(y1, ck′) ≈ min

x1∈χ i1,ck′

⎧⎪⎨⎪⎩

1

(|ρ12|2σ 22 +

∣∣∣h†1p1

∣∣∣2N0)

∣∣∣∣y1 −∣∣∣h†

1p1

∣∣∣2x1∣∣∣∣2

⎫⎪⎬⎪⎭ (18)

Table 1 compares the complexities of different recei-vers in terms of the number of real-valued multiplica-tions and additions for getting all LLR values per RE/subcarrier. Note that nr denotes the number of receiveantennas. This complexity analysis is independent of thenumber of transmit antennas as the operation of findingeffective channels bears same complexity in all receiverstructures. Moreover UEs can also directly estimatetheir effective channels if the pilot signals are also pre-coded. The comparison shows that the complexity ofthe interference-aware receiver is of the same order asof single-user receiver while it is far less than the com-plexity of the max-log MAP receiver. Figure 2 furthershows the performance-complexity trade off of differentreceivers for multi-user MIMO mode in LTE. The per-formance of the receivers is measured in terms of theSNR at the frame error rate (FER) of 10-2 whereas thecomplexity is determined from Table 1. It shows thatthe performance of the single-user receiver is severelydegraded as compared to that of the interference-awarereceiver. In most cases, the single-user receiver fails toachieve the requisite FER in the considered SNR range.

On the other hand, interference-aware receiver achievessame performance as max-log MAP receiver but withmuch reduced complexity.The interference-aware receiver is therefore not only

characterized by low complexity but also resorts tointelligent detection by exploiting the structure of resi-dual interference. Moreover, this receiver structurebeing based on the MF outputs and devoid of any divi-sion operation can be easily implemented in the existinghardware. However, the proposed receiver needs boththe channel knowledge and the constellation of interfer-ence (co-scheduled UE). As the UE already knows itsown channel from the eNodeB and the requested preco-der, it can determine the effective channel of the inter-ference based on the geometric scheduling algorithm, i.e., the precoder of the co-scheduled UE is 180° out ofphase of its own precoder. Consequently there is noadditional complexity in utilizing this receiver structureas compared to using single-user receivers except thatthe UE needs to know the constellation of interference.

4. Information theoretic perspectiveSum rate of the downlink channel is given as

I = I(Y1;X1|h†1,P) + I(Y2;X2|h†

2,P) (19)

where P = [p1 p2] is the precoder matrix,

I(Y1;X1|h†

1,P)is the mutual information of UE-1 once

it sees interference from UE-2 and I(Y2;X2|h†

2,P)is the

mutual information of UE-2 once it sees interferencefrom UE-1. Y1 is the received symbol at UE-1 while X1

is the symbol transmitted by the eNodeB to UE-1. Notethat interference is present in the statistics of Y1 and Y2.No sophisticated power allocation is employed to thetwo streams as the downlink control information (DCI)in the multi-user mode in LTE includes only 1-bitpower offset information, indicating whether a 3 dBtransmit power reduction should be assumed or not.We therefore consider equal-power distribution betweenthe two streams. For the calculation of mutual informa-tion, we deviate from the unrealistic Gaussian assump-tion for the alphabets and consider them from discreteconstellations. The derivations of the mutual

Table 1 Comparison of receivers complexity

Receiver Real multiplications Real additions

Interference-aware receiver (equal energy alphabets) 8nr + 2√M + 2M 8nr + 10M + log(M) - 4

Interference-aware receiver (non equal energy alphabets) 12nr + 4M + 72

√M 12nr + 18M + log(M) - 6

Max-log MAP receiver 2M2nr + 8Mnr 6M2nr + 4Mnr+ log(M) - M2

Single-user receiver (equal energy alphabets) 10nr + 6 10nr - 3

Single-user receiver (non equal energy alphabets) 10nr + 3M +√M/2 + 4 10nr + 3M + log(M) - 3


Page 6 of 17

information expressions for the case of finite alphabetshave been relegated to Appendix A for simplicity andlucidity.We focus on the LTE precoders but to analyze the

degradation caused by the low-level quantization andthe characteristic of EGT of these precoders, we alsoconsider some other transmission strategies. Firstly, weconsider unquantized MF precoder [23] that is given as

p =1√

|h11|2 + |h21|2[h11h21

](20)

For EGT, the unquantized MF precoder is given as

p =1√2

[1

h∗11h21/|h11||h21|

](21)

To be fair in comparison with the geometric schedul-ing algorithm for multi-user MIMO in LTE, we intro-duce a geometric scheduling algorithm for unquantizedprecoders. We divide the spatial space into four quad-rants according to the spatial angle between h†

1 and h†2,

which is given as

φ = cos−1

⎛⎝

∣∣∣h†1h2

∣∣∣||h1|| ||h2||

⎞⎠ 0◦ ≤ φ ≤ 90◦ (22)

The geometric scheduling algorithm ensures that theeNodeB chooses the second UE to be served on thesame RE as the first UE such that their channels h†

1 and

h†2 lie in the opposite quadrants.Figure 3 shows the sum rates of a broadcast channel

with the dual-antenna eNodeB and two single-antennaUEs for QAM64 alphabets. SNR is the transmit SNR,

i.e.,σ 21 ||p1||2 + σ 2

2 ||p2||2N0

whereas the two UEs have

101

102

103

104

0

5

10

15

20

25

30

35

40

Number of real−valued multiplications for LLR per RE

SN

R (

dB)

@ F

ER

=10

−2

QPSK

QAM16

QAM64

Single−user Rx Interference−Aware Rx Max−log MAP Rx

Figure 2 eNodeB has two antennas. Continuous lines indicate thecase of single-antenna UEs while dashed lines indicate dual-antennaUEs. 3GPP LTE rate 1/2 punctured turbo code is used. Simulationsettings are same as in the first part of Sec. 6.

0 10 20 30 40 50

2

4

6

8

10

12

SNR

bps/

Hz

QAM64

No Scheduling −SU RxLTE Precoders − SU RxLTE Precoders − IA RxMF EGT Precoders − IA RxMF Precoders − IA Rx

Figure 3 Sum rates of different transmission schemes for the downlink channel with dual-antenna eNodeB and 2 single-antenna UEs.‘No Scheduling - SU Rx’ indicates the case once the eNodeB uses the LTE precoders without employing the geometric scheduling strategy. Inall other cases, the eNodeB employs the geometric scheduling strategy along with the LTE precoders, MF EGT precoders and MF precoders. SURx indicates the cases when UEs employ single-user detection while IA Rx indicates the cases when UEs resort to the intelligent detection byemploying the low-complexity interference-aware receivers.


Page 7 of 17

equal-power distribution, i.e., σ 21 = σ 2

2 , MF and MF EGTprecoders are the unquantized precoders given in (20)and (21), respectively, while LTE precoders are thequantized precoders given in (1). The sum rates ofunquantized precoders along with those of LTE quan-tized precoders are shown for the case of single-userreceivers and for the case of low-complexity interfer-ence-aware receivers. The results show that under theproposed transmission strategy, the sum rate can be sig-nificantly improved (unbounded in SNR) if the low-complexity interference-aware receivers are used ascompared to the case when the UEs resort to sub-opti-mal single-user detection where rates are bounded (inSNR). The behavior of single-user detection is attributedto the fact that this detection strategy considers inter-ference as noise so the SINR is low once no geometricscheduling has been employed by the eNodeB while theSINR improves due to the reduction of interferenceonce geometric scheduling is employed. However, therates remain bounded in the SNR if the UEs resort tothe single-user detection that is due to the fact thatincreasing the SNR (transmit SNR) also increases theinterference strength thereby bounding the SINR athigh values of the transmit SNR. On the other hand,there is significant improvement in the sum rate onceUEs resort to intelligent detection by employing thelow-complexity interference-aware receivers. In thiscase, the sum rate is unbounded if the rate (constella-tion size) of each UE is adapted with the SNR. Notethat the quantized CSIT (LTE precoders) appears tohave no effect at high SNR once UEs resort to intelli-gent interference-aware detection. This behavior isbecause the rate is not adapted with the SNR in thesesimulations, i.e., the constellation size is fixed toQAM64 and is not increased with the increase in theSNR. At high SNR, the rate of each UE gets saturatedto its constellation size (six bits for QAM64) if the UEresorts to intelligent interference-aware detection. How-ever, the approach to this saturation point (slope of therate curve) depends on the quantization of channelinformation.Another interesting result is the effect of the two

characteristics of LTE precoders, i.e., low resolution andEGT. There is a slight improvement in the sum rate atmedium SNR when the restriction of low resolution(LTE quantized precoders) is relaxed, i.e., eNodeBemploys MF EGT precoders; however, there is a signifi-cant improvement in the sum rate when the restrictionof EGT is eliminated, i.e the eNodeB employs MF pre-coders. This shows that the loss in spectral efficiencydue to the employment of LTE precoders is mainlyattributed to the EGT rather than their low resolution(quantization).

5. Performance analysisWe now focus on the EGT characteristic of the LTEprecoders and carry out the performance analysis of theEGT in single-user and multi-user MIMO systems. Werestrict to the case of single-antenna UEs while the eNo-deB has two antennas. For single-user case, the receivedsignal at the k-th RE is given by

y1,k = h†1,kp1,kx1,k + z1,k (23)

For EGT, the precoder vector is given by

p1,k =1√2

[1

h21,kh∗11,k

|h21,k||h11,k|]T. So the received signal

after normalization byh11,k|h11,k| is given by

yN1,k =1√2(|h11,k| + |h21,k|)x1,k + h11,k

|h11,k|z1,k (24)

where yN1,k =h11,k|h11,k|y1,k. The PEP has been derived in

Appendix B and is given as

P(c1 → c1) ≤ 12

∏dfree

⎛⎜⎜⎜⎜⎜⎝

48(

d2

1,min

(σ 21

N0

))2

⎞⎟⎟⎟⎟⎟⎠ (25)

where

d2

1,minis the normalized minimum distance of

the constellation c1, dfree is the free distance (minimumHamming distance) of the code. Note that c1 and c1 arethe correct and error codewords, respectively. Eq. 25clearly shows full diversity of the EGT for single-userMIMO. Note that this result was earlier derived in [16]but was restricted to the case of BPSK. The same resultwas derived in [17] for EGT in MIMO systems usingthe approach of metrics of diversity order. Here, wehave generalized this result and have adopted the nat-ural approach of pairwise error probability to show thediversity order. Analysis of the EGT for multi-userMIMO system seemingly does not have closed formsolution so we shall resort to the simulations for its ana-lysis in Sec. 6.

6. Simulation resultsSimulations are divided into three parts. In the first part,we look at the performance of the proposed interfer-ence-aware receiver structure for the multi-user MIMOmode in LTE while second part is dedicated to the sen-sitivity analysis of this receiver structure to the knowl-edge of the constellation of interference. This sensitivityanalysis is motivated by the fact that the DCI formats in


Page 8 of 17

the transmission mode 5 (multi-user MIMO) do notinclude the information of the constellation of the co-scheduled UE. Third part looks at the diversity order ofthe EGT in both single-user and multi-user MIMOmodes in LTE.For the first part (Figures 4 and 5), we consider the

downlink of 3GPP LTE that is based on BICM OFDMtransmission from the eNodeB equipped with twoantennas using rate-1/3 LTE turbo code [24] with ratematching to rate 1/2 and 1/4. We deliberate on both thecases of single and dual-antenna UEs. We consider anideal OFDM system (no ISI) and analyze it in the fre-quency domain where the channel has iid Gaussian

matrix entries with unit variance and is independentlygenerated for each channel use. We assume no powercontrol in the multi-user MIMO mode so two UEs haveequal-power distribution. Furthermore, all mappings ofthe coded bits to QAM symbols use Gray encoding. Wefocus on the FER while the frame length is fixed to1,056 information bits. As a reference, we consider thefall-back transmit diversity scheme (LTE mode 2–Ala-mouti code) and compare it with the single-user andmulti-user MIMO modes employing single-user recei-vers and low-complexity interference-aware receivers.To analyze the degradation caused by the low resolutionand EGT of LTE precoders, we also look at the system

−2 −1 0 1 2 3 410

−3

10−2

10−1

100

SNR

FE

R

1bps/Hz

2 3 4 5 6 7 810

−3

10−2

10−1

100

SNR

FE

R

2bps/Hz

MU MIMOMF

MU MIMOMF EGT

MU MIMOLTE mode 5 SU MIMO

MFSU MIMOMF EGT

SU MIMOLTE mode 6

Transmit DiversityLTE mode 2IA RxIA RxIA Rx

MU MIMOLTE mode 5

SU Rx

Figure 4 Downlink fast fading channel with the dual-antenna eNodeB and two single-antenna UEs. IA Rx indicates the low-complexityinterference-aware receiver while SU Rx indicates the single-user receiver. MU MIMO and SU MIMO indicate multi-user and single-user MIMO,respectively. To be fair in comparison among different schemes, sum rates are fixed, i.e., if two users are served with QPSK with rate 1/2 in themulti-user mode, then one user is served with QAM16 with rate 1/2 in the single-user mode thereby equating the sum rate in both cases to 2bps/Hz. 3GPP LTE rate 1/3 turbo code is used with different puncturing patterns.


Page 9 of 17

performance employing the unquantized MF andunquantized MF EGT precoders. To be fair in the com-parison of the LTE multi-user MIMO mode (mode 5)employing the geometric scheduling algorithm with themulti-user MIMO mode employing unquantized MFand MF EGT precoders, we consider the geometricscheduling algorithm (Sec. 4) based on the spatial anglebetween the two channels (22). Perfect CSIT is assumedfor the case of MF and MF EGT precoding while errorfree feedback of two bits (PMI) to the eNodeB isassumed for LTE precoders. It is assumed that the UEhas knowledge of the constellation of co-scheduled UEin the multi-user MIMO mode. It is further assumedthat the UE knows its own channel from the eNodeB.So in multi-user MIMO mode, the UE can find theeffective interference channel based on the fact that theeNodeB schedules the second UE on the same RE

whose precoder is 180° out of phase of the precoder ofthe first UE. Figure 4 shows the results for the case ofsingle-antenna UEs. It shows enhanced performance ofthe multi-user MIMO mode once the UEs resort tointelligent detection by employing the low-complexityinterference-aware receivers. The performance isseverely degraded once the UEs resort to single-userdetection. An interesting result is almost the equivalentperformance of the unquantized MF EGT and low-reso-lution LTE precoders, which shows that the loss withrespect to the unquantized CSIT is attributed to theEGT rather than the low resolution of LTE precoders.Performance degradation is observed for LTE multi-userMIMO mode for higher spectral efficiencies. Figure 5shows the results for the case of dual-antenna UEs andfocuses on different LTE modes employing LTE preco-ders. It shows that single-user detection performs closeto interference-aware detection at low spectral efficien-cies once UE has two antennas; however, its perfor-mance degrades at higher spectral efficiencies. Thisbehavior is attributed to the fact that the rate with sin-gle-user detection gets saturated at high SNR due to theincreased interference strength as was shown in Sec. 4.So the performance of single-user detection degrades forhigh spectral efficiencies as these spectral efficiencies arehigher than the rate or mutual information of the sin-gle-user detection. For single-user MIMO (Mode 6),there is no saturation of the rate at high SNR as there isno interference. So mode 6 performs better than mode5 at high SNR for higher spectral efficiencies once UEsemploy single-user detection. However, if UEs resort tothe intelligent interference-aware detection, the multi-user MIMO mode shows enhanced performance overother transmission modes in LTE. No degradation ofLTE multi-user MIMO mode is observed at higher spec-tral efficiencies once UEs have receive diversity (dualantennas).In the second part of simulations, we look at the sen-

sitivity of the proposed receiver structure to the knowl-edge of the constellation of co-scheduled UE for themulti-user MIMO mode in LTE. The simulation settingsare same as of the first part except that we consider thecase when UE has no knowledge of the constellation ofco-scheduled UE. The UE assumes this unknown inter-ference constellation to be QPSK, QAM16, or QAM64,and the results for these different assumptions areshown in Figure 6. Results show that there is negligibledegradation in the performance of the proposed receiverif the interfering constellation is assumed to be QAM16or QAM64. However, there is significant degradation ifthe interference is assumed to be QPSK when it actuallycomes from QAM64. It indicates that assuming interfer-ence to be from a higher order modulation among thepossible modulation alphabets leads to the best

0 1 2 3 4 510

−4

10−3

10−2

10−1

100

SNR

FE

R

2bps/Hz

Mode 5 − IAMode 5 − SUMode 4Mode 6Mode 2

6 7 8 9 10 11 12 13 14 15

10−4

10−3

10−2

10−1

100

SNR

FE

R

4bps/Hz

Mode 5 − IAMode 5 − SUMode 4Mode 6Mode 2

Figure 5 Downlink fast fading channel with the dual-antennaeNodeB and two dual-antenna UEs. IA indicates the low-complexity interference-aware receiver while SU indicates thesingle-user receiver. 3GPP LTE rate 1/3 turbo code is used withdifferent puncturing patterns.


Page 10 of 17

compromise as this assumption includes the lower mod-ulation orders as special cases (with proper scaling).However, the converse is not true, i.e., assuming inter-ference from lower modulation order cannot includehigher order modulations. As LTE and LTE-Advancedrestrict the transmission to three modulations (QPSK,QAM16, and QAM64), assuming interference to beQAM64 (or even QAM16) leads to better performance.The proposed receiver structure, therefore, can stillexploit the discrete nature of the interference even if itdoes not know its modulation order. As the complexityof this receiver structure is independent of the constella-tion of interference, the assumption of higher ordermodulation does not add to the complexity of detection.In the third set of simulations, we look at the diversity

order of the single-user MIMO and multi-user MIMOschemes in LTE. The system settings are same as in thefirst part, but now we consider slow fading environment,i.e., the channel remains constant for the duration ofone codeword. Figure 7 shows that the MF precodershave full diversity both in multi-user MIMO and single-user MIMO modes. However, once the constraint ofEGT is imposed on the MF precoders, multi-userMIMO mode loses diversity while single-user MIMOstill exhibits full diversity, which is in conformity withthe analytical results of Sec. 5. This fundamental resultholds even when the low-level quantization of LTE isimposed on these EGT precoders. Earlier conclusion

that the performance loss in the multi-user MIMOmode in LTE is attributed to the EGT rather than thelow resolution of LTE precoders is further confirmed.These results give a general guideline for the possibleemployment of the single-user MIMO and multi-userMIMO in LTE under different environments. Once notenough diversity is available in the channel, single-userMIMO is the preferred option while multi-user MIMOis the possible choice once the channel is rich indiversity.

7. Design of LTE precoder codebook withadditional feedbackIt was shown in the information theoretic analysis andwas subsequently confirmed in the simulations that theloss in spectral efficiency due to the low-level quantizedCSIT (LTE precoders) in the multi-user MIMO mode ismore attributed to the EGT of the LTE codebook ratherthan its low resolution. It was also shown that EGTloses diversity in the multi-user MIMO mode. Focusingon these fundamental results, we now look at the designof the precoder codebook for future standardizations ofLTE. Feedback of CSIT is expected to increase in theseforthcoming wireless systems. However, the complexityassociated with the feedback overhead combined withthe low rate feedback channels would allow only a lim-ited increase in the feedback. We therefore consider thecase of two additional feedback bits for the quantized

0 2 4 6 8 10 12 14 1610

−4

10−3

10−2

10−1

100

SNR

FE

R o

f x1

QAM64−QAM64QPSK−QPSK

QAM16−QAM16

Interference (x2)assumed to be

QAM16assumed to be

QAM64assumed to be

QPSK

Interference (x2) Interference (x2)

Figure 6 Interference sensitivity for the multi-user MIMO mode in LTE. Three sets of simulations are shown. QPSK-QPSK indicates that bothx1 and x2 belong to QPSK. UE-1 does not know the constellation of interference (x2) and assumes it to be QPSK, QAM16, and QAM64.


Page 11 of 17

CSIT (precoder codebook) and look how these addi-tional bits can be efficiently employed.We consider two options for the employment of these

additional feedback bits as illustrated in Figure 8. As theLTE precoder is [1 exp(jθ)]T, so additional bits can beused to increase the angular resolution of θ, i.e., morepoints on the unit circle but restricting to EGT asshown in Figure 8a. Another option is to increase thelevels of transmission, i.e., the additional feedback bitsare used by the UE to indicate an increase of the powerlevel on either of the two antennas as [1 2 exp(jθ)]T or[2 exp(jθ) 1]T.With this new precoding codebook design, the earlier

described scheduling strategy remains same, i.e., for aUE to be scheduled in the multi-user MIMO mode, theeNodeB selects the second UE to be served on the same

time-frequency resources (co-scheduled UE) such thatthe desired signal strength is maximized while interfer-ence strength is minimized for both the UEs. So if UE-1has requested the precoder p1, the eNodeB finds theprecoding vector p2 in the codebook, which minimizestheir cross-correlation (p†

1p2) and then schedules thesecond UE with UE-1, which has requested p2 as itsdesired precoding vector. The receiver structure beingindependent of the codebook design also remains samefor these new precoding codebooks.We now look at the effect of two additional bits of

feedback for PMI on the performance. We focus on thetwo options of improved angular resolution and addi-tional levels of transmission. The simulation settings aresame as of the previous section. Figure 9 illustrates theperformance both in fast and slow fading channels.

5 10 15 20 25 30

10−3

10−2

10−1

100

SNR

FE

R

2bps/Hz

15 20 25 30 3510

−3

10−2

10−1

100

SNR

FE

R

4bps/Hz

MU MIMOMF

MU MIMOMF EGT

MU MIMOLTE mode 5

SU MIMOMF

SU MIMOMF EGT

SU MIMOLTE mode 6

Transmit DiversityLTE mode 2

Figure 7 Diversity in the single-user and multi-user MIMO modes. Downlink slow fading channel (one channel realization per codeword)with dual-antenna eNodeB and two single-antenna UEs. 3GPP LTE rate 1/3 turbo code is used with different puncturing patterns.


Page 12 of 17

These results show significant improvement in the per-formance of the multi-user MIMO mode when the addi-tional feedback bits are employed to increase the levelsof transmission as compared to the case of increasingthe angular resolution. The performance is within 1.7dB of the lower bound where lower bound is the perfor-mance curve for MF precoder without any interference.In slow fading environment, the change of the slope ofFER curve with increased levels of transmission indi-cates improved diversity as compared to the case ofincreased angular resolution. On the other hand, littlegain is observed in the single-user mode with additionalfeedback bits, which is expected as the standard LTEprecoders have been optimized for the single-user trans-mission. These results indicate that the design of preco-ders for the forthcoming versions of LTE shouldconsider increasing transmission levels rather thanenhancing the angular resolution of the precoders. This

proposed design is not merely restricted to the frame-work of LTE but gives fundamental design guidelinesfor precoding in modern wireless systems.

8. ConclusionsIn this paper, we have looked at the feasibility of themulti-user MIMO for future wireless systems that arecharacterized by low-level quantization of CSIT. Wehave shown that multi-user MIMO can deliver its pro-mised gains if the UEs resort to intelligent detectionrather than the sub-optimal single-user detection. Tothis end, we have proposed a low-complexity interfer-ence-aware receiver structure that is characterized bythe exploitation of the structure of residual interfer-ence. We have analyzed two important characteristicsof the LTE precoders, i.e., low resolution and EGT.We have shown that the performance loss of the LTEprecoders in the multi-user MIMO mode is attributedto their characteristic of EGT rather than their lowresolution. We have further shown that the EGT ischaracterized by full diversity in the single-user MIMOmode but it loses diversity in the multi-user MIMO.Based on these fundamental results, we have proposeda design of the precoder codebook for forthcomingstandardizations of LTE incorporating more levels oftransmission.

Appendix AMutual information for finite alphabetsThe mutual information for UE-1 for finite size QAMconstellation with |c1| = M1 takes the form as

I(Y1;X1|h†

1,P)= H

(X1|h†

1,P)

− H(X1|Y1,h†

1,P)

= logM1 − H(X1|Y1,h†1,P)

(26)

where H(.) = −E log p(.) is the entropy function. Thesecond term of (26) is given as

H(X1|Y1,h†1,P) =

∑x1

∫y1

∫h†1p1

∫h†1p2

p(x1, y1,h†1p1,h

†1p2) log

1

p(x1|y1,h†1p1,h

†1p2)

dy1d(h†1p1)d(h

†1p2)

=∑x1

∑x2

∫y1

∫h†1p1

∫h†1p2

p(x1, x2, y1,h†1p1,h

†1p2) log

∑x′1

∑x′2p(y1|x′

1, x′2,h

†1p1,h

†1p2)∑

x′2p(y1|x1, x′

2,h†1p1,h

†1p2)

dy1d(h†1p1)d(h

†1p2)

(27)

where x′1 ∈ χ1 and x′

2 ∈ χ2. Conditioned on the chan-nel and the precoder, there is one source of random-ness, i.e., noise. So (27) can be extended as

H(X1|Y1,h†1,P) =

1M1M2

∑x

Ez1 log

∑x’ exp

[− 1N0

∣∣∣h†1p1x1 + h†

1p2x2 + z1 − h†1p1x

′1 − h†

1p2x′2

∣∣∣2]∑

x′2exp

[− 1N0

∣∣∣h†1p2x2 + z1 − h†

1p2x′2

∣∣∣2]

=1

M1M2

∑x

Ez1 log

∑x’ exp

[− 1N0

∣∣∣h†1P(x − x’) + z1

∣∣∣2]∑

x′2exp

[− 1N0

∣∣∣h†1P(x − x′

2) + z1∣∣∣2]

(28)

where M2 = |c2|, x = [x1 x2]T, x’ = [x′

1 x′2]

T andx′2 = [x1 x′

2]T. The mutual information for UE-1 can be

rewritten as

1−1

1−1

j

−j

−j

j

(a)

2−2

2j

−2j

1

j

−j

−1

(b)

2−2

2j

−2j

1

j

−j

−1

(c)

Figure 8 Two options of using two additional bits of feedbackfor PMI. Upper row corresponds to the option of increased angularresolution of LTE precoders while lower row corresponds to theoption of enhanced levels of transmission. Square indicates theprecoder entry for the first antenna while cross indicates theprecoder entry for the second antenna.


Page 13 of 17

I(Y1;X1|h†1,P) = logM1 − 1

M1M2

∑x

Ez1 log

∑x’ p

(y1|x’,h†

1,P)

∑x′2p(y1|x′

2,h†1,P

) (29)

I(Y;X1|h†1p1,h

†1p2) = logM1 − 1

M1M2NzNh1

∑x1

∑x2

Nh1∑h1

Nz∑z1

log

∑x′1

∑x′2exp

[− 1N0

∣∣∣y1 − h†1p1x

′1 − h†

1p2x′2

∣∣∣2]∑

x′2exp

[− 1N0

∣∣∣y1 − h†1p1x1 − h†

1p2x′2

∣∣∣2]

= logM1 − 1M1M2NzNh1

∑x1

∑x2

Nh1∑h1

Nz∑z1

log

∑x′1

∑x′2exp

[− 1N0

∣∣∣h†1p1x1 + h†

1p2x2 + z1 − h†1p1x

′1 − h†

1p2x′2

∣∣∣2]∑

x′2exp

[− 1N0

∣∣∣h†1p2x2 + z1 − h†

1p2x′2

∣∣∣2](30)

The above quantities can be easily approximated usingsampling (Monte-Carlo) methods with Nz realizations of

noise and Nh1 realizations of the channel h†1 where the

precoding matrix depends on the channel. So we canrewrite (29) as (30).Similarly the mutual information for UE-2 is given as

I(Y2;X2|h†2,P) = logM2 − 1

M1M2

∑x

Ez2 log

∑x’ p

(y2|x’,h†

2,P)

∑x′1p(y2|x′

1,h†2,P

) (31)

where x′1 = [x′

1 x2]T.

3 3.5 4 4.5 5 5.5 610

−2

10−1

100

SNR

FE

R

2bps/Hz

4 6 8 10 12 14 1610

−2

10−1

SNR

FE

R

2bps/Hz

Mode 5 (1 additional bit) Mode 5

Mode 6

Enhanced Levels

Mode 5 (1 additional bit)

Angular Resolution


Enhanced Levels


Angular Resolution

LTE

LTE


Enhanced Levels & Resolution


Enhanced Levels & Resolution

Mode 5 (2 additional bits)

Enhanced Levels

Mode 6 (2 additional bits)

Enhanced Levels

MU MIMO - Full CSIT

MF Precoders

MF Precoders

SU MIMO - Full CSIT

Figure 9 Proposed precoder codebook. Downlink channel with dual-antenna eNodeB and two single-antenna UEs. Top figure shows theresults for fast fading channels while bottom figure illustrates the performance for slow fading channels. 3GPP LTE rate 1/3 turbo code is usedwith different puncturing patterns.


Page 14 of 17

For the case of single-user MIMO mode, the mutualinformation is given by

I(Y1;X1|h†1,p1) = logM1 − H

(X1|Y1,h†

1,p1

)(32)

where the second term is given by

H(X1|Y1,h†

1,p1

)=∑x1

∫y1

∫h†1p1

p(x1, y1,h

†1p1

)log

1

p(x1|y1,h†

1p1

)dy1d (h†1p1

)

=∑x1

∫y1

∫h†1p1

p(x1, y1,h

†1p1

)log

∑x′1p(y1|x′

1,h†1p1

)p(y1|x1,h†

1p1

) dy1d(h†1p1

)

=1

M1NzNh1

∑x1

Nh1∑h†1

Nz∑z1

log

∑x′1exp

[− 1N0

∣∣∣y1 − h†1p1x

′1

∣∣∣2]

exp[− 1N0

∣∣∣y1 − h†1p1x1

∣∣∣2]

(33)

where Nh1 are the number of channel realizations ofthe channel h†

1. Note that the precoding vector p1 is

dependent on the channel h†1.

Appendix BDiversity analysis of EGT in single-user MIMOConsider the system equation 24, i.e.,

yN1,k =1√2(|h11,k| + |h21,k|)x1,k + h11,k

|h11,k|z1,k (34)

The max-log MAP bit metric [20] for the bit ck’ canbe written as

�i1(yk, ck′) ≈ min

x1∈χ i1,ck′

[1N0

∣∣∣∣yN1,k − 1√2(|h11,k| + |h21,k|)x1

∣∣∣∣2](35)

The conditional PEP i.e P(c1 → c1|h1) is given as

P(c1 → c1|H1) = P

(∑k

minx1∈χ i

1,ck′

1N0

∣∣∣∣yN1,k − 1√2(|h11,k| + |h21,k|)x1

∣∣∣∣2

≥∑k′

minx1∈χ i

1,ck′

1N0

∣∣∣∣yN1,k − 1√2(|h11,k| + |h21,k|)x1

∣∣∣∣2 ∣∣H1

) (36)

where H1 indicates the complete channel from theeNodeB to UE-1 for the transmission of the codewordc1. Assume d(c1 − c1) = dfree for c1 and c1 under consid-eration for the PEP analysis, which is the worst case sce-nario between any two codewords. Therefore, theinequality on the right hand side of (36) shares the sameterms on all but dfree summation points and the summa-tions can be simplified to only dfree terms for whichck′ = ck′. Let’s denote

x1,k = arg minx1∈χ i

1,ck′

1N0

∣∣∣∣yN1,k − 1√2(|h11,k| + |h21,k|)x1

∣∣∣∣2

x1,k = arg minx1∈χ i

1,ck′

1N0

∣∣∣∣yN1,k − 1√2(|h11,k| + |h21,k|)x1

∣∣∣∣2(37)

As1N0

∣∣∣∣yN1,k − 1√2(|h11,k| + |h21,k|)x1,k

∣∣∣∣2

≥ 1N0

∣∣∣∣yN1,k − 1√2(|h11,k| + |h21,k|)x1,k

∣∣∣∣2

,

this leads to PEP being given as

P(c1 → c1|H1) ≤ P

⎛⎝∑

k,dfree

1N0

∣∣∣∣yN1,k − 1√2(|h11,k| + |h21,k|)x1,k

∣∣∣∣2

≥∑k,dfree

1N0

∣∣∣∣yN1,k − 1√2(|h11,k| + |h21,k|)x1,k

∣∣∣∣2 ∣∣H1

⎞⎠

= P

⎛⎝∑

k,dfree

√2(|h11,k| + |h21,k|)

N0(z∗1,k(x1,k − x1,k))R

≥∑k,dfree

12N0

(|h11,k| + |h21,k|)2|x1,k − x1,k|2⎞⎠

= Q

⎛⎜⎝√√√√∑

kdfree

14N0

(|h11,k| + |h21,k|)2|(x1,k − x1,k)|2⎞⎟⎠

≤ 12exp

⎛⎝−

∑k,dfree

18N0

(|h11,k| + |h21,k|)2d21,min

⎞⎠

=12

∏k,dfree

exp(

− 18N0

(|h11,k| + |h21,k|)2d21,min

)

(38)

where we have used Chernoff bound

Q(x) ≤ 12exp

(−x2

2

). Averaging over channel leads to

P(c1 → c1) ≤ 12EH1

∏k,dfree

exp(

− 18N0

(|h11,k| + |h21,k|)2d21,min

)

=12

∏k,dfree

Eh1,k exp

⎛⎜⎝⎛⎜⎝−

d2

1,min

4

⎞⎟⎠ (|h11,k| + |h21,k|)2σ 2

1

2N0

⎞⎟⎠(39)

Eq. 39 follows from the channel independence at eachRE that is the consequence of the interleaving operation.

Here we have used the notation d21,min = σ 21

d2

1,minwith

d2

1,minbeing the normalized minimum distance of the

constellation c1. Using the moment generating function(MGF) of the SNR at the output of two branch EGC asper equations (2) and (23) in [25], PEP at high SNR isupper bounded as

P(c1 → c1) ≤ 12

∏dfree

⎛⎜⎜⎜⎜⎜⎜⎝

8(

σ 21

N0

)2+

d2

1,min

(σ 21

N0

)3

4(

σ 21

N0

)2(2 +

σ 21

d2

1,min

4N0

)2 −

(

d2

1,min

2√2

)(σ 21

N0

)(2 +

d2

1,min

2

(σ 21

N0

))3/2

×

⎡⎢⎢⎢⎢⎣π − 2sin−1

⎛⎜⎜⎜⎜⎝

√√√√√√√√(

σ 21

N0

)−1+

d2

1,min

4

2(

σ 21

N0

)−1+

d2

1,min

4

⎞⎟⎟⎟⎟⎠

⎤⎥⎥⎥⎥⎦

+

4(

σ 21

N0

)2 (4 +

d2

1,min

2

(σ 21

N0

))

4(

σ 21

N0

)2(2 +

d2

1,min

4

(σ 21

N0

))2 (2 +

d2

1,min

2

(σ 21

N0

))

⎞⎟⎟⎟⎟⎟⎠

(40)


Page 15 of 17

Using the identity cos−1(x) =π

2− sin−1(x), we have

π − 2sin−1

⎛⎜⎜⎜⎜⎝

√√√√√√√√(

σ 21

N0

)−1+

d2

1,min

4

2(

σ 21

N0

)−1+

d2

1,min

4

⎞⎟⎟⎟⎟⎠ = 2cos−1

⎛⎜⎜⎜⎜⎝

√√√√√√√√(

σ 21

N0

)−1+

d2

1,min

4

2(

σ 21

N0

)−1+

d2

1,min

4

⎞⎟⎟⎟⎟⎠ (41)

Taylor series expansion [26] of cos−1(√x) is given as

cos−1(√x) =

√2 − 2

√x

∞∑k=0

(1 − √x)k(1/2)k

2k(k! + 2kk!)for | − 1 +

√x| < 2

where x! is the factorial of x while (x)n is the Poch-hammer symbol, i.e., (x)n = x (x + 1) ... (x + n - 1). Forx closer to 1, a case that shall be occurring at high SNRin (41), first term will be dominant, i.e.,

cos−1

⎛⎜⎜⎜⎜⎝

√√√√√√√√(

σ 21

N0

)−1+

d2

1,min

4

2(

σ 21

N0

)−1+

d2

1,min

4

⎞⎟⎟⎟⎟⎠ ≈

√√√√√√√√√2 − 2

√√√√√√√√(

σ 21

N0

)−1+

d2

1,min

4

2(

σ 21

N0

)−1+

d2

1,min

4

(42)

Taylor series expansion of√x at x = 1 is

√x = 1 +

x − 12

− (x − 1)2

8+(x − 1)3

16− · · ·

In the expansion of

√√√√√√√√(

σ 21

N0

)−1+

d2

1,min

4

2(

σ 21

N0

)−1+

d2

1,min

4

, first two

terms will be dominant at high SNR thereby leading to

√√√√√√√√√2 −

√√√√√√√√(

σ 21

N0

)−1+

d2

1,min

4

2(

σ 21

N0

)−1+

d2

1,min

4

≈

√√√√√√√√√√√√√√√√√2 − 2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1 +

(σ 21

N0

)−1+

d2

1,min

4

2(

σ 21

N0

)−1+

d2

1,min

4

− 1

2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

√√√√√√√√−

⎛⎜⎜⎜⎝(

σ 21

N0

)−1+

d2

1,min

4

2(

σ 21

N0

)−1+

d2

1,min

4

− 1

⎞⎟⎟⎟⎠

=1√

2 +

d2

1,min

4 +(

σ 21

N0

)

(43)

So rewriting (40), we get

P(c1 → c1) ≤ 12

∏dfree

⎛⎜⎜⎜⎜⎜⎝

2(2 +

d2

1,min4

(σ 21

N0

))2 +

d2

1,min

(σ 21

N0

)

4

(2 +

d2

1,min4

(σ 21

N0

))2

−2

(

d2

1,min

2√2

)(σ 21

N0

)(2 +

d2

1,min

2

(σ 21

N0

))3/2(2 +

d2

1,min

4

(σ 21

N0

))1/2

+

(4 +

d2

1,min

2

(σ 21

N0

))(2 +

d2

1,min4

(σ 21

N0

))2 (2 +

d2

1,min2

(σ 21

N0

))

⎞⎟⎟⎟⎟⎟⎠

(44)

At high SNR, second term converges to4

d2

1,min

(σ 21

N0

)

while the third term converges to−4

d2

1,min

(σ 21

N0

). SoPEP at high SNR is upper bounded as

P(c1 → c1) ≤ 12

∏dfree

⎛⎜⎜⎜⎜⎜⎝

32(

d2

1,min

(σ 21

N0

))2 +16(

d2

1,min

(σ 21

N0

))2

⎞⎟⎟⎟⎟⎟⎠

=12

∏dfree

⎛⎜⎜⎜⎜⎜⎝

48(

d2

1,min

(σ 21

N0

))2

⎞⎟⎟⎟⎟⎟⎠

(45)

AcknowledgementsEurecom’s research is partially supported by its industrial partners: BMW,Bouygues Telecom, Cisco Systems, France Télécom, Hitachi Europe, SFR,Sharp, ST Microelectronics, Swisscom, Thales. The research work leading tothis paper has also been partially supported by the European Commissionunder SAMURAI and IST FP7 research network of excellence NEWCOM++.

Competing interestsThe authors declare that they have no competing interests.

Received: 1 December 2010 Accepted: 14 July 2011Published: 14 July 2011


Page 16 of 17

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doi:10.1186/1687-1499-2011-40Cite this article as: Ghaffar and Knopp: Interference-aware receiverstructure for multi-user MIMO and LTE. EURASIP Journal on WirelessCommunications and Networking 2011 2011:40.

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