Massive MIMO is, in general, severely affected by pilot contamination. As
opposed to traditional detectors, we propose a group-blind detector that takes
into account the presence of pilot contamination. While sticking to the
traditional structure of the training phase, where orthogonal pilot sequences
are reused, we use the excess antennas at each base station to partially remove
interference during the uplink data transmission phase. We analytically derive
the asymptotic SINR achievable with group-blind detection, and confirm our
findings by simulations. We show, in particular, that in an
interference-limited scenario with one dominant interfering cell, the SINR can
be doubled compared to non-group-blind detection.Comment: 5 pages, 4 figure

Ferrante, Guido Carlo

Geraci, Giovanni

Quek, Tony Q. S.

Win, Moe Z.

English

arXiv

Massive MIMO is, in general, severely affected by pilot contamination. As opposed to traditional detectors, we propose a group-blind detector that takes into account the presence of pilot contamination. While sticking to the traditional structure of the training phase, where orthogonal pilot sequences are reused, we use the excess antennas at each base station to partially remove interference during the uplink data transmission phase. We analytically derive the asymptotic SINR achievable with group-blind detection, and confirm our findings by simulations. We show, in particular, that in an interference-limited scenario with one dominant interfering cell, the SINR can be doubled compared to non-group-blind detection

Geraci, G.

Quek, T. Q. S.

Win, Moe Z

DSpace@MIT

GROUP-BLIND DETECTIONWITH VERY LARGE ANTENNA ARRAYSIN THE PRESENCE OF PILOT CONTAMINATIONG. C. Ferrante◦∗, G. Geraci◦, T. Q. S. Quek◦, and M. Z. Win∗◦SUTD, Singapore, and ∗MIT, MAABSTRACTMassive MIMO is, in general, severely affected by pilot con-tamination. As opposed to traditional detectors, we proposea group-blind detector that takes into account the presence ofpilot contamination. While sticking to the traditional structureof the training phase, where orthogonal pilot sequences arereused, we use the excess antennas at each base station to par-tially remove interference during the uplink data transmissionphase. We analytically derive the asymptotic SINR achiev-able with group-blind detection, and confirm our findings bysimulations. We show, in particular, that in an interference-limited scenario with one dominant interfering cell, the SINRcan be doubled compared to non-group-blind detection.Index Terms— Group-blind detection, pilot contamina-tion, massive MIMO, interference suppression.1. INTRODUCTIONMassive MIMO is, along with network densification and in-creased bandwidth, one of the key technologies promisingto tremendously increase the rate per area in future cellularnetworks [1, 2]. The idea behind massive MIMO is to equipbase stations (BSs) with a number of antennas much largerthan the number of active users per time-frequency resourceblock [3, 4]. This allows to increase the uplink SNR throughmaximal-ratio combining, and make the matched-filter detec-tor optimal with perfect channel state information.However, the modern cellular network architecture fore-sees the use of orthogonal pilots to estimate the channel be-tween each user and the BS. The number of orthogonal pi-lots is upper bounded by the coherence time [5, 6], hencepilots are usually reused in different cells. As a consequence,channel estimation is corrupted by the interference receivedduring the training phase from users in other cells, a phe-nomenon known as pilot contamination [7, 8, 9]. As thenumber of antennas at each BS grows, the rate achievableby traditional receivers saturates due to pilot contamination[10, 11, 12, 13]. In order to mitigate pilot contaminationthrough an improved channel estimation, a nonlinear itera-tive algorithm that jointly estimates channels and transmittedsymbols has been proposed in [14]. A step toward the un-derstanding of the fundamental limits of massive MIMO hasbeen recently made by Mu¨ller et al. [15, 16, 17], showingthat pilot contamination can be removed if the power receivedfrom users within the cell is larger than that received from out-of-cell users. However, this assumption requires both powercontrol and a regular cell geometry, and may not hold in adense cellular network [18].In this paper, we propose a receiver design for the uplinkof massive MIMO multiuser cellular networks. The proposedreceiver takes into account pilot contamination, by adaptingthe group-blind detection scheme originally proposed in [19]to the case of imperfect channel knowledge due to contami-nation. Unlike [15, 16, 17], we do not assume neither powercontrol nor regular cell tessellation. Moreover, this paperdiffers from [14] as it focuses on improving data detectionrather than modifying the channel estimation phase. We pro-vide an asymptotic analysis of the SINR achievable by ourscheme, showing a gain with respect to non-group-blind re-ceivers. In particular, we show an SINR gain equal to twoin an interference-limited scenario with one dominant inter-fering cell. Simulations validate our analysis and confirm therate improvement attained by our scheme.2. SYSTEMMODEL2.1. Received signalConsider the uplink of a noncooperative multicellular net-work with L cells. Each cell is equipped with one BS havingn antennas, each BS serving K single-antenna users on thesame time-frequency signaling resource. Throughout the pa-per, the reference cell is referred to as cell 1, and interferingcells are labelled with indices l ∈ {2, . . . , L}. Users in thereference cell and in other cells will be referred to as in-cellusers and out-of-cell users, respectively. The signal receivedby the reference BS during symbol period m is:y(m) =L∑l=1K∑k=1hlk√βlk xlk(m) + n(m), (1)where: hlk = [hlk1, . . . , hlkn]T∈ Cn is the channel vec-tor between user k in cell l and the reference BS, be-ing hlkr the channel coefficient with respect to antennar; βlk > 0 is the channel gain between the reference BSarXiv:1510.03027v1  [cs.IT]  11 Oct 2015and user k in cell l, that models pathloss and shadowingeffects; xlk(m) is the symbol transmitted by user k in celll; n(m) ∈ Cn is additive white Gaussian noise (AWGN)vector. We assume {βkl} fixed during the coherence time,hlkr ∼ CN(0, 1), E[n(m)n(m′)† ] = Iδmm′ , E[hlkh†l′k′ ] =Iδll′δkk′ , E[xlkxl′k′ ] = Pδll′δkk′ , where P is the trans-mitted power, assumed equal for all users. Denote Hl =[hl1, . . . ,hlK ] ∈ Cn×K ,Rl = diag(βl1, . . . , βlK), andGl =HlR1/2l = [gl1, . . . , glK ] ∈ Cn×K . Compactly, eq. (1) canbe written as follows for the generic symbol period m:y =L∑l=1HlR1/2l xl + n=L∑l=1Glxl + n, (2)where dependence on m is implicit.2.2. Channel estimationBy using orthogonal pilots during the training phase, theMMSE estimation gˆ1k of g1k is [20, 10]gˆ1k =(∑l>1glk +√ν1k)ϕ†1kβ−11k , (3)where 1/ is equal to the effective training SNR, ν1k ∼CN(0, I), andϕ1k =β21k+∑l>1 βlk. (4)We collect in matrix form estimations Gˆl = [gˆl1, . . . , gˆlK ]and estimation errors G˜l =Gl − Gˆl.2.3. Achievable rateFollowing [5, 10], an achievable rate R1k for in-cell user kcan be derived by considering the signaly′ = Gˆ1x1 + G˜1x˜1 +∑l>1Glxl + n, (5)where x˜1 is independent on x1 and has same covariance. Letw1k denote the linear receiver for user k, the rateR1k is givenbyR1k = E[ log(1 + γ1k) ], (6)where the expectation is with respect to estimated channels,and the SINR γ1k is given in (7) at the top of the next page.3. PROPOSED GROUP-BLIND DETECTORBlind receivers were developed for multiuser detection andequalization [21], and then generalized to group-blind detec-tion in the context of CDMA [19, 22, 23, 24]. While in blindtechniques the receiver knows the signature sequence of theuser to decode only, in group-blind techniques it knows thesignature sequences of a subset of users. In the uplink of acellular network, this corresponds to a BS knowing in-cellchannels and being unaware of out-of-cell channels. Whilegroup-blind detection was originally proposed assuming per-fect knowledge of a subset of channels, we hereby extendgroup-blind detection to the case of contaminated knowledge.The proposed receiver w1k consists of two components.A first component, w˙1k, belongs to range Gˆ1 and is derivedon the basis of the signal yin = Gˆ1x1 + n; the MMSE crite-rion yieldsw˙1k = (Gˆ†1Gˆ†1 +1P I)−1gˆ1k. (8)A second component, w˘1k, belongs to a subspace orthogonalto range Gˆ1 and that lies within the signal space. Let U˘Gˆ1 bea matrix whose columns span such subspace. The componentw˘1k is derived by taking into account the whole receivedsignal. Following the MMSE criterion, a derivation similar to[19] that also accounts for imperfect channel estimation dueto pilot contamination yieldsw˘1k =− U˘ †Gˆ1(U˘ †Gˆ1C†y′U˘†Gˆ1)−1U˘ †Gˆ1C†y′w˙†1k, (9)where Cy′ is the covariance matrix of (5). The group-blinddetector w1k = w˙1k + w˘1k is, therefore, explicitly given byw1k ={I − U˘ †Gˆ1(U˘ †Gˆ1C†y′U˘†Gˆ1)−1U˘ †Gˆ1C†y′}w˙†1k. (10)We can show that simple blanking techniques [2] allow toaccurately estimateCy′ for the purpose of implementing (10).Details are omitted due to space constraints.4. ASYMPTOTIC PERFORMANCE ANALYSISWe derive the asymptotic achievable rate, as n→∞. In thislimit, the SINR is bounded by non-vanishing interferenceterms. In order to obtain these asymptotically non-vanishingterms, we use the two following properties [7, 10]:(i) channels are asymptotically orthogonal in the almostsure sense, i.e., n−1g†klgk′l′† a.s.−−→βklδkk′δll′ , where δijdenotes the Kronecker delta;(ii) in the high-SNR regime, gˆ1k ∈Sk = range{glk : l >1}.A conceptual representation of the structure of the signalspace in the high-SNR regime as n→∞ is shown onFig. 1. Properties (i) and (ii) imply that, asymptotically,the signal space S = range [G1 · · ·GL] is the direct sumS1 ⊕ · · · ⊕SK . However, the projection of gˆ1k onto vectorsin Sk does not asymptotically vanish, in general. In fact, thefollowing limit holds:1ngˆ†1kg†lj =1nϕ1kβ−11k(∑m>1g†mk +√ν†1k)glja.s.−−−→ ϕ†1kβ−11k βljδkj . (11)γ1k =|w†1kgˆ†1k|2E[w†1k(1PI + g˜†1kg˜†1k +∑j 6=kg†1jg†1j +∑l>1∑j>1g†ljg†lj)w1k∣∣∣∣ Gˆ1 ] . (7)S1 D rangefgl1gl>1Og11; Qg11 2 S1S2 D rangefgl2gl>1Og12; Qg12 2 S2Fig. 1: Conceptual representation of the asymptotic structure of thesignal space in the high-SNR regime. All channels are almostsurely orthogonal. For any fixed in-cell user k, both esti-mated and error vectors belong toSk = range{glk : l > 1}.As a consequence, in-cell user k is asymptotically interferedonly by out-of-cell users who used the same training se-quence. On the basis of the above observations, we can derivethe following results, that hold for the case of L= 2 cells. Wenote that such a scenario approximates an interference-limitednetwork with one dominant interfering cell. Proofs as wellas analytical results for L> 2 cells are omitted and will beprovided in a journal version of this manuscript. Numericalresults that validate our analysis are given in § 5.Lemma 1. Let L= 2. Asymptotically, the variable after de-tection satisfies1nw†1ky′ a.s.−−→ϕ1kx1k+{(ϕ1kβ−11k β2k)−(ϕ1kβ−11k β2k)3λ}x2k+ λ−1(ϕ1kβ−11k β2k)2(β1k − ϕ1k)x˜1k (12)where λ= (β1k − ϕ1k)2 + (ϕ1kβ−11k β2k)2.The SINR achieved with group-blind detection readilyfollows from Lemma 1.Theorem 1. Let L= 2. The SINR γ1k achieved with theproposed group-blind detector satisfies:γ1ka.s.−−−→ γ¯1k =[1 +1(1 + /β2k)2]ρ21k (13)where ρ1k = β†1kβ−12k .Let γ¯′1k denote the SINR achieved with traditional (non-group-blind) detection, i.e., when w˘1k = 0, given by [7, 10]γ¯′1k =β21k∑l>1 β2lk. (14)We define the asymptotic SINR gain η¯1k provided by the pro-posed group-blind detector asη¯1k =γ¯1kγ¯′1k. (15)For L= 2, (14) reduces to γ¯′1k = β21kβ−22k , which combiniedwith (13) and (15) yieldsη¯1k = 1 +1(1 + /β2k)2. (16)Both γ¯1k and η¯1k simplify in the limit → 0, as specified inthe following Corollary.Corollary 1. Let L= 2. The SINR gain obtained by using thegroup-blind receiver for user k satisfiesη¯1k→ 2 as → 0, (17)hence the asymtptotic SINR γ1k reduces toγ¯1k→ 2γ¯′1k as → 0. (18)The above Corollary shows that, in the high-SNR regime,the asymptotic SINR achieved with group-blind detection isdoubled compared to traditional detection.Let ∆R¯1k be the difference between the asymptotic ratesachieved by user k with and without group-blind detection,respectively, given by∆R¯1k = log(1 + γ¯1k)− log(1 + γ¯′1k). (19)In the high-SNR regime, ∆R¯1k ≈ ρ21k when β2k β1k(strong out-of-cell interference), while ∆R¯1k ≈ 1 b/s/Hzwhen β2k β1k (weak out-of-cell interference). Note thatthe case β2k β1k can occasionally occur when BSs arerandomly deployed, resulting in irregular Voronoi cells.5. NUMERICAL RESULTSIn this section, we give numerical results to validate our anal-ysis and show the performance gain achieved through group-blind detection. We assume β11 = · · ·= β1K = 1 and de-note SNR= Pβ1k = P the average received SNR from in-cell users. Solid lines on figures correspond to analytical0 50 100 150 200 250 3000:60:811:21:41:6Number of antennasAchievablerate(b/s/Hz)GBNGBFig. 2: Achievable rate (b/s/Hz) as a function of the number ofantennas n with and without group-blind detection. Scenarioparameters: L= 2, K = 1, SNR= 20 dB and β11/β21 =0 dB (strong interference).results, whereas dashed lines connect simulation outputs. Allfigures confirm that simulations converge to our closed formexpressions as the number of antennas grows.We consider a simple scenario in Fig. 2, with L= 2, K =1, SNR= 20 dB, and β11/β21 = 0 dB (strong interference).We compare the achievable rate of non-group-blind (NGB)detection vs. group-blind (GB) detection. GB detection out-performs NGB detection by ∆R¯1k ≈ 0.585 (b/s/Hz). Thisvalue matches the asymptotic rate gap following from (19),(18), and (14), that is ∆R¯1k = log2(3)− log2(2) (b/s/Hz).Figure 3 shows the achievable rate vs. the number ofantennas n for GB detection. We consider several valuesof /β2k, that model whether the estimation error is domi-nated by pilot contamination ( < β2k) or thermal noise ( >β2k). Figure 3 is based on the following scenario: L= 2,K = 1, SNR= 10 dB, and fixed β21 with β11/β21 = 10 dB(weak interference). The rate achieved with NGB detectorin the presence of negligible noise during the training phaseis plotted for comparison. The rate achieved with GB detec-tors decreases as  grows, consistently with (13). However,even when the training phase is severely affected by noise,GB detection still outperforms NGB detection with noise-freetraining phase.Finally, we consider in Fig. 4 a scenario with L= 4 cells,SNR= 10 dB, and a number of users per cell equal to eitherK = 1 or K = 10, and plot the achievable rate per user, asa function of the number of antennas n. GB detection isshowed for n>KL, that is the required minimum numberof antennas to implement the detector. In this case also, GBdetection outperforms NGB detection. Moreover, the figureshows that GB detection is much more robust to variations ofthe network load, i.e., the number of users per BS antenna.0 50 100 150 200 250 300345678Number of antennasAchievablerate(b/s/Hz)GB,  ˇ2kGB, =ˇ2k D 0:5GB, =ˇ2k D 2NGB,  ˇ2kFig. 3: Achievable rate (b/s/Hz) as a function of the number ofantennas, in presence of non-negligible noise effects duringthe training phase. Scenario parameters: L= 2, K = 1,SNR= 10 dB, and β11/β21 = 10 dB (weak interference).0 50 100 150 200 250 300012345Number of antennas nAchievablerate(b/s/Hz)NGB (K D 1) GB (K D 1)NGB (K D 10) GB (K D 10)Fig. 4: Achievable rate (b/s/Hz) with group-blind (GB) and withoutgroup-blind (NGB) detection. Scenario parameters: L= 4,K = 1 or K = 10, SNR= 10 dB, β1k/β2k = 10 dB.6. DISCUSSIONWe introduced a receiver for the uplink of multiuser massiveMIMO that accounts for the presence of pilot contamination.The proposed scheme performs group-blind detection by ex-ploiting the excess degrees of freedom provided by the largenumber of antennas per BS. We derived analytical results forthe asymptotic achievable rate in an interference-limited sce-nario with one dominant interfering cell, and confirmed ourfindings through simulations. 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Group-blind detection with very large antenna arrays in the presence of pilot contamination

G. C. Ferrante

G. Geraci

T. Q. S. Quek

M. Z. Win

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Group-blind detection with very large antenna arrays in the presence of
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Group-blind detection with very large antenna arrays in the presence of pilot contamination

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