Enhanced Normalized Conjugate Beamforming for Cell-Free Massive MIMO
aa r X i v : . [ c s . I T ] J a n INTERDONATO et al. : ENHANCED NORMALIZED CONJUGATE BEAMFORMING FOR CELL-FREE MASSIVE MIMO 1
This paper was submitted for publication in IEEE Transactions on Communications on August 28, 2020. It was finally acceptedfor publication on January 21, 2021.© 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in anycurrent or future media, including reprinting/republishing this material for advertising or promotional purposes, creating newcollective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in otherworks.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. X, XXXX
Enhanced Normalized Conjugate Beamformingfor Cell-Free Massive MIMO
Giovanni Interdonato,
Member, IEEE,
Hien Quoc Ngo,
Senior Member, IEEE and Erik G. Larsson,
Fellow, IEEE
Abstract —In cell-free massive multiple-input multiple-output(MIMO) the fluctuations of the channel gain from the accesspoints to a user are large due to the distributed topology ofthe system. Because of these fluctuations, data decoding schemesthat treat the channel as deterministic perform inefficiently. Away to reduce the channel fluctuations is to design a precodingscheme that equalizes the effective channel gain seen by theusers. Conjugate beamforming (CB) poorly contributes to hardenthe effective channel at the users. In this work, we propose avariant of CB dubbed enhanced normalized
CB (ECB), in that theprecoding vector consists of the conjugate of the channel estimatenormalized by its squared norm. For this scheme, we derivean exact closed-form expression for an achievable downlinkspectral efficiency (SE), accounting for channel estimation errors,pilot reuse and user’s lack of channel state information (CSI),assuming independent Rayleigh fading channels. We also devisean optimal max-min fairness power allocation based only onlarge-scale fading quantities. ECB greatly boosts the channelhardening enabling the users to reliably decode data relyingonly on statistical CSI. As the provided effective channel isnearly deterministic, acquiring CSI at the users does not yield asignificant gain.
Index Terms —Cell-free massive MIMO, conjugate beamform-ing, max-min fairness power control, spectral efficiency, channelhardening, downlink training.
I. I
NTRODUCTION C ELL-FREE massive multiple-input multiple-output(MIMO) [2]–[4] is a practical and scalable embodimentof network MIMO, and promises unprecedented levels ofSE by leveraging an extraordinary macro-diversity and anaggressive spatial multiplexing of the users. The practicalityand the scalability of such a system comes from decentralizingchannel estimation and precoding/combining, enabled byoperating in time division duplex (TDD). The distributeddense topology of cell-free massive MIMO enriches themacro-diversity and allows to implement a user-centric network wherein every user is in the center of a tailored
G. Interdonato was with the Dept. of Electrical Engineering (ISY),Link¨oping University, 581 83 Link¨oping, Sweden, and is now with the Dept.of Electrical and Information Engineering, University of Cassino and SouthernLazio, 03043 Cassino, Italy ([email protected])H. Q. Ngo is with the Institute of Electronics, Communications andInformation Technology (ECIT), Queen’s University Belfast, Belfast, BT39DT, U.K. ([email protected]).E. G. Larsson is with the Dept. of Electrical Engineering (ISY), Link¨opingUniversity, 581 83 Link¨oping, Sweden ([email protected]).The work of G. Interdonato and E. G. Larsson was supported in part by theSwedish Research Council (VR) and ELLIIT. The work of H. Q. Ngo wassupported by the U.K. Research and Innovation Future Leaders Fellowshipsunder Grant MR/S017666/1.Part of this work was presented at the 2016 IEEE 21st InternationalWorkshop on Computer Aided Modelling and Design of CommunicationLinks and Networks (CAMAD) [1]. virtual cell surrounded by serving cooperating access points(APs). On the other hand, since the APs are geographicallyspread out, they contribute quite differently to the effectivefading channel gain seen at each user, which is therebycharacterized by large fluctuations. Hence, the channel is farto be deterministic. This behavior does not occur in co-locatedmassive MIMO where the channel is nearly deterministicinstead, under most relevant operating conditions [5]–[7], aphenomenon known as channel hardening [8]–[10].The lower degree of channel hardening in cell-free massiveMIMO compared to co-located massive MIMO was pointedout in [11], and also analytically demonstrated in [12]–[14]under different channel model assumptions. As the channeldoes not sufficiently harden, the lack of CSI at the userconstitutes a significant limitation in the performance. In thisregard, a scalable pilot-based downlink training scheme forcell-free massive MIMO was advocated in [11] to let the usersperform data decoding based on the acquired CSI rather thanrelying on the statistical CSI.Since deriving from the law of the large numbers , the chan-nel hardening property depends on the number of antennasin the system. In general, the more antennas the more thechannel hardens. Importantly, channel hardening at the usersdoes also depend on the adopted precoding scheme as theeffective downlink channel gain is given by the inner productbetween the downlink channel vector and the precoding vector.Hence, the channel hardening can be artificially boosted byacting on the precoding scheme. In our preliminary work [1],we proposed a different flavour of the conventional conjugatebeamforming (CB) dubbed normalized
CB (NCB) for cell-free massive MIMO systems with single-antenna APs. WithNCB, the precoding factor (in this case a scalar) consists of theconjugate of the channel estimate normalized by its magnitude.This scheme enables a reduction of the uncertainty due tothe user’s lack of CSI knowledge which in turn improves theSE. Recently, the authors in [13] have extended the analysisof [1] to multi-antenna APs, providing an exact closed-formexpression for an achievable downlink SE based on the popular hardening bound [5], [6]. Another modified CB scheme forcell-free massive MIMO was proposed in [15], where theglobal CSI knowledge at the APs is exploited to compensatefor the channel fluctuations and focus the overall channel gainaround a desired mean target.
Contribution:
In this paper, we propose a variant of theNCB precoding scheme described in [1], [13] dubbed en-hanced NCB (ECB), where the vector of the channel estimatesbetween a multi-antenna APs and a given user is normalized byits squared norm. We provide an exact closed-form expression
NTERDONATO et al. : ENHANCED NORMALIZED CONJUGATE BEAMFORMING FOR CELL-FREE MASSIVE MIMO 3 for an achievable downlink SE by using the popular hardeningbound. This expression accounts for channel estimation errorsat the AP, pilot contamination due to pilot reuse, and lackof CSI at the user side. We assume independent Rayleighfading channels which is the best fading scenario for a channelhardening perspective [16] and allows us to draw insightfulconclusions by inspecting the elegant derived closed-formSE expression. Based on the latter, which depends only onthe large-scale fading quantities, we devise an optimal max-min fairness (MMF) power allocation scheme re-adaptingthe convex optimization framework used in [2]. This policydemands for a centralized coordination but, importantly, doesnot depend on the small-scale fading realizations. Hence, itcan be performed by a central processor where the channelstatistics are reasonably assumed to be available. The noveltyof this study consists of: • We provide a comprehensive SE analysis for a normal-ized CB scheme where the normalization term in theprecoding vector (which characterizes an AP-user pair)is the squared norm of the respective channel estimate,rather than just the norm of it, as in [1], [13]. Ourchoice guarantees a better channel gain equalization atthe users, hence the term enhanced
CB. This analysis isnovel in the context of cell-free massive MIMO. A similarprecoder was proposed in [17] for co-located massiveMIMO systems, but its analysis does not consider powercontrol and pilot contamination. • We devise a “local” solution to greatly boost the channelhardening. A substantial difference between the proposedECB and the modified CB in [15] is that the latter requiresCSI exchange among the APs which scales unfavorablyas the number of users and APs in the system grows.Moreover, the “local” nature of ECB, namely that eachAP only needs its own channel estimates to constructthe precoding vectors, is preferable in applications wherelatency is a concern and/or in systems with constrainedfronthaul network capacity where the additional overheaddue to the CSI exchange cannot be afforded. • We give a rigorous formulation of the MMF powercontrol optimization problem for both ECB and NCB.The latter extends the power control analysis of [1], [13]. • We derive an approximate closed-form expression of anachievable downlink SE for CB assuming downlink train-ing and multi-antenna APs along with the correspondingformulation of the MMF power control optimizationproblem. These results extend those in [11], and serveas an upper bound in the performance evaluation.
Related work:
As introduced earlier, the only studiesinvestigating the beamforming normalization are [1], [13],[15], [17]. However, cell-free massive MIMO has recentlyreceived great attention, and in general is a large researchtopic. The research conducted over the last few years aimed toanalyze this concept and evaluate its performance in practicalimplementations. [18]–[20] focus on the scalability and de-centralization aspects of cell-free massive MIMO, while [21],[22] propose an over-the-air signalling scheme to avoid theexchange of CSI for centralized baseband processing. The dynamic clustering approach enabling the user-centric net-work implementation is discussed in [23]–[25]. Optimal andheuristic downlink power allocation algorithms are devisedin [26]–[28] operating in centralized and distributed fashion,respectively. The effectiveness of minimum mean-square error(MMSE) combining with large-scale fading decoding (LSFD)and suboptimal AP selection policies has been establishedin [29], [30] and [31], respectively. Finally, a significanteffort has been spent to evaluate the performance of cell-free massive MIMO under realistic operating assumptions:finite fronthaul capacity [32], [33]; hardware impairments [34],[35]; low-resolution analog-to-digital converters [36], [37];and imperfect channel reciprocity [38].II. S
YSTEM M ODEL
Let us consider a cell-free massive MIMO system with M multi-antenna APs providing service to K single-antenna usersin the same time-frequency resources. Each AP is equippedwith N antennas, and it holds M N ≫ K . A central processingunit (CPU) masters all the APs through a fronthaul network.It is responsible for data sharing, clock synchronization andcentralized operation for resource allocation tasks.The system operates in TDD mode and we assume block-fading model. The time-frequency resources are structured incoherence blocks wherein the channel is approximately staticand frequency flat. The TDD coherence block is τ c sampleslong and is determined by the shortest user’s coherence timeand bandwidth in the system, as τ c = T c B c . Conventionally,a coherence block accommodates three phases: ( i ) uplinktraining, ( ii ) uplink data transmission , and ( iii ) downlinkdata transmission.Let g mk ∈ C N denote the channel response vector betweenuser k and multi-antenna AP m . We assume independentRayleigh fading channels, thereby g mk ∼ CN ( , β mk I N ) ,where β mk is the large-scale fading coefficient capturing thepath loss and the effects of correlated shadowing. A. Uplink Training
Uplink training takes place via pilot transmission. Thelength of the pilot, τ u , p samples, determines the trainingduration as well as the number of mutually orthogonal pilots.Ideally, every user should use an orthogonal pilot sequence toprevent interference from pilot contamination in the channelestimates. However, the share of coherence block reservedto the training is limited and pilot reuse is unavoidable. Let √ τ u , p ϕ k ∈ C τ u , p be the pilot sequence sent by user k , k ϕ k k = 1 . We assume that the pilots of any pair of users canbe either identical or orthogonal, i.e., for any k = j it holds ϕ T k ϕ ∗ j = ( , if ϕ k = ϕ j , , otherwise. (1)The overall pilot signal received at AP m is given by Y p ,m = √ τ u , p ρ u X Kk =1 g mk ϕ T k + Ω p ,m ∈ C N × τ u , p , (2) The uplink data transmission phase is out of the scope of this work andthereby its analysis is herein omitted. Table I summarizes the most relevant notation.
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TABLE IL
IST OF R ELEVANT N OTATIONS M number of APs τ c length of the coherence block N number of AP’s antennas τ u , p uplink training length K number of users τ d , p downlink training length g mk channel response vector between user k and AP m ρ u normalized SNR of the uplink pilot symbol β mk mean-square of any element of g mk ρ d , p normalized SNR of the downlink pilot symbol ˆ g mk MMSE estimate of g mk ρ d normalized SNR of the downlink data symbol γ mk mean-square of any element of ˆ g mk ϕ k uplink pilot sequence related to user k ˜ g mk channel estimation error, g mk − ˆ g mk ψ k downlink pilot sequence related to user ka kk effective downlink channel at user k x m data signal transmitted by AP m ˆ a kk MMSE estimate of a kk w mk precoding vector used by AP m to user k ˜ a kk downlink channel estimation error, a kk − ˆ a kk η mk downlink power control coefficientwhere ρ u is the normalized signal-to-noise ratio (SNR) of theuplink pilot symbol, and Ω p ,m is a matrix of additive noisewhose elements are independent and identically distributed(i.i.d.) CN (0 , . AP m de-spreads the pilot signal by usingthe pilot sequences as y p ,mk , Y p ,m ϕ ∗ k = √ τ u , p ρ u g mk + √ τ u , p ρ u K X j = k g mj ϕ T j ϕ ∗ k + ω p ,mk (3) ∈ C N , where ω p ,mk = Ω p ,m ϕ ∗ k ∼ CN ( , I N ) . Due to thepilot sequence design in (1), y p ,mk constitutes a sufficientstatistic. Provided that { β mk } are known a priori at the AP,linear MMSE channel estimation can be performed, and itis optimal—in the sense that it minimizes the mean squareerror—due to the Gaussian distribution of the channels. Chan-nel estimation is carried out locally at each AP and the MMSEestimate of g mk is given by ˆ g mk = E (cid:8) g mk y H p ,mk (cid:9) (cid:0) E (cid:8) y p ,mk y H p ,mk (cid:9)(cid:1) − y p ,mk = c mk y p ,mk , (4)where c mk , √ τ u , p ρ u β mk τ u , p ρ u P Kj =1 β mj | ϕ H k ϕ j | + 1 . (5)The MMSE channel estimate is distributed as ˆ g mk ∼ CN ( , γ mk I N ) where γ mk = √ τ u , p ρ u c mk β mk . (6)In the channel estimate, the interference from pilot contami-nation is captured by the terms that are proportional to ϕ H k ϕ j and for which this inner product gives 1. Note that, if user k and user j share the same pilot, then the respective channelestimates as well as their mean-squares are proportional toeach other: ˆ g mk = β mk β mj ˆ g mj , (7) γ mk = β mk β mj γ mj . (8) Finally, let us define the channel estimation error ˜ g mk = g mk − ˆ g mk , which is distributed as ˜ g mk ∼ CN ( , ( β mk − γ mk ) I N ) , and independent of ˆ g mk . B. Downlink Data Transmission
By leveraging the channel reciprocity deriving from theTDD operation, the channel estimates obtained in the uplinkare then employed to construct the precoding vectors. Let w mk ∈ C N be the precoding vector used by AP m in theservice of user k . We assume that w mk solely depends on localchannel estimates. Hence, there is no CSI exchange among theAPs over the fronthaul network. The data signal that AP m transmits to the users is x m = √ ρ d X Kk =1 √ η mk w mk q k , (9)where q k is the data symbol intended for user k , E (cid:8) | q k | (cid:9) = 1 and E (cid:8) q k q ∗ j (cid:9) = 0 for k = j . ρ d is the normalized SNRof the downlink data symbol, and { η mk } are power controlcoefficients satisfying the per-AP power constraint E n k x m k o ≤ ρ d , m = 1 , . . . , M. (10)By setting η mk = 0 , AP m does not participate in the serviceof user k . Hence, these coefficients are also useful to set upclustering policies aiming to preserve the scalability of thesystem.The signal received at user k resulting from the joint coherenttransmission by multiple APs is r k = M X m =1 g T mk x m + ω k = √ ρ d M X m =1 √ η mk g T mk w mk q k + √ ρ d K X j = k M X m =1 √ η mj g T mk w mj q j + ω k , (11)where ω k ∼ CN (0 , is additive noise. NTERDONATO et al. : ENHANCED NORMALIZED CONJUGATE BEAMFORMING FOR CELL-FREE MASSIVE MIMO 5
III. P
ERFORMANCE A NALYSIS - N O CSI
AT THE U SER
When evaluating the capacity that this system can achieve,we must bear in mind the lack of CSI at the user, whichemploys the channel statistics to perform data decoding. Anachievable downlink spectral efficiency for user k can beobtained by using the popular hardening bound [5], [6]. Werewrite (11) as r k = DS k q k + BU k q k + X Kj = k UI kj q j + ω k , (12)where DS k = √ ρ d X Mm =1 √ η mk E (cid:8) g T mk w mk (cid:9) , (13) BU k = √ ρ d X Mm =1 √ η mk g T mk w mk − √ ρ d X Mm =1 √ η mk E (cid:8) g T mk w mk (cid:9) , (14) UI kj = √ ρ d X Mm =1 √ η mj g T mk w mj , (15)emphasizing that user k can detect q k by exploiting only theknowledge of the channel statistics, that is the knowledgeof E (cid:8) g T mk w mk (cid:9) . Hence, DS k represents the desired signalfor user k , BU k (beamforming gain uncertainty) is a self-interference contribution capturing user’s uncertainty of theinstantaneous channel gain, while UI kj describes the inter-user interference. The term BU k constitutes a measure ofthe channel hardening degree at user k . Specifically, BU k quantifies how much the instantaneous effective downlinkchannel deviates from its mean. The smaller BU k is, the morehardening the channel offers. By treating the sum of the lastthree terms in (12) as uncorrelated effective noise, a lowerbound on the downlink capacity is given by SE k = ξ (cid:18) − τ u , p τ c (cid:19) log (1 + SINR k ) , (16)where < ξ < is the share of the coherence block reservedto the downlink, the pre-log factor − τ u , p /τ c accounts for the pilot overhead, and the signal-to-interference-plus-noise ratio(SINR) at user k is SINR k = | DS k | E {| BU k | } + P Kk = j E {| UI kj | } + 1 . (17)Expression (17) is valid for any precoding scheme and channelmodel. We next report the closed-form expression of SINR k assuming multi-antenna APs and independent Rayleigh fadingchannels for both CB [39], NCB [13] and the proposed ECB. A. Conjugate Beamforming
CB consists in setting w CB mk = ˆ g ∗ mk . By inserting thisinto (17) and computing the corresponding expectations, theeffective SINR per user is given by [39] and (18), shown atthe bottom of this page, where ς kj , X Mm =1 η mj β mk γ mj . (19)Importantly, ρ d N ς kk is the power of the beamforming gainuncertainty which equals the variance of the effective down-link channel, P Mm =1 √ ρ d η mk g T mk ˆ g ∗ mk . Finally, by inserting w mk = w CB mk into (9), the per-AP power constraint in (10)results in K X k =1 η mk γ mk ≤ N , m = 1 , . . . , M. (20)
B. Normalized Conjugate Beamforming
NCB consists in setting w NCB mk = ˆ g ∗ mk / k ˆ g mk k . By insertingthis into (17) and computing the corresponding expectations,the effective SINR per user is given by [13] and (21), shownat the bottom of this page. In this case, the power of thebeamforming gain uncertainty is equal to E n | BU NCB k | o = ρ d M X m =1 η mk (cid:2) β mk +( N − − α ) γ mk (cid:3) , (24)and the per-AP power constraint for NCB is X Kk =1 η mk ≤ , m = 1 , . . . , M. (25) SINR CB k = ρ d N (cid:18) M P m =1 √ η mk γ mk (cid:19) ρ d N K P j =1 ς kj + ρ d N K P j = k (cid:18) M P m =1 √ η mj γ mj β mk β mj (cid:19) (cid:12)(cid:12) ϕ H k ϕ j (cid:12)(cid:12) + 1 , (18) SINR
NCB k = ρ d α (cid:18) M P m =1 √ η mk γ mk (cid:19) ρ d ( N − − α ) M P m =1 η mk γ mk + ρ d K P j =1 M P m =1 η mj β mk + ρ d K P j = k Υ kj | ϕ H k ϕ j | + 1 , (21)where Υ kj , ( N − M X m =1 η mj γ mj β mk β mj + α M X m =1 M X n = m √ η mj η nj γ mj γ nj β mk β nk β mj β nj , (22) α , Γ( N + 1 / N ) . (23) IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. X, XXXX
C. Enhanced Normalized Conjugate Beamforming
ECB consists in setting w ECB mk = ˆ g ∗ mk k ˆ g mk k . (26)The reason why this normalization should enhance the nor-malization ˆ g ∗ mk / k ˆ g mk k and provide higher SE is intuitive.Consider the effective downlink channel at user k and neglectthe channel estimation error, a kk = M X m =1 √ η mk g T mk w mk ≈ M X m =1 √ η mk ˆ g T mk w mk . (27)If w mk = w ECB mk , then a kk ≈ P Mm =1 √ η mk is ideallydeterministic. Conversely, if w mk = w NCB mk as in [13], then a kk ≈ P Mm =1 √ η mk k ˆ g mk k , so that the fluctuations of thechannel gain are reduced but not entirely equalized.By inserting (26) into (17) and computing the correspondingexpectations, the effective SINR per user is given by (28) atthe bottom of this page, where Θ kj , N − N − M X m =1 η mj β mk β mj + M X m =1 M X n = m √ η mj η nj β mk β nk β mj β nj . (29) Proof:
See Appendix A .For ECB, the power of the beamforming gain uncertainty isequal to E n | BU ECB k | o = ρ d N − M X m =1 η mk (cid:18) β mk γ mk − (cid:19) . (30)By inserting w mk = w ECB mk into (9), the per-AP powerconstraint in (10) becomes K X k =1 η mk E ( k ˆ g mk k ) ≤ ⇒ K X k =1 η mk γ mk ≤ N − , (31) m = 1 , . . . , M . Importantly, equations (28), (31) are definedif N > . In fact, E (cid:8) / | x | (cid:9) does not converge if x is a scalarcircularly symmetric Gaussian random variable.IV. P ERFORMANCE A NALYSIS OF C ONJUGATE B EAMFORMING WITH D OWNLINK T RAINING
In this section we analyze the performance of CB whendownlink training is carried out (CB-DT). Downlink trainingtakes place via pilot beamforming as described in [11], andherein we extend the analysis in [11] to multi-antenna APs. Byconjugate beamforming the pilots, the estimation overhead isindependent of the number of APs. The channel estimationoverhead rather scales with K and does not require any feedback from the users to the APs as the channel is reciprocalin TDD mode.The downlink training phase is τ d , p samples long. Thisincreases the pilot overhead up to τ u , p + τ d , p . Let √ τ d , p ψ k ∈ C τ d , p be the downlink pilot sequence intended for user k , k ψ k k = 1 , and ρ d , p be the normalized SNR of the downlinkpilot symbol. AP m beamforms the pilots as X m, p = √ τ d , p ρ d , p K X k =1 √ η mk ˆ g ∗ mk ψ T k ∈ C N × τ d , p , (32)subject to the following power constraint: E n k X m, p k o = τ d , p ρ d , p Tr E K X k =1 K X j =1 √ η mk η mj ˆ g ∗ mk ˆ g T mj ψ T k ψ ∗ j = τ d , p ρ d , p Tr K X k =1 η mk γ mk I N + K X k =1 K X j = k √ η mk η mj E (cid:8) ˆ g ∗ mk ˆ g T mj (cid:9) ψ T k ψ ∗ j ( a ) = τ d , p ρ d , p N K X k =1 η mk γ mk ≤ τ d , p ρ d , p , (33)where in ( a ) we have assumed that the pilot assignment isconstrained such that ψ T k ψ ∗ j = 0 if ϕ k = ϕ j . (34)This limitation is not significant in cases of practical in-terest since the number of assignable uplink and downlinkpilot pairs satisfying (34) is larger than the number of ac-tive users, as shown in [11]. Importantly, this ensures that Tr E (cid:8) ˆ g ∗ mk ˆ g T mj (cid:9) ψ T k ψ ∗ j = 0 , ∀ j = k . From (32), (33), we have E n k X m, p k o ≤ τ d , p ρ d , p = ⇒ K X k =1 η mk γ mk ≤ N .
It is observed that the data power constraint in (20) and theabove pilot power constraint are identical. The pilot signalreceived at user k is given by y dp ,k = M X m =1 g T mk X m, p + ω dp ,k = √ τ d , p ρ d , p K X j =1 M X m =1 √ η mj g T mk ˆ g ∗ mj ψ T j + ω dp ,k (35) ∈ C τ d , p , where ω dp ,k is an additive noise vector whoseelements are i.i.d. CN (0 , . SINR
ECB k = ρ d (cid:18) M P m =1 √ η mk (cid:19) ρ d N − M P m =1 η mk (cid:18) β mk γ mk − (cid:19) + ρ d N − K P j = k M P m =1 η mj β mk γ mj + ρ d K P j = k Θ kj | ϕ H k ϕ j | + 1 , (28) NTERDONATO et al. : ENHANCED NORMALIZED CONJUGATE BEAMFORMING FOR CELL-FREE MASSIVE MIMO 7
From y dp ,k , user k separates its channel observation as ˇ y dp ,k = y dp ,k ψ ∗ k = √ τ d , p ρ d , p a kk + √ τ d , p ρ d , p K X j = k a kj ψ T j ψ ∗ k + ω dp ,k ψ ∗ k , (36)where a kj , X Mm =1 √ η mj g T mk ˆ g ∗ mj denotes the effective downlink channel seen by user k butintended for user j . Based upon ˇ y dp ,k , user k estimates a kk by MMSE estimation. Following the same approach as in [11],we can compute the MMSE downlink channel estimate ˆ a kk as well as its variance in closed form, κ k , Var { ˆ a kk } = τ d , p ρ d , p N (cid:16)P Mm =1 η mk γ mk β mk (cid:17) τ d , p ρ d , p N M P m =1 K P j =1 η mj γ mj β mk | ψ H k ψ j | = τ d , p ρ d , p N ς kk τ d , p ρ d , p N P Kj =1 ς kj | ψ H k ψ j | . (37)A closed-form expression for an approximate achievabledownlink SE can be derived by using the so-called capac-ity bound with side information [5, Section 2.3.5], and byfollowing the same methodology as in [11]: SE k = ξ (cid:18) − τ u , p + τ d , p τ c (cid:19) log (cid:16) SINR
CB-DT k (cid:17) , (38)with SINR
CB-DT k given by (39), (40) at the bottom of this page. Proof:
See Appendix B .The SINR gain over the case where the users do not haveaccess to the CSI is noticeable. Compared to (18), in (40) thecoherent gain (namely the numerator of the SINR) is increasedby ρ d κ k which is at most equal to the variance of the effectivedownlink channel, ρ d N ς kk . Importantly, downlink trainingreduces the uncertainty at the user which now can employits CSI knowledge when performing data decoding. In fact,the channel uncertainty, ρ d N ς kk , is significantly decreased by ρ d κ k . This residual self-interference represents the variance ofthe downlink channel estimation error.V. M AX -M IN F AIRNESS P OWER C ONTROL
Max-min fairness (MMF) power control is an egalitarianpolicy that ensures maximized identical SE throughout the The approximation comes from the non-Gaussian nature of { a kj } , andfrom [40]. See [11] for further details. network. This policy perfectly suits cell-free massive MIMOwhich by nature guarantees a more uniform quality of servicethan co-located massive MIMO [3]. Such a sophisticatedpower control requires a centralized approach and solvinga convex optimization problem. However, requirements onlatency, computational complexity, and fronthauling load, canbe relaxed by two factors: ( i ) if relying on closed-form SEexpressions, power control can operate at the large-scale fadingtime scale; ( ii ) if combined with clustering, the coordinationcan be confined within few APs. These two aspects are keyin all our implementations.In general, the optimization problem for the MMF powercontrol subject to per-AP power constraint can be formulatedas max { η mk ≥ } min k SINR k (41a)s.t. E n k x m k o ≤ ρ d , ∀ m. (41b)Next, we give rigorous formulations for the MMF opti-mization problem with NCB and ECB. These problems havestructural similarity to that in [2], hence they admit globaloptimal solutions that can be computed by solving a sequenceof second-order cone programs. A. Problem Formulation for NCB
Through some simple mathematical manipulations we re-shape the SINR expression in (21). Firstly, we rewrite (22) as Υ kj = ( N − M X m =1 η mj γ mj β mk β mj + α M X m =1 M X n = m √ η mj η nj γ mj γ nj β mk β nk β mj β nj + α M X m =1 η mj γ mj β mk β mj − α M X m =1 η mj γ mj β mk β mj = ( N − − α ) M X m =1 η mj γ mj β mk β mj + α M X m =1 √ η mj γ mj β mk β mj ! , (42)and by inserting (42) into (21), we obtain SINR
NCB k = ρ d α (cid:16)P Mm =1 √ η mk γ mk (cid:17) T1 k + T2 k + 1 , (43) SINR
CB-DT k ≈ ρ d E (cid:8) | ˆ a kk | (cid:9) ρ d E {| ˜ a kk | } + ρ d P Kj = k E {| a kj | } + 1 (39) = ρ d N (cid:16)P Mm =1 √ η mk γ mk (cid:17) + ρ d κ k ρ d ( N ς kk − κ k ) + ρ d K P j = k " N ς kj + N (cid:12)(cid:12) ϕ H k ϕ j (cid:12)(cid:12) (cid:18) M P m =1 √ η mj γ mj β mk β mj (cid:19) + 1 . (40) IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. X, XXXX where T1 k = ρ d K X j =1 M X m =1 η mj (cid:2) β mk + ( N − − α ) γ mk | ϕ H k ϕ j | (cid:3) = ρ d K X j =1 M X m =1 η mj ϑ mkj , (44) ϑ mkj , β mk + ( N − − α ) γ mk | ϕ H k ϕ j | , (45) T2 k = ρ d α K X j = k M X m =1 √ η mj γ mj β mk β mj ! | ϕ H k ϕ j | a ) = ρ d α K X j = k M X m =1 √ η mj γ mk ! | ϕ H k ϕ j | . (46)Note that − π/ < N − − α < , thus ϑ mkj is alwayspositive since β mk ≥ γ mk . The equality ( a ) follows from (7).By using (43) the MMF optimization problem in (41) can beformulated in epigraph form for NCB asmaximize { η mk ≥ } , ν ν (47a)s.t. k s k k ≤ γ T kk u k , ∀ k, (47b) k u ′ m k ≤ √ ρ d , ∀ m, (47c)where ν is a new variable which represents the minimum SINRamong the users that has to be maximized, and: • U = [ u , . . . , u K ] , u k = √ ρ d (cid:2) √ η k , . . . , √ η Mk (cid:3) T , and u ′ m is the m th row of U ; • γ kj = α | ϕ H k ϕ j | (cid:2) √ γ k , . . . , √ γ Mk (cid:3) T ; • s k = √ ν · (cid:2) v T k I − k , k b k ◦ u k , . . . , k b kK ◦ u K k , (cid:3) T ; • v k , (cid:2) γ T k u , . . . , γ T kK u K (cid:3) T ; • I − k is a K × ( K − matrix obtained from I k with the k th column removed; • b kj = (cid:2)p ϑ kj , . . . , p ϑ Mkj (cid:3) T .The constraints (47b), (47c) are jointly second-order coneswith respect to the power control coefficients and ν . If ν isfixed, then (47) is convex, and the globally optimal solutioncan be obtained by using interior-point methods. The globallyoptimal solution to (47) can be obtained in polynomial timevia the bisection method [41] by solving a sequence of convex(more specifically, second-order cone) feasibility problems.A detailed description of the bisection search algorithm forthe MMF power control and a more general formulation forthe MMF optimization problem can be found, for example,in [2], [42] and [6, Section 7.1.1], respectively. B. Problem Formulation for ECB
Similarly, for ECB we first realize that (29) can be written as Θ kj = N − N − M X m =1 η mj β mk β mj + M X m =1 M X n = m √ η mj η nj β mk β nk β mj β nj + M X m =1 η mj β mk β mj − M X m =1 η mj β mk β mj = (cid:18) N − N − − (cid:19) M X m =1 η mj β mk β mj + M X m =1 √ η mj β mk β mj ! = M X m =1 √ η mj β mk β mj ! − N − M X m =1 η mj β mk β mj . (48)As a result, the SINR expression in (28) is given by (49) atthe bottom of this page, where we have defined ̺ mkj , N − β mk γ mj − β mk β mj | ϕ H k ϕ j | ! , which is always positive since β mk ≥ γ mk , and N > .The MMF optimization problem in (41) can be formulatedin epigraph form for ECB asmaximize { η mk ≥ } , ν ν (50a)s.t. k s k k ≤ T u k , ∀ k, (50b) k ˆ γ ′ m ◦ u ′ m k ≤ p ρ d ( N − , ∀ m, (50c)where ν , u k and u ′ m are defined as in (47), denotes the M -dimensional vector of ones, and: • s k = √ ν · (cid:2) v T k I − k , k z k ◦ u k , . . . , k z kK ◦ u K k , (cid:3) T ; • v k , (cid:2) b T k u , . . . , b T kK u K (cid:3) T ; • ˆ γ ′ m is the m th row of ˆ Γ , ˆ Γ = [ˆ γ , . . . , ˆ γ K ] and ˆ γ k = h γ − / k , . . . , γ − / Mk i T ; • b kj = | ϕ H k ϕ j | (cid:20) β k β j , . . . , β Mk β Mj (cid:21) T ; • z kj = (cid:2) √ ̺ kj , . . . , √ ̺ Mkj (cid:3) T . C. Problem Formulation for CB with Downlink Training
The formulation of the power optimization problem for theCB scheme with downlink training and multi-antenna APsfollows that in [11, Section IV-A], with a simple adjustment: γ mk must be replaced with N γ mk everywhere in the problem SINR
ECB k = ρ d (cid:16)P Mm =1 √ η mk (cid:17) ρ d N − K P j =1 M P m =1 η mj β mk γ mj − β mk β mj | ϕ H k ϕ j | ! + ρ d K P j = k (cid:18) M P m =1 √ η mj β mk β mj (cid:19) | ϕ H k ϕ j | +1= ρ d (cid:16)P Mm =1 √ η mk (cid:17) ρ d K P j =1 M P m =1 η mj ̺ mkj + ρ d K P j = k (cid:18) M P m =1 √ η mj β mk β mj (cid:19) | ϕ H k ϕ j | +1 , (49) NTERDONATO et al. : ENHANCED NORMALIZED CONJUGATE BEAMFORMING FOR CELL-FREE MASSIVE MIMO 9 formulation. Compared to NCB and ECB, the MMF powercontrol for CB with downlink training does not rigorouslyprovide identical SINRs as it results from a sequential convexapproximation and thereby the problem solutions are sub-optimal [11]. VI. S
IMULATION R ESULTS
In this section, we compare the performance of the precod-ing schemes discussed in Section III and IV by presenting theresults of our simulations. Next, we introduce the simulationscenario and the adopted settings.
A. Simulation Scenario and Settings
In our simulations we consider a nominal area of D × D squared meters, wherein APs and users are uniformly locatedat random. A wraparound technique is used to simulate a cell-free network. A random realization of AP and user locationsdetermines a set of large-scale fading coefficients and con-stitutes a snapshot of the network. For a network snapshotthe achievable downlink SEs are computed, according to theexpressions presented in Section III and IV. The cumulativedistribution function (CDF) of the SE is obtained over manynetwork snapshots.We adopt the 3GPP Urban Microcell pathloss model definedin [43, Table B.1.2.1-1] as PL mk [dB] = − . − . (cid:18) d mk m (cid:19) , (51)where d mk is the distance (in three dimensions) between AP m and user k , and assuming a 2 GHz carrier frequency. We alsoconsider log-normal shadow fading with standard deviation σ sh and spatial correlations both at the APs and the users. Morespecifically, let q mk ∼ N (0 , be defined as q mk = √ ǫ a m + √ − ǫ b k , (52)where a m ∼ N (0 , and b k ∼ N (0 , are independentrandom variables capturing the shadow fading effects from AP m to all the users and from user k to all the APs, respectively.The parameter < ǫ < weighs these effects. The shadowfading spatial correlations are thus modeled as [43] E { a m a n } = 2 − ( d AP mn / 9 m ) , E { b k b j } = 2 − ( d UE kj / 9 m ) , (53)where d AP mn is the distance between AP m and AP n , d UE kj isthe distance between user k and user j , and 9 meters is thedecorrelation distance. Pathloss and log-normal shadow fadingenter into the large-scale fading coefficients as β mk = PL mk · σ sh q mk / . (54)Unless otherwise stated, we use the following simulationsettings: D = 500 m, σ sh = 4 dB, ǫ = 0 . , AP height10 m, user height 1.5 m, channel bandwidth B = 20 MHz,antenna gains 0 dBi. The TDD coherence block is partitionedequally between uplink and downlink, so ξ = 0 . , andit is τ C = 200 samples long, resulting from a coherencebandwidth of 200 kHz and a coherence time of 1 ms. Themaximum transmit power per AP and per user is mW TABLE IIP ER -AP POWER CONSTRAINTS FOR DIFFERENT PRECODING SCHEMES . CB, CB-DT NCB ECB N K P k =1 η CB mk γ mk ≤ K P k =1 η NCB mk ≤ N − K P k =1 η ECB mk γ mk ≤ and mW, respectively. This is normalized by the noisepower, n ( dBm ) p = − dBm, to obtain ρ d [dBm] = ρ d , p [dBm] = 10 log (200) − n ( dBm ) p , (55) ρ u [dBm] = 10 log (100) − n ( dBm ) p . (56)In all our simulations, we also consider the largest-large-scale-fading-based AP selection [39], according to which anAP participates in the service of user k if its channel issufficiently strong, and more specifically if β mk belongs tothe user- k -specific AP cluster A k which satisfies |A k | X m =1 ¯ β mk P Mn =1 β nk ≥ , (57)where |A k | is the cardinality of the set A k with min( |A k | ) =10 , and { ¯ β k , . . . , ¯ β M,k } are the large-scale fading coefficientssorted in descending order.To emphasize that the power control coefficients of theconsidered precoding schemes are subject to different per-AP power constraints, we denote by { η CB mk } , { η NCB mk } and { η ECB mk } the power control coefficients related to CB, NCBand ECB, respectively. Note that the power constraints forCB-DT and CB are identical, hence we use { η CB mk } to denotethe coefficients for CB-DT too. Table II reports the per-APpower constraints for the considered precoding schemes. As analternative to the MMF power control described in Section V,we consider a heuristic distributed power control, also knowsas maximal-ratio [2], which consists in setting the powercontrol coefficients as η CB mk = 1 N P Kj =1 γ mj , (58) η NCB mk = N γ mk η CB mk , (59) η ECB mk = ( N − γ mk η NCB mk . (60)With these settings the respective power constraints hold withequality, and thereby each AP spends all the available transmitpower. Maximal-ratio power control is an “opportunistic”policy whereby more power is allocated to the users with betterchannel conditions.Finally, since τ u , p < K , we assume that the uplink pilotsequences are assigned at random and reused throughout thenetwork. When downlink training is performed, the downlinkpilot sequences are assigned as proposed in [11] to satisfy thecondition in (34). B. Performance Evaluation
The first aspect we focus on is the impact of the precodingnormalization on the performance. In this regard, we look at
TABLE IIIE
XPLICIT CLOSED - FORM EXPRESSIONS OF THE EXPECTATIONS IN (17)
AND (39).
Coherent Gain Self-InterferenceCB ρ d N (cid:18) M P m =1 q η CB mk γ mk (cid:19) ρ d N M P m =1 η CB mk β mk γ mk NCB ρ d α (cid:18) M P m =1 q η NCB mk γ mk (cid:19) ρ d M P m =1 η NCB mk β mk + ρ d ( N − − α ) M P m =1 η NCB mk γ mk ECB ρ d (cid:18) M P m =1 q η ECB mk (cid:19) ρ d N − M P m =1 η ECB mk (cid:18) β mk γ mk − (cid:19) CB-DT ρ d N (cid:18) M P m =1 q η CB mk γ mk (cid:19) + ρ d κ k ρ d N M P m =1 η CB mk β mk γ mk − ρ d κ k Inter-user InterferenceCB, CB-DT ρ d N K P j = k " M P m =1 η CB mj β mk γ mj + N (cid:18) M P m =1 q η CB mj γ mj β mk β mj (cid:19) (cid:12)(cid:12) ϕ H k ϕ j (cid:12)(cid:12) NCB ρ d K P j = k " M P m =1 η NCB mj (cid:0) β mk +( N − − α ) γ mk | ϕ H k ϕ j | (cid:1) + (cid:18) α M P m =1 q η NCB mj γ mk (cid:19) | ϕ H k ϕ j | ECB ρ d K P j = k " N − M P m =1 η ECB mj β mk γ mj − β mk β mj | ϕ H k ϕ j | ! + (cid:18) M P m =1 q η ECB mj β mk β mj (cid:19) | ϕ H k ϕ j | two metrics: ( i ) the power of the self-interference as shareof the desired signal power which is also known as coherent gain; ( ii ) the power of the inter-user interference as share ofthe coherent gain.If downlink training is not performed, the self-interferencecorresponds to the beamforming gain uncertainty introducedin Section III. Hence, the first metric is simply E (cid:8) | BU k | (cid:9) | DS k | , (61)where E (cid:8) | BU k | (cid:9) is equal to ρ d N ς kk for CB, and givenby (24) and (30) for NCB and ECB, respectively. The coherentgain is given by the term in the numerator of the SINRexpressions in (18), (21) and (28), respectively.Importantly, the metric (61) gives a meaningful measure ofthe channel hardening, since it is a normalized variance of theeffective downlink channel for user k . The smaller this vari-ance is, the more the channel hardens. Conversely, for the CBscheme with downlink training (CB-DT) the self-interferencecorresponds to the variance of the downlink channel estimationerror, ρ d E (cid:8) | ˜ a kk | (cid:9) = ρ d N ς kk − ρ d κ k , i.e., the first term ofthe denominator in (40), the coherent gain is the term in thenumerator of (40), namely ρ d E (cid:8) | ˆ a kk | (cid:9) , and the inter-userinterference is the second term of the denominator in (40),corresponding to ρ d P Kj = k E (cid:8) | a kj | (cid:9) .Table III summarizes the closed-form expressions of coher-ent gain, self-interference and inter-user interference for allthe considered precoding schemes.In Fig. 1, we show the average (over many network snap-shots) ratio of the self-interference to the coherent gain fordifferent numbers of antennas per AP, N = { , , , } . Inthis simulation, we adopt the following settings: M = 200 APs, K = 40 users, τ u , p = τ d , p = 20 pilots and maximal-ratio Fig. 1. Average self-interference to coherent gain ratio in dB, for differentnumbers of antennas per AP. In this simulation: M = 200 , K = 40 , τ u , p = τ d , p = 20 and maximal-ratio power control. power control. The results in Fig. 1 demonstrate the outstand-ing ability of ECB to boost the channel hardening comparedto NCB and CB, reducing the normalized beamforming gainuncertainty by at least 5 dB and 10 dB, respectively. Inaddition, ECB is able to provide almost the same amountof self-interference to coherent gain ratio as CB-DT (the gapreduces as the number of antennas per AP increases), whichtells us that the CSI acquisition at the user, although reducingthe uncertainty about the channel, does not provide an addedvalue in terms of performance.Certainly, the variance of the downlink channel estimationerror depends on the level of downlink pilot contaminationwhich in turn depends on the number of orthogonal downlinkpilots. The self-interference for CB-DT is minimized if each NTERDONATO et al. : ENHANCED NORMALIZED CONJUGATE BEAMFORMING FOR CELL-FREE MASSIVE MIMO 11
Fig. 2. CDF of the self-interference to coherent gain ratio in dB, for N = 8 .The settings are identical to those in Fig. 1. user is assigned a unique orthogonal downlink pilot (nopilot reuse). Hence, it is interesting to look at how far theperformance of ECB is from that ideal case. In Fig. 2 weshow the CDF of the self-interference to coherent gain ratiofor N = 8 whose mean value has been shown in Fig. 1. Inthese results, we include the CB-DT with τ d , p = K , i.e., nodownlink pilot reuse. Compared to this ideal case, ECB onlyloses uniformly around 3 dB. However, this additional pilotoverhead significantly reduces the SE of CB-DT, as shownin (38).Another important aspect to look at is how the consideredprecoding schemes tackle the inter-user interference and howmuch coherent gain they achieve. None of the consideredschemes provides interference suppression by nature, and theamount of interference in the network remains essentially thesame regardless of the variant of CB that is adopted. In fact,the transmit power of any AP is, in any case, equal to ρ d whichis ensured by performing the maximal-ratio power controlscheme described by equations (58)-(60).From the same set of simulations used so far, we now showin Fig. 3 the power of the inter-user interference as shareof the coherent gain, for different setups: N = { , , , } .From Fig. 3 we observe that the inter-user interference tocoherent gain ratio decreases with the number of antennas perAP thanks to an increasing coherent gain. Interestingly, ECBperforms poorly when N = 2 . By substituting equations (58)-(60) into the expressions of the coherent gain in Table III,we indeed observe that the coherent gain is proportional to N and N − for CB and ECB, respectively. If N = 2 , then thecoherent gain of ECB is half the coherent gain of CB. This 3dB loss with respect to CB can be observed in Fig. 3. Clearly,this gap vanishes as N grows.Importantly, Fig. 3 tells us that the precoding normalizationhas a significant impact only on the self-interference and inabsence of CSI at the users. Hence, we can argue that anyprecoding normalization along with downlink training wouldyield negligible benefits compared to CB-DT. We will use thisimportant consideration to draw general conclusions about theusefulness of the downlink training. Fig. 3. Average inter-user interference to coherent gain ratio in dB, fordifferent numbers of antennas per AP. The simulation settings are identical tothose in Fig. 1.
In Fig. 4, we present the CDF of the achievable SE forthe considered precoding schemes. The simulation settingsare the same used so far, and we consider N = 8 andmaximal-ratio power control. Fig. 4a shows the gross SE,namely (16) and (38) without the pre-log factor capturingthe pilot estimation overhead. By doing so, we want toemphasize how ECB uniformly performs tightly close to CB-DT , regardless of the pilot overhead. Fig. 4b shows how theadditional pilot overhead negatively affects the SE, makingECB the most desirable precoding scheme.ECB outperforms NCB, especially at high percentiles butthis is due to the opportunistic nature of the maximal-ratiopower control which prioritizes the users with stronger chan-nel. In fact, if we perform MMF power control as describedin Section V a small gain can be observed for the minimum SEper user (see Fig. 5a). The reason why this gain is relativelysmall is intuitive: the power control coefficients { η mk } are partof the effective downlink channel gain, thus when optimized,they act on the normalization in different ways (whereas inmaximal-ratio power control they are proportional to γ mk )in order to maximize the minimum SE. Hence, MMF powercontrol tends to reduce the differences between NCB andECB. An additional reason why there is very little differencein SE at low percentiles between the different methods isthat the inter-user interference to coherent gain ratio, whichis basically the same for all the schemes for N = 8 , isdominant over the self-interference to coherent gain ratio, asshown in Fig. 1 and Fig. 3, hence the gain provided by ECB(and CB-DT) in terms of self-interference becomes negligible.CB-DT benefits from optimal power control for an additionalaspect: downlink pilots are beamformed and power-controlledby the same power control coefficients { η CB mk } used for the datatransmission. Hence, MMF power control adjusts the powerlevels of the downlink pilot transmissions to reduce the pilot Strictly speaking, CB-DT is not an “upper-bound”, but constitutes a real-istic performance benchmark considering distributed conjugate beamformingschemes relying only on local channel estimates. For instance, the modifiedCB proposed in [15] can achieve signal-to-interference ratio (SIR) values closeto a genie-aided receiver, but requires CSI exchange among the APs. (a) CDF of the achievable gross SE per user (b) CDF of the achievable (net) SE per userFig. 4. SE with maximal-ratio power control. The simulation settings areidentical to those in Fig. 1. Here, we consider N = 8 . (a) CDF of the achievable min SE per user with τ c = 200 . (b) CDF of the achievable min SE per user with τ c = 100 .Fig. 5. SE with MMF power control. Here M = 100 APs, K = 20 users, τ u , p = τ d , p = 10 pilots and D = 250 m. Fig. 6. Mean SE as the number of antennas per AP varies. The simulationsettings are identical to those in Fig. 4.Fig. 7. Mean SE as the number of APs varies. The simulation settings areidentical to those in Fig. 4. contamination and achieve the target SINR. However, as wecan see from Fig. 5a, this is not sufficient to outperform ECB,which is the preferable scheme.The settings adopted for the simulations in Fig. 5a consistof: M = 100 APs equipped with N = 8 antennas, K = 20 users, τ u , p = τ d , p = 10 pilots, τ c = 200 and D = 250 m. Being quite sensitive to the pilot overhead, CB-DT issignificantly affected by the length of the coherence block. Ifwe shrink the coherence block to τ c = 100 , then the resultingmin SE of CB-DT, shown in Fig. 5b, degrades more rapidlycompared to the other downlink-training-free schemes.Finally, we investigate how the mean SE varies with thenumber of antennas per APs (Fig. 6) respectively the numberof APs (Fig. 7). Increasing the number of antennas per APalways boosts the channel hardening [12], and this makesdownlink training unnecessary. In fact, Fig. 6 shows that CB-DT is preferable for very small values of N , while ECBbecomes the best scheme with N ≥ , and the gap betweenECB and CB-DT increases with N . Importantly, the abilityof NCB to help the effective downlink channel to harden isinferior compared to ECB, and NCB performs at most equallyas CB-DT. Increasing the number of APs, M , is not as crucialas increasing N for the channel to harden [12]. Fig. 7 shows NTERDONATO et al. : ENHANCED NORMALIZED CONJUGATE BEAMFORMING FOR CELL-FREE MASSIVE MIMO 13 that ECB uniformly outperforms all the other schemes and theperformance improvement increases with M , although slowerthan it would increase with N . Interestingly, NCB does notoutperform CB-DT despite N = 8 and the additional overheadthat penalizes CB-DT.In summary, ECB offers a better support for the channelto harden than NCB and its excellent performance makesdownlink training unnecessary in the considered scenarios.VII. C ONCLUSION
In this work, we have studied a variant of the conjugatebeamforming scheme, dubbed enhanced normalized conjugatebeamforming (ECB), for cell-free massive MIMO systemswith multi-antenna APs. The ECB precoding vector consistsof the conjugate of the channel estimate normalized by itssquared norm. This normalization term is the core of ourcontribution and leads to many benefits. Firstly, this precodingnormalization helps, more than any other normalization pro-posed in the literature, to achieve channel hardening, i.e., tomake the effective downlink channel gain nearly deterministic.This, in turn, makes data decoding methods based on thechannel statistics more reliable. Secondly, we have demon-strated that ECB with statistical CSI knowledge at the userscan provide better downlink spectral efficiency than conjugatebeamforming with pilot-based downlink training, even withrelatively small numbers of APs and antennas per AP. Sincethe precoding normalization significantly affects only the self-interference due to the user’s channel uncertainty, then it wouldnot appreciably increase the performance in the presence ofdownlink training. We can thereby conclude that ECB mightrender the downlink pilots unnecessary from a performanceviewpoint. This conclusion does not contrast with our previouswork [11] but rather closes the loop. In fact, the performancegap between the conventional conjugate beamforming withand without downlink training is considerable, even withmulti-antenna APs, but can be filled up by the precodingnormalization herein proposed.A
PPENDIX
A. Proof of (28)Next, we include the computation in closed form of theexpectations needed to derive the SINR expression in (28). Let g mk = ˆ g mk + ˜ g mk ∈ C N and ˆ g mk be independent of ˜ g mk .Moreover, g mk ∼ CN ( , β mk I N ) , ˆ g mk ∼ CN ( , γ mk I N ) and ˜ g mk ∼ CN ( , ( β mk − γ mk ) I N ) . It holds that, E ( g T mk ˆ g ∗ mk k ˆ g mk k ) = E ( E ( ˆ g T mk ˆ g ∗ mk k ˆ g mk k + ˜ g T mk ˆ g ∗ mk k ˆ g mk k (cid:12)(cid:12)(cid:12)(cid:12) ˆ g mk )) = 1 , (62) E ( | g T mk ˆ g ∗ mk | k ˆ g mk k ) = E ( | (ˆ g mk + ˜ g mk ) T ˆ g ∗ mk | k ˆ g mk k ) = 1 + E n | ˜ g T mk ˆ g ∗ mk | / k ˆ g mk k o = 1 + E ( k ˆ g mk k ˆ g T mk E n ˜ g ∗ mk ˜ g T mk (cid:12)(cid:12)(cid:12) ˆ g mk o ˆ g ∗ mk ) = 1 + 1 N − (cid:18) β mk γ mk − (cid:19) , (63) where in the last step we have used that E n / k ˆ g mk k o =1 / (( N − γ mk ) . Consider two different users identified bythe indices k and j , j = k . It holds that, E ( | g T mk ˆ g ∗ mj | k ˆ g mj k ) = β mk β mj E ( | g T mk ˆ g ∗ mk | k ˆ g mk k ) , if ϕ k = ϕ j ,β mk ( N − γ mj , otherwise, = β mk β mj N − N − β mk ( N − γ mj , if ϕ k = ϕ j ,β mk ( N − γ mj , otherwise,(64) E ( g T mk ˆ g ∗ mj k ˆ g mj k ) = β mk β mj , if ϕ k = ϕ j , , otherwise. (65)In these equalities, if ϕ k = ϕ j , then we exploit that g mk is independent of ˆ g mj , else if ϕ k = ϕ j we exploit therelationships among contaminated channel estimates and theirmean-squares in (7) and (8), respectively. By using the resultsabove, we can compute in closed form DS k = √ ρ d M X m =1 √ η mk E ( g T mk ˆ g ∗ mk k ˆ g mk k ) = √ ρ d M X m =1 √ η mk , (66) E n | BU k | o = ρ d M X m =1 η mk E ( | g T mk ˆ g ∗ mk | k ˆ g mk k ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ( g T mk ˆ g ∗ mk k ˆ g mk k )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ρ d N − M X m =1 η mk (cid:18) β mk γ mk − (cid:19) , (67) E (cid:8) | UI kj | (cid:9) = ρ d M X m =1 η mj E ( | g T mk ˆ g ∗ mj | k ˆ g mj k ) + ρ d M X m =1 M X n = m √ η mj η nj E ( g T mk ˆ g ∗ mj k ˆ g mj k ( g T nk ˆ g ∗ nj ) ∗ k ˆ g nj k ) = ρ d N − M X m =1 η mj β mk γ mj + ρ d N − N − M X m =1 η mj β mk β mj | ϕ H k ϕ j | + ρ d M X m =1 M X n = m √ η mj η nj β mk β nk β mj β nj | ϕ H k ϕ j | , (68)where in the last equality we exploit the independence ofchannel responses and channel estimates of different APs( n = m ), and the fact that the term ρ d N − M X m =1 η mj β mk γ mj appears in both cases whether ϕ k = ϕ j or not, as shownin (64). Hence, this term does not depend on | ϕ H k ϕ j | . Byinserting the results in (66)–(68) into (17) we obtain (28). B. Proof of (38)In this section, we include a proof for the closed-formexpression of the achievable downlink rate in (38). Thisconsists in showing how both (37) and (40) are obtained. TheMMSE downlink channel estimate is given by [11] ˆ a kk = E { a kk } + Cov { a kk , ˇ y dp ,k } Var { ˇ y dp ,k } (ˇ y dp ,k − E { ˇ y dp ,k } ) , (69)and its variance is κ k = Var { ˆ a kk } = | Cov { a kk , ˇ y dp ,k } | Var { ˇ y dp ,k } . (70)By following the same approach as in [11], we have Cov { a kk , ˇ y dp ,k } = N √ τ d , p ρ d , p M X m =1 η mk β mk γ mk , (71) Var { ˇ y dp ,k } = N τ d , p ρ d , p K X j =1 M X m =1 η mj β mk γ mj | ψ H k ψ j | +1 , (72)where equations (71), (72) hold when imposing (34). Byinserting equations (71), (72) into (70), we obtain (37). Themean-square of the downlink channel estimate is given by E (cid:8) | ˆ a kk | (cid:9) = Var { ˆ a kk } + | E { ˆ a kk } | = κ k + N M X m =1 √ η mk γ mk ! , (73)where E { ˆ a kk } = E { a kk } = N M X m =1 √ η mk γ mk , as the MMSE estimator is unbiased under the regularityassumptions . The mean-square of the downlink channel es-timation error is given by E (cid:8) | ˜ a kk | (cid:9) = E (cid:8) | a kk − ˆ a kk | (cid:9) = E (cid:8) | a kk | (cid:9) + E (cid:8) | ˆ a kk | (cid:9) − E { a ∗ kk ˆ a kk } ) ( a ) = E (cid:8) | a kk | (cid:9) − E (cid:8) | ˆ a kk | (cid:9) ( b ) = N ς kk − κ k , (74)where ( a ) results from E { a ∗ kk ˆ a kk } = | E { a kk } | + Cov { a kk , ˇ y dp ,k } Var { ˇ y dp ,k }× ( E { a ∗ kk ˇ y dp ,k } − E { a kk } ∗ E { ˇ y dp ,k } )= | E { a kk } | + | Cov { a kk , ˇ y dp ,k } | Var { ˇ y dp ,k } = E (cid:8) | ˆ a kk | (cid:9) , (75)while ( b ) follows from (73) and the fact that E (cid:8) | a kk | (cid:9) = N M X m =1 η mk β mk γ mk + N M X m =1 √ η mk γ mk ! . (76) Finally, we focus on deriving E (cid:8) | a kj | (cid:9) , j = k , in closedform. Under the same assumptions on the channel vectorsconsidered in Appendix A , it holds that E (cid:8) | g T mk ˆ g ∗ mk | (cid:9) = E (cid:8) | (ˆ g mk + ˜ g mk ) T ˆ g ∗ mk | (cid:9) = E (cid:8) | ˆ g T mk ˆ g ∗ mk | (cid:9) + E n ˆ g T mk E n ˜ g ∗ mk ˜ g T mk (cid:12)(cid:12)(cid:12) ˆ g mk o ˆ g ∗ mk o = ( N + 1) N γ mk + N ( β mk − γ mk ) γ mk = N γ mk + N β mk γ mk , (77) E (cid:8) | g T mk ˆ g ∗ mj | (cid:9) = β mj β mk E (cid:8) | g T mk ˆ g ∗ mk | (cid:9) , if ϕ k = ϕ j ,N β mk γ mj , otherwise = N γ mk β mj β mk + N β mk γ mj , if ϕ k = ϕ j ,N β mk γ mj , otherwise = N γ mj β mk β mj | ϕ H k ϕ j | + N β mk γ mj , (78) E (cid:8) g T mk ˆ g ∗ mj ( g T nk ˆ g ∗ nj ) ∗ (cid:9) = E (cid:8) g T mk ˆ g ∗ mj (cid:9) E (cid:8) g T nk ˆ g ∗ nj (cid:9) = N β mk β mj β nk β nj γ mj γ nj , if ϕ k = ϕ j , , otherwise = N β mk β mj β nk β nj γ mj γ nj | ϕ H k ϕ j | . (79)In these equalities, if ϕ k = ϕ j , then we exploit that g mk is independent of ˆ g mj , else if ϕ k = ϕ j we exploit therelationships among contaminated channel estimates and theirmean-squares in (7) and (8), respectively. Moreover, in (79),we exploit the independence of channel responses and channelestimates of different APs ( n = m ). By using the results inequations (78), (79), we can compute in closed form E (cid:8) | a kj | (cid:9) = M X m =1 η mj E (cid:8) | g T mk ˆ g ∗ mj | (cid:9) + M X m =1 M X n = m √ η mj η nj E (cid:8) g T mk ˆ g ∗ mj ( g T nk ˆ g ∗ nj ) ∗ (cid:9) = N M X m =1 η mj β mk γ mj + N M X m =1 √ η mj γ mj β mk β mj ! | ϕ H k ϕ j | . (80)By inserting (73), (74) and (80) into (39), we finally ob-tain (40) and in turn (38).R EFERENCES[1] G. Interdonato, H. Q. Ngo, E. G. Larsson, and P. Frenger, “On the perfor-mance of cell-free massive MIMO with short-term power constraints,”in
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Giovanni Interdonato (Member, IEEE) received theM.Sc. degree in computer and telecommunicationsystems engineering from the University Mediter-annea of Reggio Calabria, Italy, in 2015, and thePh.D. degree in electrical engineering with special-ization in communication systems from Link¨opingUniversity (LiU), Sweden, in 2020. From October2015 to October 2018, he was researcher at theradio network department at Ericsson Research inLink¨oping, and a Marie Sklodowska-Curie researchfellow of the H2020 ITN project . Heis currently a postdoctoral researcher at the Department of Electrical andInformation Engineering, University of Cassino and Southern Lazio, Italy. Hismain research interests include distributed (cell-free) Massive MIMO systems,and 5G New Radio communication protocols.He has filed about twenty Massive MIMO related patent applications, andreceived a scholarship from the Ericsson Research Foundation in 2019.
Hien Quoc Ngo (Senior Member, IEEE) receivedthe B.S. degree in electrical engineering from the HoChi Minh City University of Technology, Vietnam,in 2007, the M.S. degree in electronics and radioengineering from Kyung Hee University, South Ko-rea, in 2010, and the Ph.D. degree in communicationsystems from Link¨oping University (LiU), Sweden,in 2015. In 2014, he visited the Nokia Bell Labs,Murray Hill, New Jersey, USA. From January 2016to April 2017, Hien Quoc Ngo was a VR researcherat the Department of Electrical Engineering (ISY),LiU. He was also a Visiting Research Fellow at the School of Electronics,Electrical Engineering and Computer Science, Queen’s University Belfast,UK, funded by the Swedish Research Council.Hien Quoc Ngo is currently a Lecturer at Queen’s University Belfast, UK.His main research interests include massive (large-scale) MIMO systems, cell-free massive MIMO, physical layer security, and cooperative communications.He has co-authored many research papers in wireless communications and co-authored the Cambridge University Press textbook
Fundamentals of MassiveMIMO (2016).Dr. Hien Quoc Ngo received the IEEE ComSoc Stephen O. Rice Prize inCommunications Theory in 2015, the IEEE ComSoc Leonard G. AbrahamPrize in 2017, and the Best PhD Award from EURASIP in 2018. He alsoreceived the IEEE Sweden VT-COM-IT Joint Chapter Best Student JournalPaper Award in 2015. He was an
IEEE Communications Letters exemplaryreviewer for 2014, an
IEEE Transactions on Communications exemplaryreviewer for 2015, and an
IEEE Wireless Communications Letters exemplaryreviewer for 2016. He was awarded the UKRI Future Leaders Fellowship in2019. Dr. Hien Quoc Ngo currently serves as an Editor for the IEEE Transac-tions on Wireless Communications, IEEE Wireless Communications Letters,Digital Signal Processing, Elsevier Physical Communication (PHYCOM), andIEICE Transactions on Fundamentals of Electronics, Communications andComputer Sciences. He was a Guest Editor of IET Communications, specialissue on “Recent Advances on 5G Communications” and a Guest Editor ofIEEE Access, special issue on “Modelling, Analysis, and Design of 5G Ultra-Dense Networks”, in 2017. He has been a member of Technical ProgramCommittees for several IEEE conferences such as ICC, GLOBECOM, WCNC,and VTC.
Erik G. Larsson (Fellow, IEEE) received the Ph.D.degree from Uppsala University, Uppsala, Sweden,in 2002. He is currently Professor of Communi-cation Systems at Link¨oping University (LiU) inLink¨oping, Sweden. He was with the KTH RoyalInstitute of Technology in Stockholm, Sweden, theGeorge Washington University, USA, the Universityof Florida, USA, and Ericsson Research, Sweden.His main professional interests are within the areasof wireless communications and signal processing.He co-authored
Space-Time Block Coding for Wire-less Communications (Cambridge University Press, 2003) and
Fundamentalsof Massive MIMO (Cambridge University Press, 2016).Currently he is a member of the
IEEE Transactions on Wireless Com-munications steering committee. He served as chair of the IEEE SignalProcessing Society SPCOM technical committee (2015–2016), chair of the
IEEE Wireless Communications Letters steering committee (2014–2015),General and Technical Chair of the Asilomar SSC conference (2015, 2012),technical co-chair of the IEEE Communication Theory Workshop (2019),and member of the IEEE Signal Processing Society Awards Board (2017–2019). He was Associate Editor for, among others, the
IEEE Transactions onCommunications (2010-2014), the
IEEE Transactions on Signal Processing (2006-2010), and the