Massive MIMO under Double Scattering Channels: Power Minimization and Congestion Controls
Trinh Van Chien, Hien Quoc Ngo, Symeon Chatzinotas, Björn Ottersten, Merouane Debbah
MMassive MIMO under Double Scattering Channels: PowerMinimization and Congestion Controls
Trinh Van Chien ∗ , Hien Quoc Ngo † , Symeon Chatzinotas ∗ , Bj¨orn Ottersten ∗ , and Merouane Debbah ξ ∗ Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Luxembourg † School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, UK ξ CentraleSup´elec, Universit´e Paris-Saclay & Lagrange Mathematical and Computing Research Center, Paris, France
Abstract —This paper considers a massive MIMO systemunder the double scattering channels. We derive a closed-form expression of the uplink ergodic spectral efficiency (SE)by exploiting the maximum-ratio combining technique withimperfect channel state information. We then formulate andsolve a total uplink data power optimization problem that aimsat simultaneously satisfying the required SEs from all the userswith limited power resources. We further propose algorithmsto cope with the congestion issue appearing when at least oneuser is served by lower SE than requested. Numerical resultsillustrate the effectiveness of our proposed power optimization.More importantly, our proposed congestion-handling algorithmscan guarantee the required SEs to many users under congestion,even when the SE requirement is high.
I. I
NTRODUCTION
Wireless communications has sustained an exponentialdemand growth in data throughput over the last decades. How-ever, mobile traffic will increase as foreseen in a short timewith . billion devices by [1]. To handle this, massiveMIMO, a disruptive technology, does not only inherit all themultiplexing and diversity gains of the conventional MIMObut also offers large degree-of-freedoms as a consequenceof equipping base stations (BSs) with many antennas [2].Massive MIMO, therefore, provides unprecedented spectraland energy efficiency gains of modern wireless networks withonly utilizing the contemporary time and frequency resources.In massive MIMO, the closed-form SE expression can beobtained in certain scenarios such as rich scattering envi-ronments modeled by uncorrelated Rayleigh fading [3] andreferences therein. Nonetheless, practical channels usuallyinvolve spatial correlation, which is modeled by correlatedRayleigh fading when the gathered energy at the antenna arraycomes from many directions likely leading to the full ranksof covariance matrices with an overwhelming probability[4]–[6]. For rank deficiency occurring in poor scatteringconditions, the authors in [7] proposed the double scatteringchannel to characterize by the structure of scattering in thepropagation environment and the spatial correlations aroundthe transceiver. The first work numerically studying the uplinkergodic SE of cellular massive MIMO systems with thedouble scattering channels was found in [8]. For theoreticalanalysis, the authors in [9], [10] computed the asymptotic The work of T. V. Chien, S. Chatzinotas, and B. Ottersten was supportedby FNR, Luxembourg under the COREproject C16/IS/11306457/ELECTIC(Energy and CompLexity EffiCienT mIllimeter-waveLarge-Array Communi-cations). The work of H. Q. Ngo was supported by the UK Research andInnovation Future Leaders Fellowships under Grant MR/S017666/1. ergodic SE of a single-cell massive MIMO system with thedifferent linear precoding techniques when the number of BSantennas, scatterers, and users grow large with the same rate.To the best of our knowledge, no prior work analyzes theperformance of massive MIMO systems with a finite numberof BS antennas, users, and scatterers.Many resource allocation tasks in massive MIMO com-munications can be implemented on the large-scale fadingtime scale [11]. Notice that the key component of massiveMIMO communications is allowing many users to access andshare the radio resource at the same time with high qualityof service. The max-min fairness optimization is promisingto provide uniform service to all the users [2]. However, forlarge-scale networks with many base stations and users, thefairness level will approach a zero rate. In contrast, one caninclude separate SE constraints in the optimization problemsto simultaneously maintain the quality of service for all theusers [12]. Since users were randomly distributed, many userlocations with poor channel conditions leads the optimizationproblems to be infeasible [13].By exploiting the double scattering channel model, thispaper considers a massive MIMO system where a set oforthogonal pilot signals are reused by all the users. A newuplink ergodic SE expression is derived in closed form fora finite number of antennas at each base station (BS) anddifferent number of scatterers observed by every user andBS. After that, we formulate and solve a total uplink dataenergy minimization problem subject to the required SE fromevery user and the power constraints. For user locations andshadow fading realizations, where our optimization problemis feasible, the global optimum can be obtained in polynomialtime. We further propose two low computational complexityiterative algorithms that tackle the infeasible optimizationproblem by relaxing the SE constraints of unsatisfied users.Numerical results manifest the closed-form SE expressionoverlapping Monte-Carlo simulations. The effectiveness ofthe proposed data power control algorithms are comparedwith the interior-point methods.
Notation : Upper-case/lower-case bold face letters are usedto denote matrices and vectors, respectively. I M is the identitymatrix of size M × M . E {·} denotes the expectation of arandom variable. (cid:107) · (cid:107) is Euclidean norm. tr( · ) is the trace ofa matrix. The regular and Hermitian transposes are denotedby ( · ) T and ( · ) H , respectively. Finally, CN ( · , · ) denotes thecircularly symmetric complex Gaussian distribution. a r X i v : . [ c s . I T ] F e b I. M
ASSIVE
MIMO
WITH D OUBLE S CATTERING C HANNELS
We consider an uplink massive MIMO system comprising L cells, where cell l has one BS equipped with M antennasserving K single-antenna users. A quasi-static channel modelis used, where the time-frequency plane is divided intocoherence blocks. Each coherence block has τ c symbols forwhich the τ p symbols are dedicated to the pilot training phaseand the remaining τ c − τ p symbols are used for the uplink datatransmission (the downlink data transmission is neglected).The channel between user k in cell l and BS l (cid:48) is modeledby [8], which is h l (cid:48) lk = (cid:113) β l (cid:48) lk /S l (cid:48) lk (cid:16) R l (cid:48) lk (cid:17) / G l (cid:48) lk (cid:16) (cid:101) R l (cid:48) lk (cid:17) / g l (cid:48) lk , (1)where β l (cid:48) lk is the large-scale fading coefficient. S l (cid:48) lk is thenumber of scatterers generating the channel between BS l (cid:48) and user k in cell l . The matrix R l (cid:48) lk ∈ C M × M representsthe correlation between the BS antennas and its scatterers; G l (cid:48) lk ∈ C M × S l (cid:48) lk includes the corresponding small-scale fadingcoefficients. The matrix (cid:101) R l (cid:48) lk ∈ C S l (cid:48) lk × S l (cid:48) lk is the correlationmatrix between the transmit and receive scatterers and g l (cid:48) lk ∈ C S l (cid:48) lk represents the corresponding small-scale fading coeffi-cients. The elements of both G l (cid:48) lk and g l (cid:48) lk are independent andidentically distributed as CN (0 , conditioned on the traceof the covariance matrices. A. Uplink Pilot Training
In each coherence block, each BS needs instantaneouschannel state information for the uplink data detection. The τ p symbols are dedicated to the uplink pilot training, whichcan create τ p mutually orthogonal pilot signals. User k incell l uses the deterministic pilot signal φφφ lk ∈ C τ p with (cid:107) φφφ lk (cid:107) = τ p . This pilot signal is also reused by otherusers in multiple cells and we can define the pilot reuse setas P lk = { ( l (cid:48) , k (cid:48) ) : φφφ l (cid:48) k (cid:48) = φφφ lk , l = 1 , . . . , L, k (cid:48) = 1 , . . . , K } ,which contains the indices of all users sharing the same pilotsignal as user k in cell l , including ( l, k ) . Mathematically,it observes that φφφ H lk φφφ l (cid:48) k (cid:48) = τ p if ( l (cid:48) , k (cid:48) ) ∈ P lk . Otherwise, φφφ H lk φφφ l (cid:48) k (cid:48) = 0 . At BS l , the received pilot signal Y pl ∈ C M × τ p with the superscript p standing for the pilot training phase is Y pl = L (cid:88) l (cid:48) =1 K (cid:88) k (cid:48) =1 (cid:112) ˆ p l (cid:48) k (cid:48) h ll (cid:48) k (cid:48) φφφ H l (cid:48) k (cid:48) + N pl , (2)where N pl ∈ C M × τ p is additive noise with the independentand identically random elements distributed as CN (0 , σ ) .BS l estimates the channel h ll (cid:48) k (cid:48) from user k (cid:48) in cell l (cid:48) from y l,pl (cid:48) k (cid:48) = Y pl φφφ l (cid:48) k (cid:48) = (cid:88) ( l (cid:48)(cid:48) ,k (cid:48)(cid:48) ) ∈P l (cid:48) k (cid:48) (cid:112) ˆ p l (cid:48)(cid:48) k (cid:48)(cid:48) τ p h ll (cid:48)(cid:48) k (cid:48)(cid:48) + N pl φφφ l (cid:48) k (cid:48) . (3)The processed received signal y l,pl (cid:48) k (cid:48) ∈ C M has sufficientstatistics to obtain a channel estimate of the origin h ll (cid:48) k (cid:48) byutilizing linear MMSE (LMMSE). This channel model was initiated for conventional MIMO systems undera far-field region and dedicated sub -GHz bands for mobile services. Inmassive MIMO communications, the far-field effects are still observed sincemany antenna components can be installed in a small compact array [14]. Lemma 1.
By utilizing the LMMSE estimation, the channelestimate ˆ h ll (cid:48) k (cid:48) ∈ C M from user k (cid:48) in cell l (cid:48) and BS l is ˆ h ll (cid:48) k (cid:48) = (cid:112) ˆ p l (cid:48) k (cid:48) β ll (cid:48) k (cid:48) d ll (cid:48) k (cid:48) R ll (cid:48) k (cid:48) ΨΨΨ ll (cid:48) k (cid:48) y l,pl (cid:48) k (cid:48) , (4) where ΨΨΨ ll (cid:48) k (cid:48) = (cid:16)(cid:80) ( l (cid:48)(cid:48) ,k (cid:48)(cid:48) ) ∈P l (cid:48) k (cid:48) a ll (cid:48)(cid:48) k (cid:48)(cid:48) R ll (cid:48)(cid:48) k (cid:48)(cid:48) + σ I M (cid:17) − ,with a ll (cid:48)(cid:48) k (cid:48)(cid:48) = τ p ˆ p l (cid:48)(cid:48) k (cid:48)(cid:48) β ll (cid:48)(cid:48) k (cid:48)(cid:48) d ll (cid:48)(cid:48) k (cid:48)(cid:48) and d ll (cid:48) k (cid:48) = tr (cid:0) (cid:101) R ll (cid:48) k (cid:48) (cid:1) /S ll (cid:48) k (cid:48) .The covariance matrix of the channel estimate ˆ h ll (cid:48) k (cid:48) is E (cid:110) ˆ h ll (cid:48) k (cid:48) (cid:0) ˆ h ll (cid:48) k (cid:48) (cid:1) H (cid:111) = ˆ p l (cid:48) k (cid:48) (cid:0) β ll (cid:48) k (cid:48) (cid:1) (cid:0) d ll (cid:48) k (cid:48) (cid:1) τ p R ll (cid:48) k (cid:48) ΨΨΨ ll (cid:48) k (cid:48) R ll (cid:48) k (cid:48) . (5) Proof.
The proof is based on the LMMSE estimation [15], butadapted to our framework with the channel vector in (1) andthe pilot reuse pattern with non-Gaussian random variables.The detail proof is omitted due to space limitations.Lemma 1 shows the concrete expression of the channelestimate of each user. Our channel estimation considers theinfluence of pilot contamination in multi-cell massive MIMOscenarios, which is a generalization of the previous result in[9], [10] that assumed the orthogonal pilot signals for all theusers in a single cell.
B. Uplink Data Transmission
During the uplink data transmission, user k in cell l sendsdata symbol s lk with E {| s lk | } = 1 . The received data signal y l ∈ C M at BS l is y l = L (cid:88) l (cid:48) =1 K (cid:88) k (cid:48) =1 √ p l (cid:48) k (cid:48) h ll (cid:48) k (cid:48) s l (cid:48) k (cid:48) + n l , (6)where p l (cid:48) k (cid:48) is the transmit power of user k (cid:48) in cell l (cid:48) assignedto each data symbol and n l is additive noise distributed as CN ( , σ I M ) . By utilizing a combining vector v lk ∈ C M and the use-and-then-forget channel capacity bounding tech-nique as shown in [2], the uplink ergodic SE is obtained as R lk = (1 − τ p /τ c ) log (1 + SINR lk ) , [ b/s/Hz ] , (7)where the effective signal-to-interference-and-noise ratio(SINR) is given by (8), shown at the top of the next page.The expectations in (8) are computed over all the sources ofrandomness and (7) is an achievable rate since it is a lowerbound on the channel capacity. This achievable rate can becomputed numerically for any combining scheme, but witha high cost since many instantaneous channels need to begathered such that several expectations can be numericallyestimated. C. Uplink Spectral Efficiency Analysis
If maximum ratio (MR) combining is used by each BS,i.e., (cid:0) v lk = ˆ h llk (cid:1) , ∀ l, k , we obtain the closed-form expressionfor the uplink SE in (7) as shown by Theorem 1. Theorem 1.
When BS l uses the MR combing to decode thedesired signal from user k in cell l , the uplink SE obtainedin (7) with the closed-form expression of the SINR value SINR lk = p lk z llk (cid:12)(cid:12)(cid:12) tr (cid:0) R llk ΨΨΨ llk R llk (cid:1)(cid:12)(cid:12)(cid:12) NI lk + CI lk + NO lk , (9) INR lk = p lk (cid:12)(cid:12) E (cid:8) v Hlk h llk (cid:9)(cid:12)(cid:12) (cid:80) Ll (cid:48) =1 (cid:80) Kk (cid:48) =1 p l (cid:48) k (cid:48) E (cid:8)(cid:12)(cid:12) v Hlk h ll (cid:48) k (cid:48) (cid:12)(cid:12) (cid:9) − p lk (cid:12)(cid:12) E (cid:8) v Hlk h llk (cid:9)(cid:12)(cid:12) + σ E {(cid:107) v lk (cid:107) } . (8) where NI lk , CI lk , and NO lk are respectively the non-coherentinterference, coherent interference, and noise, which are NI lk = L (cid:88) l (cid:48) =1 K (cid:88) k (cid:48) =1 p l (cid:48) k (cid:48) m ll (cid:48) k (cid:48) tr (cid:0) R llk ΨΨΨ llk R llk R ll (cid:48) k (cid:48) (cid:1) , (10) CI lk = (cid:88) ( l (cid:48) ,k (cid:48) ) ∈P lk \ ( l,k ) p l (cid:48) k (cid:48) z ll (cid:48) k (cid:48) (cid:12)(cid:12)(cid:12) tr (cid:0) R ll (cid:48) k (cid:48) ΨΨΨ llk R llk (cid:1)(cid:12)(cid:12)(cid:12) + (cid:88) ( l (cid:48) ,k (cid:48) ) ∈P lk p l (cid:48) k (cid:48) z ll (cid:48) k (cid:48) tr (cid:16)(cid:0) (cid:101) R ll (cid:48) k (cid:48) (cid:1) (cid:17)(cid:0) d ll (cid:48) k (cid:48) S ll (cid:48) k (cid:48) (cid:1) (cid:12)(cid:12)(cid:12) tr (cid:0) R ll (cid:48) k (cid:48) ΨΨΨ llk R llk (cid:1)(cid:12)(cid:12)(cid:12) + (cid:88) ( l (cid:48) ,k (cid:48) ) ∈P lk p l (cid:48) k (cid:48) z ll (cid:48) k (cid:48) tr (cid:16)(cid:0) (cid:101) R ll (cid:48) k (cid:48) (cid:1) (cid:17)(cid:0) S ll (cid:48) k (cid:48) (cid:1) × tr (cid:0) R ll (cid:48) k (cid:48) ΨΨΨ llk R llk R ll (cid:48) k (cid:48) R llk ΨΨΨ llk (cid:1) , (11) NO lk = σ ˆ p lk (cid:0) β llk (cid:1) (cid:0) d llk (cid:1) τ p tr (cid:0) R llk ΨΨΨ llk R llk (cid:1) , (12) with m ll (cid:48) k (cid:48) = β ll (cid:48) k (cid:48) d ll (cid:48) k (cid:48) ˆ p lk (cid:0) β llk (cid:1) (cid:0) d llk (cid:1) τ p , and z ll (cid:48) k (cid:48) =ˆ p l (cid:48) k (cid:48) (cid:0) β ll (cid:48) k (cid:48) (cid:1) (cid:0) d ll (cid:48) k (cid:48) (cid:1) ˆ p lk (cid:0) β llk (cid:1) (cid:0) d llk (cid:1) τ p , ∀ l (cid:48) , k (cid:48) , l. Proof.
The proof is obtained by computing the expectationsof non-Gaussian random variables in (8). The detailed proofis omitted due to space limitations.The SINR expression (9) is explicitly influenced by manyfactors such as channel covariance matrices, the number ofscatterers, pilot reuse, channel estimation quality, which arehidden in the general formulation (8). Specifically, the numer-ator of (9) shows the contribution of both channel estimationquality and covariance matrix of user k in cell l . Moreover, theeffectiveness of the array gain is verified since the numeratorscales up with the number of antennas. The first part in the de-nominator of (9) demonstrates the degradation of the receivedsignal quality due to non-coherent interference. The secondpart presents the contributions of coherent interference causedby reusing the pilot signals among the users. Unlike previousworks with many scatterers [5], this part also points out thata small number of scatterers have significant contributionsto increase non-coherent interference. The last part in thedenominator of (9) represents additive noise effects.III. U PLINK T OTAL D ATA E NERGY C ONSUMPTION M INIMIZATION
This section expresses an uplink energy consumption min-imization problem by assuming that user k in cell l requestsa SE ξ lk > , ∀ l, k, and has a maximum power P max ,lk > .Investigating this optimization problem, we manifest thefeasibility and infeasibility under the limited power budget. A. Problem Formulation
The main goal of 5G-and-beyond systems is to provide thehigh SEs to all users with a minimal power consumption. In this paper, we formulate a total data energy optimizationproblem for the uplink data transmission as follows minimize { p lk ≥ } ( τ c − τ p ) L (cid:88) l =1 K (cid:88) k =1 p lk subject to R lk ≥ ξ lk , ∀ l, k,p lk ≤ P max ,lk , ∀ l, k, (13)where P max ,lk is the maximum power level that user k in cell l can allocate to each data symbol. Problem (13) constrains onthe rate requirement and limited power budget of each user.By setting ν lk = 2 ξlkτcτc − τp − and removing the constant τ c − τ p in the objective function, problem (13) is converted from theSE constraints into the equivalent SINR constraints as minimize { p lk ≥ } L (cid:88) l =1 K (cid:88) k =1 p lk subject to p lk z llk (cid:12)(cid:12) tr (cid:0) R llk ΨΨΨ llk R llk (cid:1)(cid:12)(cid:12) NI lk + CI lk + NO lk ≥ ν lk , ∀ l, k,p lk ≤ P max ,lk , ∀ l, k. (14)We stress that problem (14) jointly optimizes the powers tosatisfy the requested SINRs from all the users. This problemcan be either feasible or infeasible for a given set of userlocations and shadow fading realizations. B. Feasible and Infeasible Problems
When problem (14) has a non-empty feasible domainmeaning that the network can simultaneously provide therequired SEs to all the users conditioned on the powerconstraints. The global optimal solution to problem (14) canbe then found. Indeed, it is straightforward to show that (14)is a linear program on standard form [16]. We hence enableto solve (14) to the global optimality in polynomial time,for instance, utilizing a general interior-point optimizationtoolbox as CVX [17]. It should be noticed that all the KL users will spend non-zero data powers at the global optimumwhen (14) is feasible owning to the non-zero SE requirements.There may be a situation that all the users cannot besimultaneously served by the SE requirements. Only oneunfortunate user served with a lower SE suffices to createan empty feasible domain for the total transmit power opti-mization problem. Alternatively, problem (14) lacks a feasiblesolution [16, Section 4.1]. The unsatisfied SE is caused byhigh mutual interference in cellular networks and/or extremelocations as the cell edge leading to some users having aweak channel. Moreover, a user may require a too high SEand the system cannot provide this service even spendingmaximum data power. A feasible solution might still existfor most of the users with the required SEs by expectingthat only one or a few users are unsatisfied . It is sufficientto remove or reduce the required SEs of those unsatisfiedusers to convert an infeasible problem to a feasible one.This paper develops the power allocation strategies to handleuch infeasible instances by allowing the corresponding SINRconstraints to be violated.IV. C ONGESTION SOLUTION BASED ON ALTERNATINGOPTIMIZATION
This section proposes the two algorithms attaining a fixed-point solution to problem (14) with either empty or non-emptyfeasible domain.
A. Spending Maximum Transmit Power on Unsatisfied Users
For the glorification of simplification in comprehension,problem (14) with a non-empty feasible domain is firstconsidered. We stack all the data powers into a vector p = [ p , . . . , p LK ] T ∈ R LK + , then the SINR constraint ofuser k in cell l is reformulated as p lk ≥ I lk ( p ) , where I lk ( p ) is so-called a standard interference function, which is I lk ( p ) = ν lk NI lk ( p ) + ν lk CI lk ( p ) + ν lk NO lk z llk (cid:12)(cid:12) tr (cid:0) R llk ΨΨΨ llk R llk (cid:1)(cid:12)(cid:12) . (15)In (15), the detailed expressions of NI lk ( p ) and CI lk ( p ) have been already expressed in (10) and (11), but we hereemphasize them as the functions of data power variablesstacked in p . We now introduce the definition of a standardinterference function for which an algorithm to obtain a fixedpoint solution is proposed. Definition 1 (Standard interference function) . A function I ( z ) is a standard interference function for all z (cid:23) , if thefollowing properties hold: a ) Positivity I ( z ) > , ∀ z > . b ) Monotonicity I ( z ) ≥ I ( z (cid:48) ) if z (cid:23) z (cid:48) . c ) Scalability: αI ( z ) > I ( α z ) , ∀ α > , for all scalar α > . The positivity property is because of the inherent mutualinterference and thermal noise in the system, which impliesa non-zero value data powers when users request non-zeroSEs. The monotonicity property ensures that we can scaleup or down (15) by adjusting the data powers. Finally, thescalability property suggests a method to uniformly scaledown the data power coefficient of user k in cell l at eachiteration by utilizing a positive constant α . We now constructa policy to update the data power of every user k in cell l inTheorem 2. Theorem 2.
By assuming that the feasible domain is non-empty and ≤ I lk ( p ) ≤ P ,lk always holds for all p in the feasible domain. For the initial values of data powers p lk (0) = P max ,lk , ∀ l, k , there exist data powers for whicheach interference function I lk ( p ) is non-increasing alongiterations and converges to a fixed point. Particularly, thedata power of user k in cell l , denoted by p lk ( n ) , can beupdated at iteration n as p lk ( n ) = I lk ( p ( n − , ∀ l, k. (16) Proof.
The proof is to testify I lk ( p ) , ∀ l, k, defined in (15)being standard interference, so the updated power policy (16)ensures that this iterative approach converges to a fixed point.The detailed proof is omitted due to space limitations. The notation z (cid:23) z (cid:48) indicates element-wise inequality z n ≥ z (cid:48) n , ∀ n =1 , . . . , KL with z n and z (cid:48) n being the elements of z and z (cid:48) , respectively. Algorithm 1
Data power allocation to problem (14) byspending maximum transmit power on unsatisfied users
Input : Define P max ,lk , ∀ l, k ; Select p lk (0) = P max ,lk , ∀ l, k ;Compute the total data power P tot (0) = (cid:80) Ll =1 (cid:80) Kk =1 p lk (0) ;Set initial value n = 1 and tolerance (cid:15) .1. User k in cell l computes the standard interferencefunction I lk ( p ( n − using (15).2. If I lk ( p ( n − > P max ,lk , update p lk ( n ) = P max ,lk .Otherwise, update p lk ( n ) = I lk ( p ( n − .3. Repeat Steps , with other users, then compute the ratio γ ( n ) = | P tot ( n ) − P tot ( n − | /P tot ( n − . If γ l ( n ) ≤ (cid:15) → Set p ∗ lk = p lk ( n ) , ∀ l, k, and Stop.Otherwise, set n = n + 1 and go to Step . Output : A fixed point p ∗ lk , ∀ l, k .Every user has its own standard interference functionsatisfying the three fundamental properties in Definition 1and utilizing it to update the data power as in (16). Theanalysis in Theorem 2 is based on the assumption thatproblem (14) has the global optimum for which all usersare served with their required SEs. The power constraintsin (14) ( p lk ≤ P max ,lk , ∀ l, k ) are tackled by the fact if I lk ( n − > P max ,lk , then the congestion issue appears andleads to an obvious selection p lk ( n ) = P max ,lk . We thereforedefine the constrained standard interference function used atiteration n − as ˆ I lk ( p ( n − I lk ( p ( n − , P max ,lk ) . (17)For a cellular massive MIMO system with the power budgetconstraints and the initial data power vector p (0) with theentries p lk (0) = P max ,lk , ∀ l, k, iteration n updates the datapower of user k in cell l as p lk ( n ) = ˆ I lk ( p ( n − . (18)Combining (17) and (18), we observe that if ˆ I lk ( p ( n − P max ,lk , the update p lk ( n ) = P max ,lk maintains the non-increasing objective function of problem (14). Otherwise, itholds that ˆ I lk ( p ( n − I lk ( p ( n − , and hence user k incell l consumes less power than the maximum. This procedurewill be applied to all the KL users, which results in analternating approach summarized in Algorithm 1. Notice that,when users cannot be served by the required SEs, one stilllets them utilize the maximum power. This policy aims atmaximizing the SE of a particular user, however producingmore mutual interference to the other users. B. Softly Removing Unsatisfied Users
Instead of allowing potential unsatisfied users to spendfull data power, one can reduce their power with the goalto degrade mutual interference to the others. This policymight ameliorate the number of satisfied users in the entirenetwork. At first, every user improves the transmission qualityby spending more power to each data symbol. This targetcan be achieved by, for example, simply constructing thestandard inference functions as in the previous subsection.If at the limited power budget, the required SE cannot beachieved, unsatisfied users will reduce data power. We then lgorithm 2
Data power allocation to problem (14) by softlyremoving unsatisfied users
Input : Define P max ,lk , ∀ l, k ; Select p lk (0) = P max ,lk , ∀ l, k ;Compute the total data power P tot (0) = (cid:80) Ll =1 (cid:80) Kk =1 p lk (0) ;Set initial value n = 1 and tolerance (cid:15) .1. User k in cell l computes the standard interferencefunction I lk ( p ( n − using (15).2. If I lk ( p ( n − ≤ P max ,lk , p lk ( n ) = I lk ( p ( n − .Otherwise, p lk ( n ) = P ,lk /I lk ( p ( n − .3. Repeat Steps , with other users, then compute the ratio γ ( n ) = | P tot ( n ) − P tot ( n − | /P tot ( n − . If γ l ( n ) ≤ (cid:15) → Set p ∗ lk = p lk ( n ) , ∀ l, k, and Stop.Otherwise, set n = n + 1 and go to Step . Output : A fixed point p ∗ lk , ∀ l, k .mathematically suggest an update of the data powers alongiterations as follows. Theorem 3.
From initial values p lk (0) = P max ,lk , ∀ l, k, ifthe data power of user k in cell l is updated at iteration n as p lk ( n ) = (cid:40) I lk ( p ( n − , if I lk ( p ( n − ≤ P max ,lk , P ,lk I lk ( p ( n − , if I lk ( p ( n − > P max ,lk , (19) then the iterative approach converges to a fixed point.Proof. The proof is first to confirm that the updated powerpolicy in (19) follows a so-called two-sided function and theconvergence is then established. The detailed proof is omitteddue to space limitations.This theorem provides a procedure to minimize the totaltransmit power in the network and coping with the congestionissue. If I lk ( p ( n − is less than P max ,lk , then the datapower of user k in cell l is updated based on (16), sameas Algorithm 1. The main distinction is to prevent anyunsatisfied user from transmitting full power whenever thecongestion issue happens, i.e. I lk ( p ( n − > P max ,lk . Bydoing this power update, the mutual interference from thisunsatisfied user to the others should be reduced, and hencethere is chance for the remaining users to get their requiredSEs. The proposed optimization approach is summarized inAlgorithm 2 and its properties are stated in Remark 1. Remark 1.
The proposed algorithms work in both feasibleand infeasible domain such that a fixed point to problem (14) can be obtained. For realizations of user locations that resultin feasible domains, the fixed point obtained by those algo-rithms is the global optimum. The main difference betweenthe two algorithms appears whenever the congestion issuehappens: While Algorithm 1 allocates the maximum datapower to users when their SINR constraints are not satisfied,Algorithm 2 reduces the data power. Thus, the fixed pointobtained by each algorithm may be different from each other.
V. N
UMERICAL R ESULTS
A massive MIMO system is considered with L = 4 square cells in a km area, each serving K = 5 users. Fig. 1. The CDF of the uplink SE per user [b/s/Hz] with Monte-Carlosimulation and closed-form expression.Fig. 2. The CDF of the power consumption per user [mW] for feasiblesystems with the different required SEs at the users and M = 100 . All the users are uniformly distributed within its cell withthe distance to the BS no less than m. Each coherenceblock has τ c = 200 symbols and τ p = 5 orthogonal pilotsignals with ˆ p lk = P max ,lk = 200 mW, ∀ l, k . The userswith same index in all cells share the same pilot sequence.The system bandwidth is MHz and the noise varianceis − dBm with the noise figure of dB. The large-scale fading coefficient [dB] of user k in cell l and BS l (cid:48) is modeled based on the 3GPP LTE specifications [18] as β l (cid:48) lk = − . − . ( d l (cid:48) lk / km )+ z l (cid:48) lk , where d l (cid:48) lk > mis the distance between user k in cell l and BS l (cid:48) ; z l (cid:48) lk is theshadow fading coefficient following a Gaussian distributionwith zero mean and standard deviation dB. There are scatterers per communication link. The covariance matricesare computed by using [8]. In the proposed algorithms, (cid:15) = 0 . . For feasible systems, the global optimum obtainedby interior point methods [12] are included for comparison.Figure 1 shows the cumulative distribution function (CDF)of SE per user [b/s/Hz] to verify the correctness of theclosed-form expression of the uplink SE for each user. Allusers spend full power for the data transmission. Particularly,the closed-form expression result matches very well Monte-Carlo simulation result for all the considered number of BSantennas. Fig. 1 also illustrates the SE per user gets better ig. 3. The CDF of the power consumption per user [mW] for infeasiblesystems with the different required SEs at the users and M = 100 .Fig. 4. The satisfied SE probability versus the different required SE per userfor a system with M = 100 . when each BS is equipped with more antennas. Each user canbe served by a data rate increasing from . to . [b/s/Hz]on the average if the number of BS antennas increases from to .The CDF of the data power consumption [mW] consumedby each user is shown in Fig. 2 for feasible systems. Theproposed algorithms provide a unique fixed point that is theglobal optimum as what has been obtained by the interior-point methods. Furthermore, data power escalates when usersrequire higher SEs. With a required SE of . [b/s/Hz], eachuser only spends . mW on average. But it drastically growsto . mW if the required SE is . [b/s/Hz].Figure 3 displays the CDF of the data power consumption[mW] per user for infeasible systems which is the maininterest of this paper when working with multiple access inmassive MIMO, since there is no global optimum to obtain.All the users consume non-zero powers at the fixed pointsidentified Algorithms 1 and 2. The data power consumptionper user obtained by Algorithm 1 are . and . higherthan the ones by Algorithm 2 for the given required SEs.Figure 4 plots the satisfied SE probability defined as thefraction of the number of large-scale fading realizations inwhich the users can be served by the required SEs. If eachuser requires an SE . [b/s/Hz], all the benchmarks providean overwhelming satisfied SE probability. The interior-point methods perform worse with higher SE requirements, espe-cially only . users satisfied the required SE [b/s/Hz].In contrast, the proposed algorithms still offer a satisfied SEprobability of more than .VI. C ONCLUSION
This paper has analyzed the system performance of massiveMIMO systems with an arbitrary number of BS antennas,users, and scatterers by utilizing the double scattering channelmodel, rather than the asymptotic regime as in previousworks. The closed-form expression of the uplink SE per userwas computed in closed form. We proposed two algorithmsto handle effectively the congestion issue that often happenssince multiple users are simultaneously connecting to thenetwork and sharing the same time and frequency resources.R
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