Effects of Channel Aging in Massive MIMO Systems

Transcription

1 338 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 5, NO. 4, AUGUST 03 Effects of Channel Aging in Massive MIMO Systems Kien T. Truong and Roert W. Heath Jr. Astract: Multiple-input multiple-output (MIMO) communication may provide high spectral efficiency through the deployment of a very large numer of antenna elements at the ase stations. The gains from massive MIMO communication come from the use of multi-user MIMO on the uplink and downlink, ut with a large excess of antennas at the ase station compared to the numer of served users. Initial work on massive MIMO did not fully address several practical issues associated with its deployment. This paper considers the impact of channel aging on the performance of massive MIMO systems. The effects of channel variation are characterized as a function of different system parameters assuming a simple model for the channel time variations at the transmitter. Channel prediction is proposed to overcome channel aging effects. The analytical results on aging show how capacity is lost due to time variation in the channel. Numerical results in a multicell network show that massive MIMO works even with some channel variation and that channel prediction could partially overcome channel aging effects. Index Terms: Channel aging, channel prediction, large-scale antenna systems, massive multi-input multi-output (MIMO), outdated channel state information (CSI). I. INTRODUCTION Massive multiple-input multiple-output (MIMO) is a new reakthrough communication technique. The key ideas of massive MIMO are to deploy a very large numer of antennas at each ase station and to use multi-user MIMO (MU-MIMO) transmission to serve a much smaller numer of users [], []. In a typical envisioned deployment scenario, each ase station has hundreds of antennas to simultaneously serve tens of singleantenna users. The large excess of antennas at the ase station makes it possile to design low-complexity linear signal processing strategies that are well matched to the propagation channel to maximize system capacity. Although the theory of massive MIMO is now estalished (see [] and references therein) and preliminary system level simulations are promising [3], further investigation under practical settings is needed to understand the real potential of this technique. Prior work on massive MIMO communication considers the impact of channel estimation error due to noise or pilot contamination. An important oservation in prior work is that pilot contamination puts deterministic limits on the signal-to- Manuscript received Feruary 8, 03. This research was supported y Huawei Technologies. K. T. Truong is with the MIMO Wireless Inc, Austin, TX 78704, kientruong@utexas.edu. R. W. Heath Jr. is with the University of Texas at Austin, Austin, TX 787, rheath@utexas.edu. He is also President and CEO of MIMO Wireless Inc. The terms of this arrangement have een reviewed and approved y the University of Texas at Austin in accordance with its policy on ojectivity in research. Digital oject identifier 0.09/JCN /3/$0.00 c 03 KICS interference-plus-noise ratio (SINR) and hence the achievale rates [4] [9]. In addition to estimation errors, another reason for channel state information (CSI) inaccuracy is channel aging. Due to time variation of the propagation channel and delays in the computation, the channel varies etween when it is learned at the ase station and when it is used for eamforming or detection. The impact of channel aging has not yet een fully characterized in prior work on massive MIMO. Although channel aging has een studied in other MIMO cellular configurations, like in multicell transmission [0], such results are not directly applicale to massive MIMO systems. In this paper, we incorporate the practical impairment known as channel aging into massive MIMO systems on oth the uplink and the downlink. For performance analysis, we adopt the approach using deterministic equivalents developed in [] and [] for cellular networks where the numer of antennas at each ase station is much larger than the numer of active users per cell. Although the existing framework in [] and [] allows for taking into account certain practical effects like antenna correlation, their main focus is to develop an analytical framework ased on random matrix theory. By introducing time variation into the framework developed in [] and [], our results are a natural, ut not straightforward, generalization of those in [] and []. Specifically, we provide asymptotic analysis on the impact of channel aging on oth the uplink and the downlink achievale rates when the maximal ratio comining (MRC) receiver or the matched filtering (MF) precoder is used. The analysis allows for the characterization of the performance loss due to channel aging. Our analysis shows that channel aging mainly affects the desired signal power to a user and the inter-cell interference due to pilot contamination (corresponding to users in other cells that share the same pilot as the user). We also report on one approach for mitigating channel aging effects ased on the finite impulse response (FIR) Wiener predictor. The idea is to use current and past oservations to predict future channel realizations and thus, reduce the impact of aging. We incorporate prediction into the deterministic equivalent analysis under some assumptions. Our work provides a foundation for incorporating etter predictors into massive MIMO in the future. We numerically investigate channel aging effects and channel prediction enefits in a multi-cell massive MIMO network with realistic parameters. Our simulation results show how channel aging degrades the performance of massive MIMO systems on oth the uplink and the downlink. Notaly, our results show that massive MIMO still works even when there is some time variation in the channel. For example, the achievale rate in the aged CSI case is still aout half of that in the current CSI case if the normalized Doppler shifts are as large as 0. on the uplink and on the downlink. Our results also show that y exploiting temporal correlation in the channel the proposed linear FIR channel predictor could partially overcome the effects of aging, though further work is needed to fully investigate the potential of pre-

2 TRUONG AND HEATH: EFFECTS OF CHANNEL AGING IN MASSIVE MIMO SYSTEMS 339 diction. The remainder of this paper is organized as follows. Section II descries the system model. Section III generalizes the framework in [] and [] to incorporate channel aging and shows how to overcome channel aging ased on linear FIR prediction. Section IV provides an analysis of the achievale rates on the uplink and on the downlink in the presence of channel aging and/or channel prediction. Section V numerically investigates the effects of channel aging and the enefits of channel prediction in a multicell network. Section VI concludes the paper and provides suggestions for future work. II. SYSTEM MODEL Consider a cellular network with C cells. Each cell has a ase station and U randomly distriuted active users. Let C := {,,,C} e the set of indices of the cells. Let U c := {,,,U} e the set of indices of active users in cell c C and U := U U U C e the set of indices of all active users in the network. Each ase station is equipped with N t antennas and each active user is equipped with a single antenna. A distinguishing feature of massive MIMO systems is that the numer of antennas at each ase station is much larger than the numer of served users, i.e., N t U. The network operates in a time division duplex (TDD) protocol, i.e., each node uses a single frequency for oth transmission and reception of signals. Since each node cannot transmit and receive on the same frequency at the same time, the transmission and reception at each node are spaced apart y multiplexing signals on a time asis. All ase stations and active users are perfectly synchronized in time and frequency. As in 3rd generation partnership project (3GPP) long term evolution (LTE)/LTE-Advanced standards [3], a frequency reuse of one is assumed. Extensions to other frequency reuse factors are straightforward []. We assume that the channels are frequency flat, i.e., a single frequency and or sucarrier; the extension to orthogonal frequency division multiplexing (OFDM)-ased frequency selective channels with multiple sucarriers follows in a similar manner. We consider a quasi-static lock fading channel model where the channel andwidth is much smaller than the coherence andwidth and the channel coefficients do not change within one symol, ut vary from symol to symol. Let h cu [n] C e the channel vector from user u in cell c to ase station at the nth symol. Define H c [n] := [h c [n], h c [n],,h cu [n]] C U as the comined channel matrix from all users in cell c to ase station. For analysis, we assume thath cu [n] is modeled as [] and [] h cu [n] :=R / cu v cu[n] () where v cu [n] C is a fast fading channel vector and R cu C is a deterministic Hermitian-symmetric positive definite matrix. We assume that v cu [n] is uncorrelated wide-sense stationary complex Gaussian random processes with zero mean and unit variance, i.e., v cu [n] CN(0,I ). The deterministic matrix is independent of symol index n and is determined as R cu =E[h cu [n]h cu[n]] () BS p User k h* pqk h pqk BS q BS antenna elements Desired channel Interfering channel Fig.. The massive MIMO system under consideration. There are C cells, each has one ase station (BS) and U single-antenna users. Each ase station has N t antennas. h cu [n] C is the column channel vector from user u in cell c to ase station at time n. Due to channel reciprocity, we assume that h cu [n] is the channel vector from ase station to user u in cell c at time n. for alln. R cu may include many effects like pathloss, shadowing, uilding penetration losses, spatial correlation, and antenna patterns. Fig. illustrates the system model under consideration. On the uplink (or reverse link), the users simultaneously send data to their serving ase stations. Let x r,cu [n] e the transmitted symol sent y user u in cell c C on the uplink at time n, where E[ x r,cu [n] ] =. The suscript r is used to denote reverse link. The transmitted symols sent y the users are mutually independent. Define x r,c [n] := [x r,c [n], x r,c [n],,x r,cu [n]] T C U as the transmitted symol vector y the U users in cell c. The users use the same average transmit power of p r during the uplink data transmission stage. Let z r, [n] C e spatially white additive Gaussian noise at ase station, where z r, [n] CN(0,σ I N t ). Base stationoserves y r, [n] = C p r H c [n]x r,c [n]+z r, [n]. (3) c= Base station applies a linear detector W [n] C U to y r, [n] to detect x r,c [n], where the uth column of W [n] is denoted as w u [n]. Denote z r,u = w r,u [n]z r,[n] as spatially filtered Gaussian noise. The post-processing received signal for detectingx r,cu [n] is ỹ r,u [n] = wu [n]h u[n]x r,u [n] + r,u [n] pr z desired signal noise + wu [n]h ck[n]x r,ck [n]. (4) (c,k) (,u) } {{ } interference On the downlink (or forward link), each ase station uses MU-MIMO transmission strategies to roadcast data to its associated users. Since the ase stations simultaneously send data, downlink transmission forms an interfering roadcast channel. The suscript f is used to denote forward link. Denote x f, [n] := [x f, [n], x f, [n],, x f,u [n]] T C U as the data symols that ase station sends to its ] served U users, where E[x f, [n]] = 0 and E [x f, [n]x f, [n] = I. Base station

3 340 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 5, NO. 4, AUGUST 03 uses a linear precoding matrix F [n] C U to map x f, [n] to its transmit antennas. The signal vector transmitted y this ase station is λ F [n]x f, [n], where λ is the normalization factor to satisfy the average transmit power constraint λ := E[trF [n]f (5) [n]]. The ase stations use the same average transmit power of p f during the downlink data transmission stage. Let z f,u [n] CN(0,σu ) e complex Gaussian noise at user u in cell. Define the comined noise vector at the users in cellasz f, [n] := [z f, [n], z f, [n],, z f,u [n]] T C U. Let f u [n] e the uth column off [n]. Useruin cell oserves y f,u [n] = p f λ h u[n]f u [n]x f,u [n] +z f,u [n] desired signal noise + pf λc h cu [n]f ck[n]x f,ck [n]. (6) (c,k) (,u) } {{ } interference The ase stations estimate the channels ased on pilots, or training sequences, sent y the users. Let τ e the length of training period. The suscript p is used to denote the pilot transmission stage, or the training stage. Suppose that all cells share the same set of U pair-wisely orthogonal pilot signals Ψ := [ψ ; ;ψ U ] C U τ, where ψ u C τ for u =,,U. The training sequences are normalized so that ΨΨ = I U. The users use the same average transmit power of p p the training stage. The received training signal at ase stationis Y p, [n] = ( C ) p p τ H c [n] Ψ+Z p, [n] (7) c= where Z p, [n] C τ is spatially white additive Gaussian noise matrix at ase station during the training stage. Base stationcorrelatesy p, [n] withψto otain Ỹ p, [n] = pp τ Y p,[n]ψ. (8) This gives the following noisy oservation of the channel vector from useru U to ase station ỹ p,u [n] =h u [n] + cu [n] + c h pp τ Z p,[n]ψ u desired z p, [n] interference noise where z p, [n] CN(0,σ I N t ) is the post-processed noise at ase station. The interference channels during the training stage are those from the users in the other cells using the same pilot. The effect of these interference channels on channel estimation error and then on system performance is called pilot contamination []. Similarly, that of noise is called noise contamination. Note that this model for training and channel estimation in the presence of pilot contamination and noise contamination is proposed and used widely in prior work [9], [], []. (9) Base station applies minimum mean square error (MMSE) estimation to the right-hand side of (9) to estimate h u [n] for u U. Define R u := C c= R cu as the sum of the covariance matrices from the ase stations to user u in cell. The MMSE estimate ofh u [n] is [4] where ĥ u [n] =R u Q u ỹ p,u [n] (0) ) σ Q u =( p p τ I N t + R u. () Since R cu is independent of index n for all B,c C, and u U c, then Q u is independent of index n. To compute the channel estimates, ase station needs to know the deterministic correlation matrices R u and R u for u U. The sum correlation can e estimated from the received signal using standard covariance estimation techniques; it is assumed to e known perfectly in the analysis along with R u. The distriution ofĥu[n] is ĥu[n] CN(0,Φ u ), where [] Φ cu =R u Q u R cu, B,c C,u U c. () By settingc = in (), we otainφ u. Note thatφ cu is independent of indexnfor all B,c C, and u U c. Due to the orthogonality property of the MMSE estimation, the oserved channel can e decomposed as h u [n] = ĥu[n]+ h u [n] (3) where h u [n] CN(0,R u Φ u ) is the channel estimation error and is uncorrelated withĥu[n]. Because h u [n] and ĥ u [n] are jointly Gaussian, they are statistically independent. III. INCORPORATING CHANNEL AGING EFFECTS We present a method for incorporating channel aging effects into the existing framework in susection III-A. We then derive the optimal linear FIR Weiner channel predictor to overcome the aging effects in susection III-B. A. Channel Aging In principle, the channel changes over time due to the movements of antennas and those of ojects (or people) in the propagation medium. To analyze the impact of channel aging, we need a time-varying model for the channel. For simplification and tractaility, we assume that the channel temporal statistics are the same for all antenna pairs. Further, we assume that every user moves with the same velocity, so that the time variation is not a function of the user index. While this is not practical, from an analysis perspective, the performance will e dominated y the user with the most varying channel. Consequently, we assume every user has the same (worst-case) variation. Similar assumptions for channel temporal correlation are made in prior work on MIMO wireless channels [5] [7]. Let h[n] e the univariate random process modeling the fading channel coefficient from a ase station antenna to a user antenna. We model the random process h[n] as a complex Gaussian process with zero mean (we do not consider a line-of-sight

4 TRUONG AND HEATH: EFFECTS OF CHANNEL AGING IN MASSIVE MIMO SYSTEMS 34 component). Under this assumption, the time variation of the channel is completely characterized y the second order statistics of the channel, in particular, the autocorrelation function of the channel, which is generally a function of propagation geometry, velocity of the user, and antenna characteristics [8]. A commonly-used autocorrelation function is the Clarke-Gans model, which is often called the Jakes model and assumes that the propagation path consists of a two-dimensional isotropic scatter with a vertical monopole antenna at the receiver [9]. In this model, the normalized (unit variance) discrete-time autocorrelation of fading channel coefficients is [9] r h [k] =J 0 (π T s k ) (4) where J 0 ( ) is the zeroth-order Bessel function of the first kind, T s is the channel sampling duration, is the maximum Doppler shift, and k is the delay in terms of the numer of symols. The maximum Doppler shift is given y = vf c (5) c where v is the velocity of the user in meters per second (mps), c = mps is the speed of light, and f c is the carrier frequency. As the delay k increases or the user moves faster, the autocorrelationr h [k] decreases in magnitude to zero though not monotonically since there are some ripples. Note that other models for the autocorrelation function can e used; the choice of (4) primarily impacts the simulations. To generate realizations of the channel model, we adopt the approach of using an autoregressive model of order L, denoted as AR(L), for approximating the temporally correlated fading channel coefficient process h[n] [8]. Specifically, we assume that L h[n] = a l h[n l]+w[n] (6) l= where {a l } L l= are the AR coefficients and w[n] is temporally uncorrelated complex white Gaussian noise process with zero mean and variance σw,(l). Given the desired autocorrelation functions r h [k] in (4) for k 0, we can use the Levinson- Durin recursion to determine {a l } L l= and σ w,(l). More details on how to simulate temporally correlated fading channels are referred to [8]. Note that increasing the AR model order L improves the accuracy of channel modeling ut also increases the complexity of the associated analysis. For analysis and to design simplified predictors, we use an AR() approximate model for the fading channel coefficients. This allows us to incorporate channel aging into analysis that already includes channel estimation error. It is reasonale to design predictors ased on the AR() model ecause it only requires estimating the parameters of the AR() model; designing more elaorate predictors is a topic of future work. We denote α = J 0 (π T s ) as a temporal correlation parameter that corresponds to r h [] in (4). We assume that α is known perfectly at the ase stations. Let h cu [n] e the channel vector etween a user and a ase station at time n. Under the AR() model, for any,c C andu U c, h cu [n] =αh cu [n ]+e cu [n] (7) whereh cu [n ] is the channel in the previous symol duration ande cu [n] C is an uncorrelated channel error due to channel aging. We assume thate cu [n] is uncorrelated withh cu [n ] and is modeled as a stationary Gaussian random process with i.i.d. entries and distriution CN(0,( α )R cu ) [7]. Note that the channel model in (7) is also known as the stationary ergodic Gauss-Markov lock fading channel model and has een used in prior work on multiuser MIMO [0] []. It follows from (7) that E[h cu [n q]h cu[n k]] =α k q R cu. (8) Now, we estalish a model for the comined effects of channel estimation errors and aging. We denote n as the index of the channel sample where the channel is estimated. This means that ased on the channel estimate ĥu[n] for all users u U, ase station B designs the precoder F [n + D] or the decoder W [n+d], which is actually used at time (n+d). For illustration, we assume that the CSI at the ase stations is outdated y a channel sample duration, i.e.,d =. We refer to this as a one frame delay; extensions to larger delays, i.e., D >, are straightforward. It follows from (3) and (7) that the true channel at time(n + ) can e decomposed as h u [n+] =αh u [n]+e u [n+] (9) =αĥu[n]+α h u [n]+e u [n+] ẽ u [n+] (0) whereẽ u [n+] CN(0,R u α Φ u ) is mutually independent of ĥu[n]. Note that while pilot contamination affects the estimation error h u [n], moility and processing delay affect the aging errore u [n+]. B. Channel Prediction Channel prediction is one natural approach to overcome the channel aging effects. In this section, we focus on predicting h u [n + ] ased on the current and previous received training signals. Effectively, we have the prolem of predicting an autoregressive multivariate random process in the presence of noise. For simplicity, we assume that the interference from other ase stations during training periods can e treated as uncorrelated additive Gaussian noise with zero mean and constant variance. In practice, these interference channels change over time as the user moves. Let {V u,q } p q=0, where V u,q C, e the optimal pth order Wiener linear predictor that minimizes the MSE in the prediction of h u [n + ] ased on ỹ p,u [n],ỹ p,u [n ],,ỹ p,u [n p]. For notational convenience, we define V u := [V u,0 V u, V u,p ] C (p+) and ȳ p,u [n] := [ỹ p,u [n] ỹ p,u [n ] ỹ p,u [n p]] C (p+). For,c C andu U, define δ(p,α) :=[α α p ], () α α p α α p (p,α) :=......, () α p α p

5 34 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 5, NO. 4, AUGUST 03 [ ] T u (p,α) := (p,α) R u + σ p p τ I N t(p+), (3) Θ cu (p,α) :=[δ(p,α) R u ]T u (p,α)[δ(p,α) R cu ]. (4) Theorem provides the results for the optimal pth order linear Wiener predictor. Theorem : The optimalpth linear Wiener predictor is V u =α[δ(p,α) R u ]T u (p,α). (5) Proof: Based on the orthogonality principle [3],V u can e found y solving the following prolem Equivalently, E[(h u [n+] V u ȳ p,u [n])ȳp,u [n]] =0. (6) E[h u [n+]ȳ p,u[n]] =V u E[ȳ p,u [n]ȳ p,u[n]]. (7) Thus, the optimalpth order linear Wiener predictor is V u =R h Ỹ []R [0]. (8) Ỹ According to (9) and the independence etween h u and h cu forc, the cross-correlation etween the true channel and the training signals is It follows that R hỹ [k +] :=E[h u [n+]ỹ u[n k]] =α k+ R u. (9) R h Ỹ [] :=E[h u[n+]ȳ p,u[n]] =α[δ(p,α) R u ]. (30) Moreover, the autocorrelation function of training signals is Consequently, Rỹ[k q] :=E[ỹ u [n q]ỹu [n k]] (3) =α k q C c= R cu +δ[k q] σ p p τ. (3) RỸ[0] :=E[ȳ p,u [n]ȳp,u [n]] (33) Rỹ[0] Rỹ[] Rỹ[p] Rỹ[] Rỹ[0] Rỹ[p ] = (34) Rỹ[p] Rỹ[p ] Rỹ[0] =T u (p,α). (35) Sustituting (30) and (35) into (8), we otain (5). The predicted channel is h u [n+] = p V u,q ỹ p,u [n q] = V u ȳ p,u [n]. (36) q=0 The resulting MMSE is ǫ p =E[ h u [n+] V u ȳ p,u [n] F] (37) =tr(e[(h u [n+] V u ȳ p,u [n])h u[n+]]) (38) =tr(r u α Θ cu (p,α)). (39) The covariance matrix of h u [n+] isα Θ u (p,α). We have the following orthogonal decomposition h u [n+] = h u [n+]+ h u [n+] (40) where h u [n+] is uncorrelated channel prediction error vector with covariance matrix ofr u α Θ cu (p,α). Moreover, we have T u (0,α) = Q u and Θ u (0,α) = Φ u, thus h u [n+ ] = αĥu[n] whenp = 0. IV. PERFORMANCE ANALYSIS In this section, we consider three different scenarios: i) Current CSI, ii) aged CSI, and iii) predicted CSI. The superscripts ( ) (c),( ) (a), and( ) (p) are used to denote current CSI, aged CSI, and predicted CSI. For oth the uplink and the downlink, we first derive achievale SINR expressions for different scenarios for the general setting. Next, we provide some asymptotic results ased on the approach using deterministic equivalents in [] for the cases when N t is large for MRC receivers on the uplink or MF precoders on the downlink. While these results are asymptotic in the sense that they are derived under an assumption thatn t, simulations in [] show that the fit etween simulation and approximation is good, even for small numers of antennas (around50). The approximations are derived under some technical assumptions, that essentially we summarize as (i) the maximum eigenvalue of any spatial correlation matrix is finite, (ii) all spatial correlation matrices have non-zero energy, and (iii) the intercell interference matrix including channel estimation errors are finite. From a practical perspective, the assumptions are reasonale. From the perspective of doing the calculations, only (ii) is prolematic. Essentially, one has to rememer not to use zero-valued correlation matrices in the expressions. The key ingredient in this asymptotic analysis is the deterministic equivalent SINRs. The resulting expressions are a function of channel covariance matrices R cu, the SNR, and various quantities computed from them. We want to emphasize that the analysis does not include overhead, for training or other purposes. Also, we use the notation without temporal index to refer to the deterministic equivalents asn t. A. Uplink Transmission Recall that we assume ase station knows R u, Ru for u U and α. Moreover, depending on the CSI assumption, ase stationhas the following CSI ĥ u [n+], current CSI g u [n+] = αĥu[n], aged CSI (4) h u [n+], predicted CSI. We can rewriteỹ r,u [n+] as ỹ r,u [n+] = wu[n+]g u [n+]x r,u [n+] +wu[n+](h u [n+] g u [n+])x r,u [n+]

6 TRUONG AND HEATH: EFFECTS OF CHANNEL AGING IN MASSIVE MIMO SYSTEMS (c,k) (,u) w u[n+]g ck [n+]x r,ck [n+] + pr z r,u [n+]. (4) Applying the method commonly used in prior work [8], [], [], [4], we derive a standard ound on the ergodic achievale uplink rates ased on the worst-case uncorrelated additive noise. The idea is to treat ỹ r,u [n + ] as the received signal of a single-input single-output (SISO) system with the effective channel ofg u [n+] while the remaining terms act like uncorrelated additive Gaussian noise. As a result, the desired signal power is S r,u = w u[n+]g u [n+]. (43) The interference plus noise power is I r,u = w u[n+](h u [n+] g u [n+]) + σ p r w u [n+] + (c,k) (,u) The post-processed SINR in this case is given y w u [n+]h ck[n+]. (44) η r,u = S r,u I r,u. (45) The uplink ergodic achievale rate of useruin cellis R r,u =E[log (+η r,u )]. (46) Note that the expectation in the expression of the ergodic achievale uplink rate of user u in cell is over the realizations of the desired channel as in (45) η r,u is computed for only one realization of the desired channel. We now consider the asymptotic results when N t. Lemma summarizes the key results used for deriving asymptotic deterministic equivalents in the paper. Theorem and Theorem 3 present the expressions of the deterministic equivalent SINR for the aged CSI and predicted CSI for the MRC receiver w u [n+] = g u [n+]. Lemma : Consider A C N N with uniformly ounded spectral norm (with respect to N). Consider x and y, where x,y C N, x CN(0,Φ x ), and y CN(0,Φ y ), are mutually independent and independent ofa. Then, we have ( E[ N x Ax N x Ax N traφ x 0, N (47) N x Ay 0, N (48) ) ( N traφ x ) ] 0, (49) N N (x Ay) raφ xaφ y 0. (50) N Proof: If we denote x = N Φ / x, then x (0, N I N). Similarly, if we denote ỹ = x N Φy / y, then ỹ (0, N I N). Note that x and ỹ are mutually independent and independent ofa,φ x, andφ y. We can rewrite N x Ax = x Φx / AΦx / x. Applying Lemma 4 (i) in [] for x and à = Φ /, we otain x AΦ/ x x Φ / x AΦ / x x N trφ/ x AΦ / x 0. (5) N It follows that N x Ax N traφ x 0, (5) N which is exactly (47). Using the same technique, we can prove (48), (49), and (50). Theorem : With aged CSI, the deterministic equivalent SINR for user u in cell is where η (a) r,u (α) = α A (a) r,u B (a) r,u +C(a) r,u +D(a) r,u +α E (a) r,u (53) A (a) r,u = trφ u, (54) B (a) r,u =tr(r u α Φ u )Φ u, (55) C (a) r,u =σ trφ u, (56) p r D (a) r,u = trr ck Φ u, (57) (c,k) (,u) E (a) r,u = trφ cu. (58) c Proof: Sustituting g u [n+] = αĥu[n] into (43), we otain the signal power (scaled y ) as α Applying Lemma, we have Let S (a) r,u = α ĥ u [n]ĥu[n]. (59) ĥ u [n]ĥu[n] trφ u 0. (60) (a) S r,u e the deterministic equivalent signal power. Thus, S (a) r,u α trφ u 0. (6) Sustitutingg u [n+] = αĥu[n] andh u [n+] g u [n+ ] = ẽ u [n+] into (44), we otain the interference plus noise power (scaled y ) as α I (a) r,u = ĥ u [n]ẽ u[n+] + + N (c,k) (,u) t Applying Lemma, we have ĥ u [n]ẽ u[n+] σ p r ĥ u [n]ĥu[n] ĥ u[n]h ck [n+]. (6) A(c,k,,u)

7 344 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 5, NO. 4, AUGUST 03 trφ u (R u α Φ u ) 0, (63) ĥ u N [n]ĥu[n] trφ u 0. t N t (64) Ifk u, thenh ck [n+] andĥu[n] are mutually independent, thus fork u, Ifk = u, define A(c,k,,u) ẑ p,cu [n] := c c trr ck Φ u 0. (65) h c u[n]+ pp τ z p,[n]. (66) As a result, ẑ p,cu [n] CN(0,Q u R cu) and ẑ p,cu [n] is independent ofh cu [n+]. It follows from (0) and (7) that ĥ uh cu [n+] = αĥ uh cu [n]+ĥ u[n]e cu [n+] (67) Thus, we have A(c,u,,u) = αh cu [n]q ur u h cu [n] +αẑ p,cu Q ur u h cu [n] +ĥ ue cu [n+]. (68) ( α lim +α ẑ p,cu Q ur u h cu [n] + h cu [n]q ur u h cu [n] ĥ ue cu [n+] ) 0. (69) Applying Lemma, we have h cu [n]q ur u h cu [n] trφ cu ẑ p,cu Q ur u h cu [n] (trφ cu R u trq u R u Φ u R cu ) ĥ u e cu[n+] ( α ) trφ cu R u 0, (70) 0, (7) 0. (7) It follows from (69) (7) that A(c,u,,u) ( α trφ cu +trr cu Φ u ) α trq u R u Φ u R cu 0. (73) Let Ī(a) r,u e the deterministic equivalent interference plus noise power. From (6) (65) and (73), we have Ī (a) r,u ( σ trφ u +α trφ u (R u Φ u ) p r +( α )trφ u R u + (c,k) (,u) trr ck Φ u +α trφ cu α ) trq u R u Φ u R cu c c 0. (74) Applying (74) and (6) into (45) after neglecting the terms that vanish asymptotically, we otain η (a) r,u as in (53). The four terms in the denominator of η (c) r,u characterize the following effects: i)b (a) r,u for channel estimation error, ii)c(a) r,u for post-processed local noise at ase station, iii) D (a) r,u for post-processed traditional intra-cell and inter-cell interference, and iv) α E (a) r,u for inter-cell interference due to pilot contamination. Because the current CSI scenario is a special case of the aged CSI scenario when α =, the deterministic equivalent SINR with current CSI is η (c) r,u = η(a) r,u (), which is the result in Theorem 3 in []. Moreover, η (a) r,u (α) η(c) r,u since α and A (a) r,u,b(a) r,u,c(a) r,u, and D(a) r,u are nonnegative. We notice that channel aging affects only desired signal, channel estimation error, and inter-cell interference due to pilot contamination. Specifically, we can show that η (a) r,u is an increasing function of α in [0,]. This means that when user u in cell moves faster, i.e., α increases, the post-processed uplink SINR is degraded more. Intuitively, when user u in cell moves, this movement not only affects the desired channel to ase station ut also affects the interference channels corresponding to the users sharing the same pilot in other cells, i.e., user u in cells c. Quantitatively, this movement decreases oth the desired signal power and the inter-cell interference power due to pilot contamination α times in the relative comparison with the no channel aging case. Nevertheless, this movement does not affect the local noise power and the traditional intra-cell and inter-cell interference from the other users, those do not share the same pilot with user u in cell during the training stage. This explains how channel aging degrades the uplink ergodic achievale rate. Theorem 3: With predicted CSI otained y using the optimal pth order linear Wiener predictor, the deterministic equivalent SINR for useruin cell is where η (p) r,u (p,α) = α A (p) r,u B (p) r,u +C(p) r,u +D(p) r,u +α E (p) r,u (75) A (p) r,u = trθ u(p,α), (76) B (p) r,u =tr(r u α Θ u (p,α))θ u (p,α), (77) C (p) r,u =σ trθ u (p,α), (78) p r D (p) r,u = trr ck Θ u (p,α), (79) (c,k) (,u) E (p) r,u = c α p trθ cu (p,α). (80) Proof: With predicted CSI, we haveg u [n+] = h u [n+ ] and h u [n +] h u [n+] = h u [n +]. Sustituting

8 TRUONG AND HEATH: EFFECTS OF CHANNEL AGING IN MASSIVE MIMO SYSTEMS 345 g u [n+] = h u [n+] into (43), we otain the signal power (scaled y ) as S (p) S (p) r,u = h u [n+] h u [n+]. (8) Let r,u e the deterministic equivalent signal power. Applying Lemma, we have h u [n+] h u [n+] α 4 trθ u (p,α) 0 (8) whereθ u (p,α) is given in (4). Thus, S (p) r,u α4 trθ u (p,α) 0. (83) From (44) and since h u [n+] is independent of h u [n+], we otain the interference plus noise power (scaled y ) as I (p) r,u = h u[n+] h u [n+] + + N (c,k) (,u) t Applying Lemma, we have h u[n+] h u [n+] σ p r h u [n+] h u[n+]h ck [n+]. (84) B(c,k,,u) α tr(r cu α Θ u (p,α))θ u (p,α) 0, (85) h u [n+] α trθ u (p,α) 0. (86) If k u, then h u [n + ] and h ck [n + ] are mutually independent. Thus, B(c,k,,u) α trr ck Θ u (p,α) 0. (87) Ifk = u, it follows from (36) and (9) that ( C ) h u [n+] =V u h c u[n]+ τ z p,[n]. (88) pp c = As a result, h u [n+] and h cu [n+] are not mutually independent. We have B(c,u,,u) = ȳp,u[n]v uh cu [n+] (89) ( p ) = ỹu [n m]v u,m h cu [n+]. m=0 (90) It follows from (7) that h cu [n+] =α p+ h cu [n p]+v cu [n+] (9) where v cu [n+] CN(0,( α (p+) )R cu ). Also, it follows from (9) and (9) that form = 0,,,p, p m ỹu[n m] = α p m h cu[n p]+ α q e cu [n m q] q=0 +ẑ p,cu [n m]. (9) Sustituting (9) and (9) into (90), then applying Lemma and removing asymptotically negligile terms, we otain B(c,u,,u) α(p+) ( p ) h cu [n p] α p m Vm h cu [n p] It follows that m=0 0. (93) B(c,u,,u) α(p+) trθ cu (p,α) 0. (94) Applying (84) (87), (94), and (6) into (45), we otain η (p) r,u as in (75). Note that A (p) r,u,b(p) r,u,c(p) r,u,d(p) r,u, and E(p) r,u in (75) have the same role as A r,u,b r,u,c r,u,d r,u, and E r,u in (53), respectively. Notaly, we have η (p) r,u (0,α) = η(a) r,u (α). B. Downlink Transmission We assume that the users do not have any information on instantaneous channels on the downlink. We use the technique developed in [4], which is also used in [8] and [], to derive an expression for downlink achievale rates. Specifically, user u in cell knows only E[h u [n + ]f u[n + ]]. Similar to the uplink analysis, we consider the worst-case uncorrelated additive noise. From (6), the received signal at user u in cell (scaled y / p f ), is rewritten as y f,u [n+] = λ E[h u [n+]f u[n+]]x f,u [n+] + λ (h u[n+]f u [n+] E[h u [n+]f u[n+])x f,u [n+] + z f,u [n+] p f + λc h cu [n+]f ck[n+]x f,ck [n+]. (95) (c,k) (,u) The signal power at useruin cell(scaled y p f ) is S f,u = λ E[h u [n+]f u[n+]]. (96) The interference plus noise power at user u in cell (scaled y p f ) is I f,u = λ var[h u [n+]f u[n+]]+ σ u p f

9 346 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 5, NO. 4, AUGUST 03 + N (c,k) (,u) t λ c E[ h cu [n+]f ck[n+] ]. (97) We otain the achievale SINR and rate of useruin cell as η f,u = S f,u I f,u, (98) R f,u =log (+η f,u ). (99) For deterministic equivalent analysis as N t, we focus only on the MF precodersf u [n+] = g u [n+] for all C and u U. Theorem 4 and Theorem 5 present the downlink deterministic equivalent SINR at useruin cellin the aged CSI and predicted CSI scenarios. Theorem 4: With aged CSI, the downlink deterministic equivalent SINR at useruin cell is where η (a) f,u (α) = α 4 A (a) f,u α B (a) f,u +C(a) f,u +α D (a) f,u +α4 E (a) f,u λ (a) = ( α (00) U trφ u ), (0) u = A (a) (a) f,u = λ trφ u, (0) B (a) (a) f,u = λ tr(r u α Φ u )Φ u, (03) C (a) f,u =σ u p f, (04) D (a) f,u = (c,k) (,u) λ (a) c trr cu Φ cck, (05) E (a) f,u = λ (a) c trφ cu. (06) c Proof: With aged CSI, g u [n + ] = αĥu[n], thus the eamforming vector in this case is f (a) u [n + ] = αĥu[n] for C andu U. It follows from (5) that λ (a) λ (a) = [ U α E u=ĥ u [n]ĥu[n] ]. (07) Let := lim λ (a) e the deterministic equivalent transmit power normalization factor at ase station C. Using (07) and Lemma, we otain λ (a) α ( U ) trφ u u= 0. (08) Sustituting (0) andf (a) u [n+] = αĥu[n] into (96), we otain S (a) f,u = λ (a) αe[(αĥ u[n]+ẽ u[n+])ĥu[n]] (09) = λ (a) α 4 E[ĥ u [n]ĥu[n]]. (0) S (a) Let f,u := lim S (a) f,u e the deterministic equivalent signal power. Using (08) and Lemma, we otain S (a) f,u α 4 λ(a) trφ u 0. () Sustitutingf (a) u [n+] = αĥu[n] into (97), we otain f,u = α λ (a) var[h u[n+]ĥu[n]]+ σ u p f I (a) + (c,k) (,u) Using (0), we have α λ (a) c var[h u[n+]ĥu[n]] E[ h cu [n+]ĥcck[n] ]. () C(c,k,,u) E[ ẽ u [n+]ĥu[n] ] 0. (3) Applying Lemma and then using (3), we otain α λ (a) var[h u[n+]ĥu[n]] α λ(a) tr(r u α Φ u )Φ u 0. (4) Ifk u, thenh cu [n+] andĥcck[n] are mutually independent. Thus, whenk u, we have C(c,k,,u) trr cu Φ cck 0. (5) If k = u, then h cu [n + ] and ĥccu[n] are not mutually independent ecause oth depend onh cu [n]. We have C(c,u,,u) = E[ (αh cu[n]+e cu [n+])ĥccu[n] ] (6) = α E[ h cu [n]r ccuq cu (h cu [n]+z p,cu ) ] +E[ e cu [n+]ĥccu[n] ] (7) = α E[ h cu [n]r ccuq cu h cu [n] ] +α E[ h cu[n]r ccu Q cu z p,cu ] +E[ e cu [n+]ĥccu[n] ]. (8) Applying Lemma, we otain + α ( C(c,u,,u) (α trφ ccu +trr cu Φ ccu ) ) trr cu Φ cu Q cu R ccu 0. (9) Sustituting (08), (4), (5), and (9) into (), and then sustituting the resulting expression and () into (98), and we otain η (a) f,u (α) as in (00). The terms in (00) characterize the same effects as their counterpart in (53). Settingα =, we otain the deterministic equivalent SINR for the current CSI scenario as η (c) f,u = η(a) f,u (),

10 TRUONG AND HEATH: EFFECTS OF CHANNEL AGING IN MASSIVE MIMO SYSTEMS 347 which is the result in Theorem 4 in []. We can show that η (a) f,u (α) is an increasing function ofαin[0,]. Finally, we also notice that channel aging affects desired signal, channel estimation error, and inter-cell interference due to pilot contamination on the downlink. Theorem 5: With predicted CSI, the downlink deterministic equivalent SINR at useruin cell is η (p) f,u (p,α) = where λ (p) = ( α 4 A (p) f,u α B (p) f,u +C(p) f,u +α D (p) f,u +α4 E (p) f,u (0) U α trθ u (p,α)), () u = A (p) (p) f,u = λ trθ u (p,α), () B (p) (p) f,u = λ tr(r u α Θ u (p,α))θ u (p,α), (3) C (p) f,u =σ u p f, (4) D (p) f,u = (c,k) (,u) λ (p) c trr cu Θ cck (p,α), (5) E (p) f,u =αp λ (p) c trθ cu (p,α). (6) c Proof: With predicted CSI, we haveg u [n+] = h u [n+ ] and hencef (p) u [n+] = h u [n+] for C andu U. It follows from (5) that λ (p) = [ U E h u= u [n+] h u [n+] ]. (7) Let λ (p) := lim λ (p) e the deterministic equivalent transmit power normalization factor at ase station C. Using (7) and Lemma, we otain ( ) λ U (p) α trθ u (p,α) 0. (8) u= Sustituting (0) and f (p) u [n + ] = h u [n + ] into (96), we otain S (p) f,u = λ (p) E[( h u[n+]+ h u[n+]) h u [n+]] (9) = λ (p) E[ h u[n+] h u [n+]]. (30) S (a) Let f,u := lim S (a) f,u e the deterministic equivalent signal power. Using (30) and Lemma, we otain S (p) f,u α4 λ (p) trθ u (p,α) 0. (3) Sustitutingf (p) u [n+] = h u [n+] into (97), we otain I (p) f,u = λ(p) var[h u [n+] h u [n+]]+ σ u p f + (c,k) (,u) Using (40), we have λ (p) c var[h u [n+] h u [n+]] E[ h cu [n+] h cck [n+] ]. (3) D(c,k,,u) E[ h u[n+] h u [n+] ] 0. (33) Applying Lemma and then using (33), we otain λ (p) var[h u[n+] h u [n+]] α λ (a) tr(r u α Θ u (p,α))θ u (p,α) 0. (34) If k u, then h cu [n + ] and h cck [n + ] are mutually independent. Thus, D(c,k,,u) α trr cu Θ cck (p,α) 0. (35) If k = u, then h cu [n + ] and h ccu [n + ] are not mutually independent ecause oth depend onh ccu [n]. We have D(c,u,,u) = ȳp,cu [n]v ccu h cu[n+] (36) ( p ) = ỹ cu [n m]vccu,m h cu [n+]. m=0 It follows from (7) that (37) h cu [n+] =α p+ h cu [n p]+v cu [n+] (38) where v cu [n+] CN(0,( α (p+) )R cu ). Also, it follows from (9) and (38) that form = 0,,,p, p m ỹcu [n m] =αp m h cu [n p]+ q=0 α q e cu [n m q] +ẑ p,cu [n m]. (39) Sustituting (38) and (39) into (90), then applying Lemma and removing asymptotically negligile terms, we otain D(c,u,,u) α(p+) ( p ) h cu[n p] α p m Vccu,m h cu [n p] m=0 Applying Lemma and then using (5) and (4), we have 0. (40) D(c,u,,u) α(p+) trθ cu (p,α) 0. (4)

11 348 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 5, NO. 4, AUGUST 03 Tale. Simulation parameters. Parameter Description Inter-site distance 500 m Numer of users per cell PLNLOS = log 0 (d) where d > Path-loss model km is the transmission distance in kilometers and f c = GHz is the carrier frequency. Penetration loss 0 db Antenna array configuration at users antenna omni with 0 dbi gain CSI delay ms to 0 ms User dropping Uniformly distriuted within a cell Shadowing model Not considered User assignment Each user is served y the ase station in the same cell. BS antenna gain 0 dbi BS total transmit power 46 dbm UE speed 3 to 0 km/h Frequency carrier GHz Bandwidth 0 MHz Thermal noise density 74 dbm/hz BS noise figure 5 db UE noise figure 9 db Sustituting (8), (34), (35), and (4) into (3), and then sustituting the resulting expression and (3) into (98), we otain η (p) f,u (p,α) as in (0). Note that η (p) f,u (0,α) = η(a) f,u (α). V. NUMERICAL RESULTS In this section, we present numerical results on the effects of channel aging and the enefits of channel prediction in a multicell network. Specifically, the simulated network consists of 7 cells as illustrated in Fig.. We assume the cells have the same numer of users. Also, we assume that all the cells have the same ase station configurations, e.g., the numer of antennas at each ase station and how the antennas are deployed in the cell. Several key simulation parameters are provided in Tale. This set of parameters corresponds to an interference-limited scenario. We are interested in computing the average achievale sum-rates of the users in the center cells on the downlink and on the uplink. Because we notice that similar oservations can e made for the uplink and for the downlink, we present only selected results due to space constraints. To incorporate spatial correlation in the simulations, we consider a circular array geometry for massive MIMO with a single scattering cluster, of fixed spread, and randomly chosen. Moreover, the spatial correlation is chosen to e independent of the temporal correlation so that we can separate the channel aging and spatial correlation effects. Indeed, there are many potential array geometries for massive MIMO. While patch antennas seem attractive, other miniaturized antenna designs may alter- Fig.. Base stations are located at the center of cells and are illustrated y the red squares. We are interested in the average achievale sumrates of the users in the center cell. natively e employed (patch antennas are not used on moile devices for example). At this point, there is no defacto geometry and channel model that is considered a standard for massive MIMO simulations. We chose to use a circular array ecause they are commercially deployed in other systems that exploit reciprocity, i.e., time division synchronous code division multiple access (TD-SCDMA) in China, and it is possile to compute the spatial correlation matrix efficiently using algorithms developed in prior work [5]. Specifically, the approach in [5] provides a closed-form approximate expression of the spatial correlation matrix etween a user and a ase station. In the following simulations, we consider mainly the uniformly circular array (UCA) configuration and assume that the antenna array configuration has a uniformly chosen angle-of-arrival (or departure), and a given angle of spread (AoS) σ as. Moreover, we assume that the angles of arrival (AoAs)/angles of departure (AoDs) are distriuted according to a certain power azimuth spectrum (PAS). The PAS is modeled y the truncated Laplacian proaility density function (PDF), which is given y { βas σas e φ/σ as, ifφ [ π,π] P φ (φ) = (4) 0, otherwise where φ is the random variale descriing the AoA/AoD with ( ). respect to the mean angle φ 0 and β as = e π/σ as Note that ecause our simulations also consider random user locations (unlike prior work that usually considers a fixed location of users and interferers), the complexity of computing the spatial correlation matrix ecomes the ottleneck in our simulations (according to the MATLAB profile function). Our approach does work with other more complex antenna and correlation models at the expense of computational complexity. We use normalized Doppler shifts, which are defined as T s, to characterize channel aging. Larger normalized Doppler shifts correspond to large velocities of the users or large CSI delays. Fig. 3 shows the downlink average achievale sum-rates of the users in the center cell as a function of the normalized

12 TRUONG AND HEATH: EFFECTS OF CHANNEL AGING IN MASSIVE MIMO SYSTEMS antennas 8 6 Current CSI T s = 0. T s = 0. Average sum rates [ps/hz] antennas 4 antennas Average sum rates [ps/hz] T s = 0.3 T s = 0.4 T s = Normalized Doppler shift f T D S Numer of ase station antennas Fig. 3. The downlink average achievale sum-rates of the users in the center cell as a function of normalized Doppler shifts for different numers of antennas at each ase station. Fig. 4. The uplink average achievale sum-rates of the users in the center cell as a function of the numer of antennas at a ase station for different normalized Doppler shifts. Each cell has active users, that are uniformly distriuted in the cell area. Doppler shifts for N t {4,48,7}. We notice the trend that the downlink average achievale sum-rates of the users in the center cell decreases in magnitude to zero though not monotonically since there are some ripples. At first, it decreases with the increasing T s until getting to T s 0.4. Moreover, channel aging reduces the downlink average achievale sum-rates of the users in the center cell y half at T s 0.. Finally, increasing N t does not help improve the value of T s at which the downlink average achievale sum-rates of the users in the center cell gets to zero for the first time. Fig. 4 presents the uplink average achievale sum-rates of the users in the center cell as a function of the numer of antennas at a ase station for different normalized Doppler shifts. We notice that increasingn t improves the uplink average achievale sumrates of the users in the center cell. Moreover, when T s = 0.4, the uplink average achievale sum-rates of the users in the center cell is negligile even for a large numer of antennas at a ase station, e.g., N t = 7. Also, we oserve that for the simulation setting, the uplink average achievale sum-rates of the users in the center cell when T s = 0. is always more than half of that in the case of current CSI. In the previous experiments, we consider the achievale sumrates of the users in the center cell. Fig. 5 shows the cumulative distriution function (CDF) of the downlink achievale rate of the users in the center cell for different normalized Doppler shifts. Notaly, channel aging affects the peak rates significantly. Similar to the other experiments, we notice that the channel aging corresponding to T s = 0. gracefully degrades the rate distriution of the users in the center cell. We now investigate the enefits of FIR channel prediction when each ase station is equipped with 0 antennas. To reduce the computational complexity associated with spatial correlation matrix computation and to separate etween spatial correlation and temporal correlation, in this experiment, we consider only the spatially uncorrelated channel model. Fig. 6 shows the uplink average achievale sum-rates of the users in the center cell as a function of different normalized Doppler shifts without pre- CDF increasing T S Current CSI T S = 0. T S = 0. T S = User data rate [ps/hz] Fig. 5. The CDF of the downlink achievale rate of users in the center cell for different normalized Doppler shifts. diction and with FIR prediction of p = 5 and p = 5. We notice that channel prediction does help cope with channel aging, however, for these values of p, the relative gain is not large, especially at large normalized Doppler shifts. Moreover, the larger value ofpmakes use of more oservations in the past to provide a higher gain. Alternatively, the Kalman filter, which is used to approximate the non-causal Wiener filter, may cope etter with channel aging effects [6]. The investigation of the Kalmanfilter ased channel prediction in massive MIMO systems is left for future work. VI. CONCLUSION In this paper, we proposed a new framework to incorporating practical effects like channel aging into massive MIMO systems. Channel aging causes the mismatch etween the channel

13 350 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 5, NO. 4, AUGUST 03 Average sum rates [ps/hz] Current CSI Aged CSI without prediction Predicted CSI with p = 5 Predicted CSI with p = 5 Increasing p Normalized Doppler shift f T D S Fig. 6. The uplink achievale rate of the typical user as a function of different normalized Doppler shifts without prediction and with FIR prediction of p = 50. when it is learned and that when it is used for eamforming or detection. We derived the optimal causal linear FIR Wiener predictor to overcome the issues. We also provided analysis of achievale rate performance on the uplink and on the downlink in the presence of channel aging and channel prediction. Numerical results showed that although channel aging degraded performance of massive MIMO, the decay due to aging is graceful. Simulations also showed the potential of channel prediction to overcome channel aging. The work focuses only on channel aging. Other practical effects in massive MIMO need to e studied as well. For example, the large arrays in massive MIMO are likely to e closely spaced, leading to correlation in the channel. Future work can investigate the impact of spatial correlation on massive MIMO performance, especially the comparison etween collocated antennas and distriuted antennas. Our initial results along these lines are found in [7]. In this paper, we consider only the MRC receivers on the uplink and the MF precoder on the downlink. Thus, another interesting topic is to analyze the performance in when other types of receivers/precoders are used. Finally, future work can investigate the use of more complicated channel predictors to overcome channel aging effects. REFERENCES [] T. L. Marzetta, Noncooperative cellular wireless with unlimited numers of ase station antennas IEEE Trans. Wireless Commun., vol. 9, no., pp , Nov. 00. [] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, Scaling up MIMO: Opportunities and challenges with very large arrays, IEEE Signal Process. Mag., vol. 30, no., pp , Jan. 03. [3] Samsung, Rel- and onward, in Workshop on TD-LTE Enhancements and Evolution for Rel- and Beyond, Apr. 0. [4] K. Appaiah, A. Ashikhmin, and T. L. Marzetta, Pilot contamination reduction in multi-user TDD systems, in Proc. IEEE ICC, May 00, pp. 5. [5] A. Ashikhmin and T. L. Marzetta, Pilot contamination and precoding in multi-cell large scale antenna systems, in Proc. IEEE Int. Symp. Inf. Theory, Camridge, MA, 0, pp [6] F. Fernandes, A. Ashikhmin, and T. Marzetta, Interference reduction on cellular networks with large antenna arrays, in Proc. IEEE ICC, Ottawa, Canada, 0, pp [7] B. Gopalakrishnan and N. Jindal, An analysis of pilot contamination on multi-user MIMO cellular systems with many antennas, in Proc. Int. Workshop Signal Process. Adv. Wireless Commun., June 0, pp [8] J. Jose, A. Ashikhmin, T. L. Marzetta, and S. 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[3] 3GPP TR 36.84, Evolved universal terrestrial radio access (E-UTRA): Further advancements for E-UTRA physical layer aspects, Mar. 00. [4] S. Verdu, Multiuser Detection. Camridge University Press, 998. [5] J. P. Kermoal, L. Schumacher, K. I. Pedersen, P. E. Mogensen, and F. Frederiksen, A stochastic MIMO radio channel model with experimental validation, IEEE J. Sel. Areas Commun., vol. 0, no. 6, pp. 6, Aug. 00. [6] W. Weichselerger, M. Herdin, H. Ozcelik, and E. Bonek, A stochastic MIMO channel model with joint correlation of oth link ends, IEEE Trans. Wireless Commun., vol. 5, no., pp , Jan [7] M. Vu and A. Paulraj, On the capacity of MIMO wireless channels with dynamic CSIT, IEEE J. Sel. Areas Commun., vol. 5, no. 7, pp , Sept [8] K. E. Baddour and N. C. Beaulieu, Autoregressive modeling for fading channel simulation, IEEE Trans. Wireless Commun., vol. 4, no. 4, pp , July 005. [9] W. C. Jakes, Microwave Moile Communications. New York: Wiley, 974. [0] J. Zhang, R. W. Heath, Jr., M. Kountouris, and J. G. Andrews, Mode switching for the multi-antenna roadcast channel ased on delay and channel quantization, EURASIP J. Adv. Signal Process., 009. [] G. Caire, N. Jindal, M. Koayashi, and N. Ravindran, Multiuser MIMO achievale rates with downlink training and channel state feedack, IEEE Trans. Inf. Theory, vol. 56, no. 6, pp , June 00. [] A. Adhikary, H. C. Papadopoulos, S. A. Ramprashad, and G. Caire, Multi-user MIMO with outdated CSI: Training, feedack and scheduling, in Proc. Allerton Conf. Commun. Control Comput., Sept. 0, pp [3] S. Haykin, Adaptive Filtering Theory. nd Ed. Englewood Cliffs, NY: Prentice-Hall, 99. [4] T. L. Marzetta, How much training is required for multiuser MIMO?, in Proc. Asilomar Conf. Signals Systems Comput., Oct. 006, pp [5] A. Forenza, D. J. Love, and R.W. Heath, Jr., Simplified spatial correlation models for clustered MIMO channels with different array configurations, IEEE Trans. Veh. Technol., vol. 56, no. 4, pp , July 007. [6] M. H. Hayes, Statistical Digital Signal Processing and Modeling. John Wiley and Sons, Inc., 996. [7] K. T. Truong and R. W. Heath, Jr., Impact of spatial correlation and distriuted antennas for massive MIMO systems, in Proc. Asilomar Conf. Signals Syst. Comput. (to appear), Pacific Grove, CA, Nov. 03. Kien T. Truong received the B.S. degree in Electronics and Telecommunications from Hanoi University of Technology, Hanoi, Vietnam, in 00, and the M.Sc. and Ph.D. degrees in Electrical Engineering from the University of Texas at Austin, Austin, TX, USA, in 008 and 0, respectively. From 00 to 006, he was with the Department of Wireless Communications, Research Institute of Posts and Telecommunications, Hanoi, Vietnam. He was a 006 Vietnam Education Foundation (VEF) Fellow. He is a Consultant at MIMO Wireless Inc. His research interests include massive MIMO communication, link adaptation and interference management for wireless cooperative communications, and capacity analysis of wireless ad hoc networks. He was Co-Recipient of the 03 EURASIP Journal on Wireless Communications and Networking Best Paper Award.

14 TRUONG AND HEATH: EFFECTS OF CHANNEL AGING IN MASSIVE MIMO SYSTEMS 35 Roert W. Heath Jr. received the B.S. and M.S. degrees from the University of Virginia, Charlottesville, VA, in 996 and 997, respectively, and the Ph.D. from Stanford University, Stanford, CA, in 00, all in Electrical Engineering. From 998 to 00, he was a Senior Memer of the Technical Staff, then a Senior Consultant at Iospan Wireless Inc, San Jose, CA where he worked on the design and implementation of the physical and link layers of the first commercial MIMO-OFDM communication system. Since January 00, he has een with the Department of Electrical and Computer Engineering at the University of Texas at Austin where he is a Professor and Director of the Wireless Networking and Communications Group. He is also President and CEO of MIMO Wireless Inc. and Chief Innovation Officer at Kuma Signals LLC. His research interests include several aspects of wireless communication and signal processing: Limited feedack techniques, multihop networking, multiuser and multicell MIMO, interference alignment, adaptive video transmission, manifold signal processing, applications of stochastic geometry, and millimeter wave communication techniques. He has een an Editor for the IEEE Transactions on Communication, an Associate Editor for the IEEE Transactions on Vehicular Technology, Lead Guest Editor for an IEEE Journal on Selected Areas in Communications special issue on limited feedack communication, and Lead Guest Editor for an IEEE Journal on Selected Topics in Signal Processing special issue on Heterogenous Networks. He currently serves on the Steering Committee for the IEEE Transactions on Wireless Communications. He was a Memer of the Signal Processing for Communications Technical Committee in the IEEE Signal Processing Society and is a former Chair of the IEEE COMSOC Communications Technical Theory Committee. He was a Technical Co-Chair for the 007 Fall Vehicular Technology Conference, General Chair of the 008 Communication Theory Workshop, General Co-Chair, Technical Co-Chair and Co-Organizer of the 009 IEEE Signal Processing for Wireless Communications Workshop, Local Co-Organizer for the 009 IEEE CAMSAP Conference, Technical Co-Chair for the 00 IEEE International Symposium on Information Theory, the Technical Chair for the 0 Asilomar Conference on Signals, Systems, and Computers, General Chair for the 03 Asilomar Conference on Signals, Systems, and Computers, General Co-Chair for the 03 IEEE GloalSIP conference, and is Technical Co-Chair for the 04 IEEE GLOBECOM conference. He was a Co-Author of Best Student Paper Awards at IEEE VTC 006 Spring, WPMC 006, IEEE GLOBECOM 006, IEEE VTC 007 Spring, and IEEE RWS 009, as well as Co-Recipient of the Grand Prize in the 008 Win- Tech WinCool Demo Contest. He was Co-Recipient of the 00 and 03 EURASIP Journal on Wireless Communications and Networking Best Paper Awards and the 0 Signal Processing Magazine Best Paper Award. He was a 003 Frontiers in Education New Faculty Fellow. He is the Recipient of the David and Doris Lyarger Endowed Faculty Fellowship in Engineering, an IEEE Fellow, a licensed Amateur Radio Operator, and is a registered Professional Engineer in Texas.