Throughput Optimization for Training-Based
Large-Scale Virtual MIMO Systems
Zhiyan Wang∗, Xiaojun Yuan†, and Ying Jun (Angela) Zhang∗,
∗Department of Information Engineering, The Chinese University of Hong Kong, Shatin, New Territories †School of Information Science and Technology, ShanghaiTech University, Shanghai, China
Email:∗{wz012,yjzhang}@ie.cuhk.edu.hk,†[email protected]
Abstract—We consider large-scale virtual multiple-input
multiple-output (MIMO) systems, in which a large number of user terminals communicate with a large number of cooperative bases stations (BSs). We focus on a training-based scheme and investigate the throughput maximization over various system parameters including pilot symbols, the time allocation coefficient
α, the power allocation coefficient γ, and the user number K. Our
main contribution is to derive simple throughput expressions by utilizing the random matrix theory, based on which closed-form optimal solutions of (K, γ, α) are obtained. We show that, for a large but finite coherent time T , the optimal K for throughput optimization satisfiesK > T2 and converges to T2 as the signal-to-noise ratio goes to infinity.
I. INTRODUCTION
Multiple antenna (a.k.a. MIMO: input multiple-output) techniques have been extensively studied to improve the spectral efficiency of mobile communication systems, and are envisioned to be ubiquitously supported in future wireless networks to accommodate the exponential growth of wireless service demands. Compared with conventional MIMO, large-scale virtual MIMO (VMIMO) systems exhibit two prominent features: first, mobile terminals and base stations can cooperate to form a VMIMO system via relaying and centralized data processing; second, such cooperation is of a large scale, so as to harvest significant spatial multiplexing and diversity gains. Large-scale VMIMO (LS-VMIMO) covers a broad range of cutting-edge MIMO techniques under intensive research. For instance, a distributed cellular network [1], [2] is a special LS-VMIMO system in which geometrically separated BSs are allowed to cooperate by connecting them to a common data-processing center. Massive MIMO [3], [4] provides another example in which large-scale antenna arrays are deployed at base stations. More generally, in wireless ad-hoc networks and future cellular systems, cooperation among base stations and mobile terminals can be realized in a large area, which forms a large-scale virtual antenna array that jointly processes the signals from or to the mobile terminals [5], [6].
Large-scale cooperation, however, requires channel state information (CSI) of all user-BS links at the base stations to detect user signals and to carry out precoding, which results in a considerable pilot and processing overhead, especially for systems operating in frequency-division duplexing. Therefore, it is of pressing interest to investigate the impact of such an overhead on the overall system performance, which is the focus of this paper.
A. Contributions
In this paper, we investigate the fundamental throughput limit of a training-based LS-VMIMO system, in which each transmission frame consists of a training phase for acquiring CSI and a data-transmission phase for delivering information. We assume that the system consists of N single-antenna BSs and K single-antenna user terminals, forming an N×K VMIMO channel with coherent time T (during which the channel is assumed to be constant), where N , K, and T are large but finite. We characterize the system throughput by uti-lizing the random matrix theory, and optimize the throughput over the system parameters including pilot symbols, the time allocation coefficient α (which specifies the fraction of the training phase in a transmission frame), the power allocation coefficient γ, and the user number K. The major contributions of this work are listed below:
1) We derive the optimal pilot design to maximize the system throughput for any given (K, γ, α).
2) We characterize the throughput by using the random matrix theory and derive closed-form optimal (K, γ, α) for throughput maximization.
3) We show that, for a large but finite T , the optimal K always satisfies K > T2, and converges to T2 as the signal-to-noise (SNR) goes to infinity.
Our analytical results can provide insights and guidelines for practical design of LS-VMIMO systems.
B. Related Work
Existing work related to the throughput analysis of MIMO systems includes [7]–[11]. In [7], the authors analyzed the throughput of a conventional training-based MIMO system with the user number set to K=αT (so that the channel of each mobile terminal is on average estimated by using one pilot symbol). Based on that, it was shown in [7] that the degrees of freedom of the system is limited by min(N, K,T/2) in the high SNR regime. In [8], the authors studied the throughput optimization of a conventional MIMO system for the case of
K≤αT . Compared with [7] and [8], the work in this paper
has the following novelties. First, the optimal pilot design for a general setup of (α, γ, K, T ) (particularly when K > αT ) is unknown prior to this work. Such knowledge is of essential importance to provide a complete solution to the through-put maximization problem. Second, under the large MIMO assumption, we establish simple throughput expressions by using the random matrix theory, based on which closed-form
optimal solutions of (K, γ, α) are derived. In [9] and [10], the authors studied massive MIMO systems and focused on throughput improvement and complexity reduction ensured by the use of a large excess of base-station antennas over active user terminals. Nevertheless, such a massive MIMO setup in general performs far from the throughput limit of the corresponding cellular network. In [11], the authors analyzed the mutual information of a large-scale MIMO system by using the random matrix theory, but without considering the impact of the channel estimation overhead. This gap, however, is filled in by the work in this paper.
II. SYSTEMMODEL
We consider communications over a virtual MIMO channel, where K single-antenna user terminals send private informa-tion to N single-antenna base stainforma-tions. All base stainforma-tions are able to fully cooperate by connecting to a common data-processing center via perfect links. We assume that both K and N are very large but finite.
The channel is assumed to be block-fading, i.e., the channel keeps invariant within the coherent time T . The corresponding channel model for a frame of T symbols is given by
Y = K k=1 hkxTk + W, (1) or equivalently Y = HX + W, (2)
where Y ∈CN×T represents the received signal matrix at the base stations, andX = [x1,· · ·, xK]T∈ CK×T is the transmit
signal matrix with the k-th row given byxTk, andH ∈ CN×K is the corresponding channel matrix with the (i, j)-th element
Hij connecting the j-th user to the i-th base station, andW∈ CN×T is the white Gaussian noise matrix with each element of power N0. The power constraint of user k is
1
Txk2≤ P0, k∈ IK{1, 2, · · · , K}. (3)
The channel is assumed to follow independent Rayleigh fad-ing, i.e. the elements of H are independently and identically drawn from CN (0, 1).
We adopt a training-based scheme where each transmission frame consists of two phases. In the first phase, pilot sym-bols known to the receiver are transmitted, based on which the channel H is estimated. In the second phase, data are transmitted and detected based on the estimated channel. The details of these two phases are described below.
A. Training Phase
Assume that a period of αT is assigned to transmit pilot symbols and αT is an integer, where α∈(0, 1) is a coefficient to be optimized. From (2), the channel model for the training phase is
Yp= HXp+ Wp, (4)
where Xp∈ CK×αT is the pilot symbol matrix with the k-th
row denoted by xTp,k, and Wp is the corresponding AWGN.
The power of user k in the training phase satisfies
1
αTxp,k2≤ γkP0, k∈ IK, (5)
where γk is the power allocation coefficient of the user k. Throughout this paper, we set γk = γ, k ∈ IK, i.e., every user has a same power budget for the training phase. This is partially justified by the fact that the user links are statistically symmetric to each other.
The base stations useXp andYp to generate an estimate
of the channelH, denoted as H = f(Xp,Yp). The minimum
mean-square error (MMSE) estimate ofH is given by H = Yp X† pXp+ N0IαT −1 X† p. (6)
The MMSE matrix is given by RMMSE E vec(H− H) vec(H− H) † (7) = IN ⊗ MH, (8)
where vec(H− H) is the vector obtained by sequentially stack-ing the columns of (H− H)T, and “⊗” denotes the Kronecker product, and MH = IK− Xp X† pXp+ N0IαT −1 X† p. (9)
B. Data Transmission Phase
In the data transmission phase, the users transmit data and the base stations carry out coherent detection based on the channel estimate obtained in the training phase. The channel model is written as
Yd= HXd+ V, (10a)
where
V (H − H)Xd+ Wd, (10b)
and Xd ∈ CK×(1−α)T is a zero-mean data matrix, and Wd
is the corresponding AWGN. The power consumption at user
k is expressed as
1
(1−α)Txd,k2= γP0, (11)
wherexTd,kis the k-th row ofXd and γis the corresponding
power coefficient. Then, to meet the average power constraint, we obtain from (3), (5) and (11) that
γ= 1 − αγ
1 − α . (12)
The covariance matrices ofXd andV are respectively given
by RXd 1 (1−α)TE XdX†d = γP 0IK (13a) RV (1−α)T1 E VV† = σv2IN, (13b) where the equivalent noise power is given by
σv2= γP0tr(MH) + N0. (14)
In the above, the second equality in (13a) follows from the fact that the user signals are independent of each other; the second equality in (13b) follows by notingV in (10b) and H in (6).
Recall the signal model in (10). The interference-plus-noise termV is in general correlated with the signal HXd, which
complicates the analysis. It is known that the “worst-case” noise for the additive channel in (10a) follows an independent Gaussian distribution [12]. That is, the instantaneous
achiev-able rate over (10a) is lower bounded by
I (Xd; Yd| H) = logIN+ R−1V HR XdH†
= logIN+ (1−αγ)P(1−α)σ20
v H H
† , (15)
where “log” denotes logarithm with base 2, I (Xd; Yd| H)
is calculated by assuming that the elements of Xd are
in-dependently drawn from CN (0, γP0), and those of V are
independently drawn from CN (0, σ2
v). Then, by considering the two phases and averaging over the channel fading, we obtain an achievable throughput of the system as
R (Xp, K, γ, α) = (1 − α) E
logIN+(1−αγ)P(1−α)σ2v0H H
†, (16) where the expectation E is taken over H.
C. Problem Statement
This paper is focused on maximizing the throughput in (16) over the system parameters (Xp, K, γ, α). This problem can
be formulated as maximize Xp,K,γ,α R (Xp, K, γ, α) (17a) subject to xp,k2≤ αγP0T, k∈ IK (17b) 0 ≤ α ≤ 1 (17c) 0 ≤ γ ≤ 1 α. (17d)
Note that a similar throughput optimization problem has been previously considered for conventional MIMO systems; see, e.g., [7], [8]. The difference is that individual power constraints (17b) are considered in our VMIMO setup. Also, as aforementioned, the work in [7] and [8] only provides partial solutions to the problem. The problem in (17) is difficult to solve analytically, mainly due to the randomness of H. In what follows, we characterize the randomness of H by using the random matrix theory and tackle the problem whenH is of a large but finite size.
III. PILOTDESIGN ANDSIMPLIFIEDPROBLEM
FORMULATION A. Optimal Pilot Design
We start with the design of pilot symbols. We focus on minimizing the equivalent noise power σ2v in (14). Together with (9), we formulate this problem as
minimize Xp tr IK− Xp X† pXp+N0IαT−1X†p (18a) subject to xp,k2≤ αγP0T, k∈ IK. (18b) It can be shown that maximizing the throughput in (15) is equivalent to minimizing σ2
v. We leave the details to [14] due to space limitation. The solution to (18) is presented below. Theorem 1. The optimal training matrixXpfor (18) satisfies
the following conditions:
XpX†p = αγP0TIK, K ≤ αT (19a)
X†
pXp = γP0KIαT, K > αT (19b)
xp,k2≤ αγP0T, k∈ IK. (19c)
The corresponding minimum noise power is given by
σv2opt= ⎧ ⎨ ⎩ 1 αT γρ0+1Kγ ρ 0+ 1 N0, K ≤ αT (K−αT )γρ 0+1 Kγρ0+1 Kγ ρ 0+ 1 N0, K > αT, (20)
where ρ0= P0/N0 denotes the average SNR of each user in
a transmission frame. Proof: See Appendix A.
Remark 1. For K≤ αT , Xp is a wide matrix. Then (19c) is
met provided (19a) holds. Note that a similar optimal training matrix has been previously derived for conventional MIMO systems (limited to the case of K≤ αT ) in [8].
Remark 2. For K > αT , Xp is a tall matrix. In this case, (19b) doesn’t necessarily imply (19c). To meet the conditions in (19b) and (19c) simultaneously, Xp can be formed by
extracting αT columns of a K×K normalized discrete– Fourier–transform (or Hadamard) matrix.
B. Asymptotic Analysis
We now simplify the throughput expression in (16) by utilizing the random matrix theory. To this end, substituting (20) into (16), we obtain
R (K, γ, α) = (1−α) ElogIN+ (1−αγ)P0
(1−α)(σ2v)optH H†
, (21)
where H is obtained by substituting (19) into (6), yielding H = 1 N0+αγP0T αγP0TH+WpX†p, K≤ αT 1 γP0K+N0 HXpX†p+WpX†p , K > αT. (22a) (22b) Let λ be an eigenvalue of 1τH H†, where τ is a normaliza-tion factor defined as
τ N−KNtrMH= ⎧ ⎨ ⎩ αT γρ0K (αT γρ0+1)β, K≤αT αT γρ0K (Kγρ0+1)β, K > αT, (23a) (23b) with the ratio β KN.
Remark 3. In general, the number of non-zero λs is deter-mined by min(N, K), which is related to β. In this paper, we assume 0 < β ≤ 1, which implies that there are K non-zero
λs. The analysis can be extended to the case when β > 1. We omit the details due to page limit.
Proposition 1. As K, N → ∞ with a fixed ratio KN = β ∈ (0, 1], the distribution of λ is given by
fK,β(λ) = ⎧ ⎨ ⎩ √ (λ−a)+(b−λ)+ 2πλ , K ≤ αT √ (λ−a)+(b−λ)+ 2πλ , K > αT, (24a) (24b) where a = (1−√β)2, b = (1+√β)2, a= (1 −βαT /K)2,
b= (1+βαT /K)2 and (a)+ max{0, a}.
Sketch of proof: (24a) for K≤ αT follows the result given
by Theorem 2.35 in [13]. (24b) for K > αT is derived by taking the inverse of the η-transform of fβ,K(λ). The proof is omitted due to page limit.
By applying (24) to (21), we obtain the asymptotic through-put expression given in (25). We use numerical results to demonstrate the accuracy of (25) as an approximation of (21). Denote by R1 the real throughput in (21) and by
R (K, γ, α) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (1 − α) K b a log1 + αT ·K·γγρ20 (Kγ+γαT )ρ0+1· λ β √ (λ−a)+(b−λ)+ 2πλ dλ, K≤ αT (1 − α) K b a log1 + αT ·K·γγρ20 K(K−αT )γγρ2 0+K(γ+γ)ρ0+1· λ β √ (λ−a)+(b−λ)+ 2πλ dλ, K > αT. (25a) (25b)
R2 the approximation in (25). Define the relative gap as Δ = |R1−R2|/R1. In numerical evaluation, we set γ = 1,
β = 1, ρ0 = 10 dB and T = 200, and vary the parameters
α and K to see the trend of the gap, as illustrated in Fig. 1.
Note that the expectation in (21) is evaluated by averaging over randomly generated channels. We see that the gap is less than 1% when K > 20. Thus, the asymptotic expression in (25) is a good approximation of (21) when the user number is relatively large, say, K > 20. In the remaining of this paper, we always assume that K, N, and T are large enough (but finite), so that (25) is a good approximation of the throughput in (21). That is, we focus on the throughput optimization based on (25). 0 25 50 75 100 0 1 2 3 4 User Number K Gap in percentage (%) α = 0.2 α = 0.4 α = 0.6 α = 0.8
Fig. 1. The gapΔ vs. the number of users
IV. OPTIMALDESIGN
In this section, we determine the optimal design of the parameters (K, γ, α) for the following problem:
maximize K,γ,α R (K, γ, α) (26a) subject to 0 ≤ α ≤ 1, 0 ≤ γ ≤ 1 α (26b) where R(K, γ, α) is given in (25). A. Optimizing over K
We first optimize the user number K for any fixed α and
γ. The result is presented in the following theorem.
Theorem 2. The optimal K to (26) satisfies K
T → α, as ρ0→ ∞. (27)
Proof: See Appendix B.
Remark 4. Following the proof of Theorem 2, we can further
show that the limit of KN → α still holds if we fix ρ0but let T
go to infinity. We leave the detailed proof to the long version of this paper.
Remark 5. The tradeoff involved in optimizing K are
elab-orated as follows. On one hand, for the data transmission phase, it is known from information theory that the throughput (of a multiple access channel) increases monotonically in the user number K (given the other conditions unchanged). On the other hand, for the training phase, the channel estimation accuracy decreases monotonically in K, as more channel coefficients need to be estimated for a larger K. Theorem 2 reveals that the best tradeoff occurs at K = αT for a sufficiently large ρ0.
As commented in Remark 4, the optimality of K = αT holds even when ρ0 is fixed, provided that T is large enough.
We demonstrate this using numerical results as follows. Let
ρ0=10 dB, T =200, β=1 and γ=1. We choose some
repre-sentative values of α ∈ {0.1, 0.3, 0.5, 0.7, 0.9} ⊂ (0, 1). For each α, we plot the throughput by varying K, as illustrated in Fig. 2. We see that the throughput achieves the maximum at
K = αT in all the cases studied in Fig. 2.
0 20 40 60 80 100 120 140 160 180 200 0 100 200 300 400 User number K Throughput R (bit/channel use) α =0.3 α =0.1 α =0.9 α =0.7 α =0.5
Fig. 2. The throughput R against the user number K for T = 200 and ρ0= 10 dB. The time allocation coefficient α is set to be 0.1, 0.3, 0.5, 0.7 and 0.9, respectively.
B. Optimizing over γ
We now optimize the power allocation parameter γ. We always set K = αT in what follows. Then, the throughput R in (25) is a function of α and γ. Define
g(K, γ, α) = αT Kγγρ20
(Kγ+γαT )ρ
0+1. (28)
From (25a), we see that, for any given α∈[0, 1] with K=αT , maximizing the throughput R is equivalent to maximizing
g(αT, γ, α). Thus, the problem is formulated as
maximize
γ g (αT, γ, α) (29a) subject to 0 ≤ γ ≤ 1/α. (29b) The solution to (29) can be determined by solving the Karush-Kuhn-Tucker conditions [15], with the result presented below. Proposition 2. The optimal γ to (29) is given by
γopt= 1 α 1 +1 − (2α−1)ρ0T 1−α+αρ0T (30) C. Optimizing over α
With K = αT and the optimal γ in (30), the optimal α is given as
αopt= arg max
0≤α≤1R
αT, γopt, α, (31) which is solvable by a one-dimension search over α ∈ [0, 1] or by finding the zero of the derivative of R with respect to
α. Further, we have the following properties on αopt.
Proposition 3. The optimal αopt in (31) satisfies
αopt> 12, and (32a)
αopt→1
2, as ρ0→ ∞. (32b)
Proof: The proof is omitted due to space limitation. Remark 6. Proposition 3 shows that the maximum
through-put of the training-based MIMO system is achieved at (α=1
2, K=T2) in the high SNR regime, which coincides with
the previous result in [7] (that the optimal DoF of the system is achieved at (α=1
2, K=T2)). Further, Proposition 3 reveals
that, for a finite SNR, the maximum throughput is in general achieved at (α>12, K>T
2). This implies that more users can
be simultaneously served and more time resource shall be allocated to the training phase.
We now present numerical results to demonstrate the throughput improvement by optimizing α at finite SNR, as compared with the case of setting α = 12 in [7]. Define the throughput improvement as
ΔR= R (αoptT , γopt, αopt) − R T 2, γopt,12 RT2, γopt,1 2 . (33)
Let K = αT , γ=γopt, β=1 and ρ0=−20 dB. As illustrated
in Fig. 3, αopt is always greater than 0.5 and approaches 0.5
as T increases. A throughput gain ΔR=40%, 30%, and 24% is obtained corresponding to the block length T = 100, 150, and 200, respectively. Furthermore, we investigate ΔR as a function of the SNR in Fig. 4. We see that ΔR → 0 as
ρ0→ ∞, which verifies the asymptotic optimality of α = 1 2
at high SNR. Again, we observe a considerable throughput improvement by optimizing α, especially in the relatively low SNR regime.
V. CONCLUSION
In this paper, we studied the throughput of training-based LS-VMIMO systems. We derive the optimal design of pilot symbols for arbitrary system parameters (K, γ, α). Further-more, based on the random matrix theory, we derive simple
0 0.25 0.5 0.75 1 0 5 10 15 20 25 α Throughput R (bit/channel use) T = 100 T = 150 T = 200 24% 30% 40%
Fig. 3. ThrouguputR vs. α with SNR ρ0= −20 dB;
−200 −15 −10 −5 0 5 10 15 20 10 20 30 40 50 ρ0 (dB) Throughput improvement Δ R (%) T = 100 T = 150 T = 200
Fig. 4. Throughput improvementΔRvs. SNRρ0
throughput expressions, with which the optimal (K, γ, α) for throughput maximization are obtained in closed-form. We ana-lytically show that the optimal number of users satisfies K >T2 in the general SNR regime. Numerical results demonstrate that a considerable throughput improvement is obtained by optimizing (K, γ, α) at relatively low SNR, as compared with the previously known result.
APPENDIXA PROOF OFTHEOREM1
We replace the individual power constraints in (18b) by a total power constraint of trX†pXp
≤ αγP0T K. Then the problem in (18) is relaxed as minimize Xp trIK− Xp X† pXp+N0IαT−1X†p (34a) subject to trX†pXp ≤ αγP0T K. (34b)
In general, the solution to (34) leads to a smaller σ2
v than (18) does, as the latter searchesXp over a strictly smaller feasible
region. By noting tr Xp X† pXp+N0IαT−1X†p = trIαT − N0 X† pXp+N0IαT−1 , (35) we see that (34) can be written as
minimize Xp trX†pXp+N0IαT−1 (36a) subject to tr X†pXp ≤ αγP0T K, (36b) or equivalently, minimize Xp min(K,αT ) i=1 1 λi+N0 + (αT −K)+ N0 (37a) subject to min(K,αT ) i=1 λi≤ αγP0T K, (37b) where λi, i ∈ {1, 2, · · · , min(K, αT )} is the i-th non-zero eigenvalue ofX†pXp. It can be readily shown that the optimal
solution to (37) is
λ1=λ2=· · · = λmin(K,αT )= γP0K, (38) and the other λ’s, if exist, are all zero. This implies XpX†p=αγP0TIK for K≤αT and X†pXp=γP0KIαT for
K>αT . Finally, if Xp satisfying the above conditions is in
the feasible region of the original problem (18), such anXpis
also the optimal solution to (18). This gives the extra condition in (19c), which concludes the proof.
APPENDIXB PROOF OFTHEOREM2
We first consider the case of K ≤ αT . Note that
αT Kγγρ2
0
(Kγ+γαT )ρ
0+1is monotonically increasing in K, and so is R
in (25a). Thus, the maximum of R for K ≤ αT is achieved at K = αT .
We next consider the remaining case of K≥αT . Denote
q (K) = αT ·K·γγρ20
K(K−αT )γγρ2
0+K(γ+γ)ρ0+1. (39)
Then the throughput R in (25b) can be written as
R = (1−α)
b
a
K log1+q(K)λβ √(λ−a2πλ)(b−λ)dλ. (40)
Let ¯H ∈ CN×Kbe a matrix whose elements are independently and identically drawn from CN
0, 1 NKtr( H H†) . Then, we can rewrite (16) as R = (1−α) ElogIN+(1−αγ)P0 (1−α)σ2v HQ ¯¯ H† , (41a)
where the expectation is taken over ¯H, and
Q= 1
(αT P0)2XpX
†
p. (41b)
From (19b), we see that Q in (41b) satisfies Q I, which further implies ¯HQ ¯H† ¯H ¯H†. Then, we obtain
IN+q(K)λβHQ ¯¯ H† ≤I N+q(K)λβH ¯¯H† . (42) Thus, (41a) is upper bounded by
(1 − α) ElogIN+ (1−αγ)P0
(1−α)σv2 H ¯¯H†
. (43) Letting K → ∞ with KN, we obtain that R in (40) is upper
bounded by
Rup=(1−α)
b a
K log1+q (K)βλ √(λ−a)(b−λ)2πλ dλ. (44)
Note that R = Rup at K = αT . Then, it suffices to show
that Rup achieves the maximum at K = αT . To see this, we
note that Rup goes to infinity at K = αT as ρ0 → ∞. For
K > αT , denote x = KT ∈ (α, ∞). Then lim ρ0→∞ K log1 + q(K)λβ = T log1+ α x−α·βλ x−α 1 + α x−α·λβ α ≤ αλ βT log e + αT log 1 + α x−α·λβ .
Thus, from (44), Rup is bounded away from infinity for any
x > α, or equivalently, K > αT . Therefore, the maximum Rup is achieved at K = αT , and so is R, which concludes the proof.
ACKNOWLEGEMENT
This work was supported by grants from the University Grants Committee of the Hong Kong Special Administrative Region, China (Project No. 418712 and AoE/E-02/08).
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