ISIT 2006, Seattle, USA, July 9 - 14, 2006
On Achievable Sum Rates of A Multiuser MIMO Relay Channel Taiwen Tang ∗ , Chan-Byoung Chae † , Robert W. Heath, Jr. ‡ and Sunghyun Cho ∗†‡ Email: {ttang, cbchae, rheath}@ece.utexas.edu Department of Electrical & Computer Engineering Wireless Networking & Communications Group The University of Texas at Austin 1 University Station C0803, Austin, TX 78712-0240 § Email:
[email protected] Samsung Advanced Institute of Technology P.O. Box, 111, Suwon, Korea, 440-600
Abstract— In this paper, we investigate a multiple input multiple output (MIMO) multiuser relay channel, where a source with multiple antennas sends data to multiple users via a relay with multiple antennas. The relay applies linear processing to the received signal and forwards the processed signal to multiple users. In our system model, the direct links from the source to the users are neglected. We propose algorithms to compute achievable sum rates of this system based on dirty paper coding. An achievable sum rate defines a sum rate that can be achieved in the MIMO multiuser relay channel with zero error probability for any user, hence it is also a lower bound of the capacity of this channel. These algorithms also produce coefficients of the precoder at the source node and the coefficients of the linear processing unit at the relay. Simulations show that the proposed system architecture and algorithms achieve sum rate performance that is close to the derived performance upper bound.
I. I NTRODUCTION Recently using fixed relays in cellular systems has received significant interest. Fixed relays are low cost and low transmit power elements that receive and forward data from the base station to the users via wireless channels, and vice versa. Using fixed relays boosts coverage in cellular networks when carefully placed at the cell edge or in regions with significant shadowing. Because they implement a subset of base station functions, fixed relays are a low cost and low complexity solution to providing coverage far from the best station at the cell edge [1]. Unfortunately, it is likely that only a few fixed relays will be available in each cell. Consequently, each fixed relay will need to support multiple users. This motivates developing point-to-multipoint relaying solutions, where the relay forwards data to and from multiple users. The main challenge in the point-to-multipoint fixed relay is providing a high capacity link between the base station and relay, while at the same time providing multiple lower capacity links to multiple users. A natural solution to this problem is to exploit the advantages of MIMO communication This material is based in part upon work supported by Samsung Advanced Institute of Technology.
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to both provide a high capacity link between the base station and relay and to deliver data streams to multiple users from the relay. Consequently, in this paper we propose a multiuser MIMO relay channel. We assume the practical case where the relay node does not decode the signal from the source, but simply process the received signal with matrix multiplication. Further we neglect the direct connection between the source and the users. This is realistic for the case where relays are used to improve coverage and is helpful because it simplifies the analysis. Under these assumptions the multiuser MIMO relay channel can be modeled as concatenation of a point-topoint MIMO link and a MIMO broadcast channel [2]–[4]. Our main contribution is several different algorithms for computing the transmit precoder, relay linear processing matrix, and sum rate under the assumption of zero-forcing dirty paper coding and Gaussian signals. We compare our lower bounds with a simple upper bound on the MIMO relay channel capacity with decode-and-forward and optimal time sharing under the assumption that the direct path is neglected, finding that the performance loss is not significant. Prior work on relay channels focuses on the standard relay channel where relays are used to help send data from a single source to a single destination [5]–[9]. The multiuser relay channel considered in this paper supports data transmission from a source to multiple destinations via a relay. This system does not require decoding and re-encoding at the relay, thus may have lower implementation complexity at the relay than decode-and-forward relaying. It also differentiates from the prior work on MIMO relays with linear processing for single user transmission [10], [11] by considering the multiuser scenario. Compared to the prior work on MIMO broadcast channel [2]–[4], we still apply the dirty paper coding to derive achievable rates of the multiuser MIMO relay channel, however, we conduct a joint design of the precoder at the source node and the linear processing matrix at the relay for this channel. The design needs to satisfy the power constraints at both the source node and the relay node, which differs from the MIMO broadcast channel.
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The noise power at the ith user is simply denoted by 6RXUFH1RGH
vi = hTi,2 WWH h∗i,2 σ12 + σ22
5HOD\1RGH
where (·) denotes conjugate operation. The multiuser relay channel model resembles the MIMO broadcast channel. The major difference is that there exists an additional relay power constraint and a matrix W in the model.
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Fig. 1. The multiuser MIMO relay channel. The relay does not perform decoding-and-forwarding and it applies linear processing to the received signal.
II. S YSTEM M ODEL The system block diagram is illustrated in Fig. 1. The source is deployed with Mb transmit antennas and communicates with a relay node that has Mr antennas. A multiple input multiple output (MIMO) channel is thus created between the source and the relay, which is denoted by H1 . The transmitted signal at the antennas of the source node is denoted by u, which has a dimension Mb × 1. The total power at the source node is denoted by Pt . The transmit power constraint can be written as (1) E(u2 ) ≤ Pt where E stands for expectation. We assume that the relay uses all Mr antennas for transmit and receive. A linear signal processing unit denoted by a matrix W of a size Mr × Mr is employed to process the received signal at the relay. The noise added to the received signal at the relay node is represented by n1 . The number of users in the system is denoted by Nu . The relay forwards the signal from the source to multiple users, each with one receive antenna. The channel between the relay and the ith user can be represented by a vector hi,2 of size Mr × 1. User i observes the following linear combination of the transmitted signals xi
(4)
∗
=
hTi,2 W(H1 u + n1 ) + n2,i
=
hTi,2 WH1 u + hTi,2 Wn1 + n2,i
(2)
where xi is the scalar received signal at user i. The noise term n2,i is a scalar at the ith user and n1 is the noise vector at the relay. The entries of the noise vector n1 and the noise term n2,i follow i.i.d. complex Gaussian distribution with zero mean and variance σ12 and σ22 respectively. The received signal at the relay is processed by the relay matrix W. Denote ˜s = WH1 u + Wn1 . The relay power constraint is equivalent to E˜s2 ≤ Pr , where Pr is the total transmit power at the relay node. This power constraint can be written as 2 H Tr{W(H1 E(uuH )HH 1 + Iσ1 )W } ≤ Pr .
(3)
III. ACHIEVABLE R ATES OF THE M ULTIUSER MIMO R ELAY C HANNEL A well-known result about MIMO broadcast channel is that the sum capacity of this channel can be achieved using dirty paper coding [3], [4]. The result follows the duality between the MIMO broadcast channel and the MIMO multiple access channel [3]. The duality result in [3], however, may not be applicable to the multiuser MIMO relay channel due to the fact that there is an additional power constraint at the relay. As the first step for deriving sum capacity of this system, our objective in this paper is to derive achievable rates with dirty paper coding, which are also lower bounds of the sum capacity. We denote the transmitted signal vector intended for multiple users by s. The source node uses a linear precoding matrix F. The input signal vectors are independent inputs that are all with zero mean and unit variance. The signal after the linear precoding satisfies the total power constraint Tr{FFH } ≤ Pt
(5)
where Tr{·} denotes the trace of a matrix. The sum rate using dirty paper coding can be expressed as a function of the precoding matrix F and the relay processing matrix W. A brute-force approach is to directly optimize the sum rate with respect to the matrices F and W, however, this approach optimizes large number of parameters and has very high computational cost. Further, in this formulation, the optimizers may not be unique, finding global optimum solution is very difficult. To resolve this problem, we introduce certain design structure for the parameters F and W. This leads to achievable sum rates that can be computed using low complexity algorithms. A. SVD Relay Design We propose a design of the eigen-space of the relaying matrix. The proposed algorithm formulates a standard-form optimization problem that has a unique optimizer. It can be transformed into a convex optimization and solved using standard optimization algorithms. We stack the channel vectors between the users and the ¯2 = relay into a matrix H2 = [h1,2 , ..., hNu ,2 ]T . Define H ΠH2 , which is a permutated version of the channel matrix and Π denotes the permutation. The quantity Mu is defined as Mu = min{Mr , Mb , Nu }. The (Mu + 1)th to the Mbth diagonal elements are zeros. Thus a maximum of Mu users can be served simultaneously using this scheme. Applying the QR decomposition and singular value de¯ 2 and H1 composition (SVD) to the channel matrices H respectively, ¯ 2 = G2 Q2 (6) H
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where G2 is a lower triangular matrix and Q2 is unitary. Performing the singular value decomposition, H1 = U1 Σ1 V1H
(7)
where both U1 and V1 are unitary, and the nonzero diagonal elements of the matrix Σ1 are placed in a descending order √ √ ( ν1 ≥ ... ≥ νmin{Mb ,Mr } ). We construct a diagonal matrix K such that KKH = diag{k1 , ..., kMr }. In our design, the linear processing matrix W is expressed with respect to K as W=
H QH 2 KU1 .
(8)
We jointly design the precoder F and the diagonal relay unit K. The vector k is defined as k = [k1 , ..., kMr ]T . The precoder is structured such that FFH = V1 ΘV1H , where Θ = diag{p1 , ..., pMu , 0, ..., 0} denotes the powers allocated to the transmit streams. We stack the powers in a vector p = [p1 , ..., pMu ]T . Let d¯i = νi ki |G2 (i, i)|2 . The noise term can be written as vi =
i
|G2 (i, j)|2 kj σ12 + σ22 .
(9)
j=1 ¯
We denote the SNR at the ith user by ηi = dvi pi i . The achievable sum rate of the SVD relay design with dirty paper coding can be expressed as the following ¯ R(p, k) =
u 1 log2 (1 + ηi ) 2 i=1
M
Mu 1 −1 vi d¯−1 ≥ − log2 i pi 2 i=1 ¯ b (p, k) = R
(10)
¯ ¯ b (p, k) is a lower bound to the sum rate R(p, k). where R We use it as the cost function in the relay design. The reason is that with this cost function, we can formulate a geometric program that has unique optimizer and can be solved using standard optimization algorithms [12] [13]. Notice that with this particular relay structure, the relay power constraint equates the following 2 H Tr{W(H1 FFH HH 1 + Iσ1 )W } =
Mu
ki (νi pi + σ12 ). (11)
i=1
Using the transformation pt = log(p) and kt = log(k), we are able to transform the problem in (12) into a convex optimization [12]. The convex optimization can be solved by standard iterative algorithms, e.g., using interior point methods [13]. Due to the page limit, we do not elaborate on the algorithm. We select the permutation that provides the highest sum rate over all permutations.
B. Relay Water-filling Design The design algorithms proposed earlier formulate optimization problems that can be solved using geometric programming for all user permutations. The search over all user permutations, however, is computationally costly. In this subsection, we propose a simplified version of the SVD Relay Design based on the assumption that equal power allocation is used at the source node. We present a reduced complexity user selection algorithm and a simplified design algorithm to determine the quantities F and W. This gives another achievable sum rate of the multiuser MIMO relay channel. a) A Lower Bound of the Sum Capacity: We assume that all the users are arranged in a permutation order Π. Then the source node selects the first T users out of all users, where T ≤ Mu and Mu = min{Mr , Mb , Nu }. This corresponds to operate on the first T rows of the aggregated channel matrix ¯ 2 in (6). Equal power is allocated to the transmitted streams, H thus each stream is loaded with power Pt /T at the source node. We first give the capacity lower bound and summarize the result in the following proposition. Proposition 2: A lower bound to the sum rate of the fixed relay system under the consideration with equal power alloca¯ l (k) = 1 T log2 (1 + η (l) ), where tion to all streams is R i i=1 2 (l) ηi is a lower bound of the SNR at the ith user (l)
ηi = ci ki , where νi |G2 (i, i)|2 Pt /T
ci =
. (14) Pr σ12 hi,2 2 νi Pt /T + σ22 +σ12 Proof: Notice that with the equal transmit power allocated to the streams at the transmitter, the noise power at the ith user can be written as
For each user permutation Π, we formulate an optimization problem to jointly maximize the sum rate with respect to the parameters p and k as ¯ b (p, k) maxp,k R Mu 2 s.t. i=1 ki (νi pi + σ1 ) ≤ Pr , Mu i=1 pi ≤ Pt , pi ≥ 0, ki ≥ 0 ∀i.
vi
=
i
|G2 (i, j)|2 kj σ12 + σ22
j=1
≤
i
|G2 (i, j)|2
j=1
(12)
Proposition 1: The optimization in (12) is a geometric program. Mu −1 is vi d−1 Proof: Notice that the cost function i=1 i pi a posynomial and the inequality constraints are also posynomials. Therefore, this problem has the structure of geometric programming [12].
(13)
≤
i
kj σ12 + σ22
j=1
Pr σ12 hi,2 + σ22 . νi Pt /T + σ12 2
(15)
i The last step follows from that hi,2 2 = j=1 |G2 (i, j)|2 via From the relay power constraint decomposition. T the QR Pt 2 j=1 kj νj T + σ1 ≤ Pr , since ν1 ≥ ... ≥ νT , we have i i Pt Pt 2 2 j=1 kj (νi T + σ1 ) ≤ j=1 kj (νj T + σ1 ) ≤ Pr .
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Therefore, the SNR for the ith user denoted by ηi can be derived as ηi
= ≥
νi ki |G2 (i, i)|2 Pt /T vi νi ki |G2 (i, i)|2 Pt /T P σ2
r 1 hi,2 2 νi Pt /T + σ22 +σ 2
.
(16)
We first assume that Mu streams with Mb transmit antennas are used in the system. The optimization problem that determines the relay power vector k is the following Mu log2 (1 + ci ki ) mink − 12 i=1 Mu 2 i=1 ki (νi Pt /Mu + σ1 ) ≤ Pr ki ≥ 0 ∀i.
1
This gives a lower bound to the receive SNR and hence leads to a lower bound to the sum capacity of the system. This bound captures the effect of noise amplification for the second link transmission. The denominator of the SNR does not depend on the relay power allocation to the other streams. b) A Greedy Reduced Complexity User Selection Algorithm: When the number of users Nu is large or Mu is large, it is costly to search over the permutations of all users. The user algorithm proposed here is a modification to the algorithm in [14] [15]. The metric for user selection is based on the lower bound in (13). The set of users is denoted by U = {1, 2, ..., Nu }. We elaborate the selection algorithm here (the description follows [15]). 1) Initialization • Set n = 1. hu,2 2 • Let r1,u = . Find a user s1 Pr σ 2 hu,2 2
1 2 ν1 Pt /Mu +σ1
+σ22
such that s1 = arg maxu∈U r1,u . • Let S1 = {s1 }. 2) While n ≤ Mu : • Increase n by 1. • Project each remaining channel vector onto the orthogonal compliment of the subspace spanned by the channels of the selected users. The projection matrix is P⊥ n
= ×
IMr − H2 (Sn−1 )H (H2 (Sn−1 ) (17) H2 (Sn−1 )H )−1 H2 (Sn−1 )
where IMr is the Mr × Mr identity matrix, and H2 (Sn−1 ) denotes the row-reduced channel matrix consisting of the channel vectors of the users selected in the first n − 1 steps H2 (Sn−1 ) = [hs1 ,2 , ..., hsn−1 ,2 ]T . •
Let τn,u = Find a user sn such that sn = arg max
u∈U \Sn−1
τn,u hu,2
2
Pr σ12 νn Pt /Mu +σ12
+ σ22
The solution to the Let us denote di = νi Pt /Mu + optimization in (21) is the standard water-filling algorithm. When ki is determined to be 0 for some streams, we want to conserve power for these channels. Mu The number of streams T can be recomputed as T = i=1 I{ki > 0}, where I{·} is the indicator function. The group of users are identified as S = {t1 , ..., tT } from the total Mu users, where ti ∈ {1, 2, ..., Mu }. With this selection of S, we perform the QR decomposition to the matrix H2 (S) = G2 Q2 and update the matrix G2 . An optimization is performed as follows T mink − 12 i=1 log2 (1 + ci ki ) T 2 (21) i=1 ki (νi Pt /T + σ1 ) ≤ Pr , ki ≥ 0 ∀i. This determines the relay coefficients k. The procedure proposed here involves a two-step water-filling. C. A Sum Rate Upper Bound For the multiuser MIMO relay channel in this paper, the MIMO channel capacity, which we denote by C1 (H1 ), can be obtained by solving the following optimization maxQ log2 |I +
. (19)
• Set Sn = Sn−1 ∪ {sn } This algorithm aims to select a group of Mu users who have good channel conditions. It has been shown in [14] that this algorithm provides a sum rate performance that is close to the sum capacity of MIMO broadcast channel. c) Relay Water-filling: After the desired set of users have been determined, we design an algorithm for the relay when ¯ l (k) is a equal power allocation is used. It follows that R concave function of k.
1 H1 QHH 1 | σ12
s.t. Q 0, Tr{Q} ≤ Pt .
(22)
This can be solved via water-filling. The achievable sum rates of the proposed schemes in the previous subsections are upper bounded by 12 C1 (H1 ), which follows from the data processing inequality [16] since the signal from the source is processed by a linear matrix multiplication, then forwards to multiple users. The normalization factor 1/2 accounts for the two phase receiving-and-forwarding operations. The capacity of the second link C2 (H2 ), which is the MIMO broadcast channel, can be obtained from the following optimization
(18)
2 hTu,2 P⊥ n .
(20)
σ12 .
maxD∈A log2 |I +
1 HH 2 DH2 | σ22
(23)
where A is the set of Nu × Nu diagonal matrices with Tr{D} ≤ Pr [4]. For comparison, we compute the achievable rate of the decode and forward scheme. Let the time-sharing factor be t (0 ≤ t ≤ 1), we have an achievable rate for the decode and forward relaying as Rdf (t) = min{tC1 (H1 ), (1 − t)C(H2 )}.
(24)
The maximum achievable rate Cdf is thus max0≤t≤1 Rdf (t). We have a closed form expression for Cdf , C1 (H1 )C2 (H2 ) . (25) C1 (H1 ) + C2 (H2 ) This is the best achievable rate for the decode and forward relaying under the assumption that the direct links are neglected.
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ISIT 2006, Seattle, USA, July 9 - 14, 2006
8.5
16
12
Upper Bound for Proposed Algorithms Relay WaterŦfilling SVD Relay Design
8 7.5
Average Sum Rate (bps/Hz)
Average Sum Rate(bps/Hz)
14
Upper Bound for Proposed Algorithms Decode and Forward (Optimal TimeŦSharing) Relay WaterŦfilling SVD Relay Design
10 8 6 4
7 6.5 6 5.5 5
2 4.5
0 5
10
15
20
25
4 2
30
SNR (dB)
4
6
8
10
12
14
16
18
20
Number of Users
Fig. 2. Average sum rate vs. SNR of the proposed schemes for Mb = 2, Mr = 3, Nu = 5 and α = 0.05.
IV. S IMULATION R ESULTS We assume that the channel between the source node and the relay H1 and the channel between the relay and the users H2 follow i.i.d. complex Gaussian distribution with zero mean and variances α and 1 respectively. The parameter α represents the path loss for the first MIMO link. Here the signal to noise Pr t ratio SNR’s are defined as SNR1 = αP σ1 and SNR2 = σ2 respectively. They are the average receive SNR’s at the receive antennas. In the simulations, we set SNR1 = SNR2 . This implies that the path losses are the same for the first and the second link. We use the reduced complexity user selection to select the desired users for the SVD relay design and relay water-filling design. The sum rate of the proposed algorithms is illustrated in Fig. 2. The proposed two algorithms (SVD relay design and relay water-filling) give sum rate performance that is close to the upper bound as derived earlier. The throughput of the relay water-filling design is similar to the SVD relay design. When the capacity of the second link becomes higher than the first link, the performance gap of the decode-forward and the upper bound increases. The reason is that using optimal time sharing, more time is allocated to the poor link. We plot the sum rate vs. the number of users in Fig. 3. The sum rate increases with the number of users. The user selection algorithm captures good multiuser diversity gain in this scenario. V. C ONCLUSIONS In this paper, we derived achievable sum rates for a multiuser MIMO relay channel, where a MIMO source node sends data to multiple users (each with one receive antenna) via a MIMO relay without direct connections. The MIMO relay uses linear processing instead of decoding to the received signal from the source, then forwarding the processed signal to multiple users. This model abstracts the downlink of celluar networks that use MIMO fixed relays for coverage enhancement. Our achievable sum rates can be obtained using dirty paper coding technique and can be computed by the proposed algorithms. Simulations show that the proposed achievable rates are close to the derived capacity upper bound.
Fig. 3. Average sum rate vs. number of users of the proposed schemes for Mb = 2, Mr = 3 and α = 0.05. The SNR is fixed to be 20 dB for both links.
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