Coordinated Beamforming for Multiuser MIMO Systems with Limited Feedforward Chan-Byoung Chae† , David Mazzarese‡ , and Robert W. Heath Jr.† † Wireless Networking and Communications Group (WNCG) Department of Electrical and Computer Engineering The University of Texas at Austin, Austin, TX, 78712, USA Email: {cbchae, rheath}@ece.utexas.edu ‡ Telecommunication R&D Center, Samsung Electronics, Suwon, Korea Email:
[email protected]
Abstract— Jointly optimized linear transmit beamforming and receive combining is a low complexity approach for communication in the multiuser MIMO (multiple input multiple output) broadcast channel. This paper proposes an iterative algorithm for jointly designing the beamforming and combining vectors, which enforces a zero interference requirement after combining. Since the optimization is performed at the base station with channel state information for all the users, the receive beamformers are quantized at the basestation and sent to the users via a lowrate feedforward control channel. Rate bounds are provided to estimate the impact of quantization loss on the achievable rate in Rayleigh channels is performed. Simulations show that the proposed approach using Grassmannian codebooks approaches the sum capacity of the MIMO broadcast channel.
I. I NTRODUCTION The sum capacity and corresponding achievable rate region for the multiuser MIMO (multiple input multiple output) broadcast channel has recently been established [1]–[3]. The optimal transmit strategy given by information theory, and required to achieve points on the boundary of the achievable rate region, is dirty paper coding (DPC) [3]. Unfortunately, DPC does not directly lead to a realizable transmission strategy [1]. There has also been substantial interest in linear beamforming techniques that avoid the need for nonlinear DPClike processing at the transmitter [4]–[9]. Channel inversion chooses beamforming vectors corresponding to the inverse of the multiuser channel at the transmitter, but only works for one receive antenna per user and further suffers from a power penalty [8]. A related strategy is block diagonalization (BD) [4], [6], for situations with multiple antennas and multiple data streams intended for each user. Block diagonalization enforces a zero interference property at each user but requires the number of receive antennas is equal to the number of data streams (unless selection is performed [10]). Another approach is coordinated beamforming, a generalization of block diagonalization, which allows more receive antennas than streams [5], [7]. Coordinated beamforming also enforces a zero interference property but requires an iterative optimization to find the transmit beamforming and received combining vectors. Other more general optimizations have been proposed in the literature, for example maximizing the jointly achievable
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signal-to-interference-plus-noise (SINR) margin under a total power constraint or minimizing the total transmission power while satisfying a set of SINR constraints, but relaxes the zero interference condition [9]. In this paper, we propose a new algorithm for coordinated beamforming under the assumption that (i) the base station has perfect knowledge of all the channels from each user and (ii) that each user knows only its own channel matrix at the receiver. Under these assumptions, each user cannot find their optimal receive combining vector and consequently must get feedforward information from the base station. We propose a limited feedback-inspired method for quantizing the receive combing vectors at the base station. The receive beamforming vectors for each user are quantized at the transmitter and the index of the best vector from a beamforming codebook is sent to each user via a low-rate feedforward control channel. We include this quantization explicitly in the proposed iterative algorithm and show that, unlike a simple application of quantization to the algorithms in [5], [7], we can mitigate inter-user interference due to quantization. Therefore, the proposed algorithms with limited feedforward still enforce zero inter-user interference even though each receiver uses the quantized combining vector. Note that this is different from the case where quantization is performed on the channel state information [11], because we can use the unquantized vectors at the transmitter. We connect the limited feedforward and feedback problems, arguing that Grassmannian codeboks are a reasonable choice for codebook design [12]. We also propose achievable sum rate bounds for the proposed coordinated beamforming with and without limited feedforward. We compare the achievable sum rate loss due to quantization and show that the analysis provides a close match to simulations and are close to the MIMO BC sum capacity. II. S YSTEM M ODEL FOR C OORDINATED B EAMFORMING Consider a multi-user MIMO broadcast channel with Nt antennas at the transmitter and Nr receive antennas of K users as shown in Fig. 1. We assume that the channel between the transmitter and user k is represented by a matrix Hk of size
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Assuming that K = Nt and that the channels are sufficiently ˜ k will be full-rank and of dimension K − 1 × K thus rich, H the null-space has dimension one and there is only one zero singular value. It is possible to extend this algorithm to work in the case where the null-space has dimension greater than one but we defer this to future work [13]. Define the singular ˜ k as value decomposition (SVD) of H H (0) ˜k = U ˜k V ˜ kD ˜ (1) v H , (3) ˜ k k
Fig. 1. Transmission model with linear processing at the transmitter and at the receivers
Nr × Nt with complex entries. For some of the analysis and the simulations we model the elements of each user’s channel matrix as independent complex Gaussian random variables with zero mean and unit variance CN (0, 1). We assume that the system uses time division duplex (TDD) in which the temporal variations of the channel are slow compared to the duration of the downlink and uplink frames. With TDD, it is reasonable to assume that the base station and the mobile users can estimate the same channel using channel sounding. Thus we assume that {Hk }K k=1 is known perfectly at the transmitter. This assumption is used in most other work on block diagonalization [4]–[7]. Let xk and nk denote the k th transmit symbol, and the additive white Gaussian noise vector with variance σk2 per entry observed at the receiver. Let mk denote the transmit beamformer (assumed unit norm, i.e. there is no power allocation) and wk denote the combining vector for the k th user. The received signal at the k th user after combining is yk = wkH Hk mk xk + wkH Hk
K
ml xl + wkH nk .
(1)
l=1,l=k
In the coordinated transmission strategies [5] [7], the transmitter chooses mk such that the subspace spanned by its columns lies in the null space of wlH Hl (∀l = k), that is, wlH Hl mk = 0 for l = 1, · · · , k − 1, k + 1, · · · , K. If chosen in this way, mk will then cause zero interference to user l, completely removing the interference term in (1). Essentially, the algorithms in [5] [7] form a equivalent channel matrix for the k th user T T T ˜ k = hT · · · hT H , (2) hk+1 · · · hK 1 k−1 where hi = wiH Hi , and then they solve for a transmit ˜ k mk = 0 to ensure that after beamformer mk that satisfies H combining, each users received signal is interference free. Assume A denotes a complex matrix, and AT , AH , and A−1 denote the transpose, Hermitian, and inverse of A, respectively. We also assume that the lower case bold letter a and the upper case bold letter A mean vector and matrix, respectively. E and [A]k denote expectation and the k-th column of matrix A.
˜ k denote the left singular matrix and the ma˜ k and D where U (0) ˜ (1) and v ˜ k , respectively, and V ˜k trix of singular values of H k are the right singular matrix and vector each corresponding to non-zero singular values and zero singular value, respectively. User k’s beamforming vector should lie in the space (0) (0) ˜ k , consequently, we take mk = v ˜ k . Asspanned by v suming that maximum ratio combining is used at the receiver, wk = Hk mk thus the solution to mk depends on {mn }n=k . This makes it difficult for each user to compute their optimal combining vector since this requires that they know channel state information from the other users. In this paper, to solve this problem, we develop joint beamforming algorithms under the assumption that only quantized information about mk is sent to each user. III. C OORDINATED B EAMFORMING WITH L IMITED F EEDFORWARD I NFORMATION A. Limited Feedforward Receive Beamforming Quantized transmit beamforming, known as limited feedback beamforming, is a well known solution for informing the transmitter about the desired transmit beamformer in single user MIMO systems [12]. We propose to use the limited feedback concept to inform each receiver about their wk or their mk . We call this limited feedforward beamforming since side information is being sent from the transmitter to the receiver instead of feedback in the control theoretic sense from the receiver back to the transmitter. To explain this concept in more detail, suppose that the transmitter and receiver have predesigned codebooks denoted by C = {c1 , c2 , · · · , c2Nb } having Nb bit codewords. Further suppose that the base station has derived the optimum set of codewords {mk }K k=1 by solving (2) and (3) iteratively. Let Q(mk ) denote the quantized value of mk . This is not a quantization function, we use it to temporarily highlight the quantization operation. We focus on the quantization of mk since wk = Hk mk and this way quantization can be performed after the channel matching operation. Under this assumption, the k th receiver observes after receive beamforming H H yk =Q(mk )H HH k Hk mk xk + Q(mk ) Hk Hk
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K
ml xl
l=1,l=k
+ Q(mk )H HH k nk .
(4)
Note in particular that the transmitter uses the optimum mk while the receiver uses the quantized Q(mk ). We propose to choose the quantization function to maximize the received SINR under the assumption of quantized received beamforming. The codebook index which has the max SINR value is chosen as follows H 2 m ˆ k = arg max SIN Rk = arg max |cH / i Hk Hk mk | ci ∈C
ci ∈C
K H H H H 2 . ( HH cH i k Hk ml ml Hk Hk )ci + ci Hk Hk ci σ l=1,l=k
ˆ k to denote the quantized We use the notation m transmit beamforming vector instead of writing K , which would be functionally , {H } Q {mn }K n n=1 n=1,n=k correct but more cumbersome notation. Note that there is still residual inter-user interference due to the quantized combining vector. We propose to solve this problem though by updating the specific choice of mk by substituting the quantized received combining vectors into (2) and solving. This will ensure that every user experiences zero interference using their quantized receive beamforming vectors at the expense of some loss of performance. The final received SINR for user k is then given by H 2 |m ˆH k Hk Hk mk | (5) σ2 where, mk is the new transmit beamformer computed from Hk and m ˆ k for k = 1, · · · , K. Consequently, our approach retains the zero-interference property of block diagonalization at the receiver with only quantized feedforward information. This will be more clear in Section III-C.
SIN Rk =
B. Codebook Construction In the proposed codebook quantization problem, the vectors for quantization are found as the solution to a joint nonlinear optimization problem. As such, finding a closed form solution for the codebook in this case is especially difficult. To solve this problem we propose to use codebooks for the case of high SINR. We will show in simulations that these codebooks provide reasonable performance. It is well known that the optimum solution for mk and wk are the right and left singular vectors that correspond to the maximum singular value of Hk . Quantization of mk thus can be solved using well-known Grassmannian codebooks in the case of an independent identically distributed complex Gaussian channel [12], [14]. This codebook choice seems reasonable in the case of interference because in the absence of interference, the right singular vector is isotropic, and Grassmannian codebooks respect that isotropic property. Given the independence of the channels between different users, it is reasonable that the right singular vector also does not have a preferred direction, thus is well modeled as isotropic. C. An Algorithm with Quantized Feedforward In this section, we summarize the proposed algorithm with quantized feedforward information. Our iterative algorithm is
inspired by zero-forcing beamforming algorithms presented in [5] [7] specialized to the case of a single data stream. The idea of our algorithm (described in detail in pseudocode) is as follows. We use the notation (l) to denote the lth iteration. First we initialize each receive beamforming vectors wk (1) to the left singular vector of Hk corresponding to the maximum singular value. Now assuming that {wk (i−1)}K k=1 is available, ˜ k , then solve for mk (i) on iteration i ≥ 2 we first compute H ˜ k , so that as the singular vector that spans the null-space of H user k does not create interference to any other user. Then let wk (i) = Hk mk (i) and repeat until the change in mk (i) ˆk is sufficiently small. Finally, perform quantization to find m and run a final iteration to find mk . This final step ensures that the transmit beamformer mk (i) of user k is chosen in such a way that user k does not create interference at the output of the receivers of the other users, as we introduced in section III-A. A summary of the proposed algorithm is given below. Pseudo code of the proposed algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20:
STEP 1: Initialization for k = 1 : K Hk = Uk Dk VkH wk (1) = u1,k : the principal singular vector end STEP 2: Iterative Update Computation of the effective channel vectors Repeat i for k = 1 :HK hk = wk (i − 1)Hk end Update of transmit and receive beamformers for k = 1h: K iT T T ˜ k = hT . . . hT H hk+1 . . . hK 1 k−1 ˆ (1) ˜H ˜k V ˜k = U ˜ kD ˜ H ˜ (0) v k
k
˜ k(0) mk (i) = v wk (i) = Hk mk (i) end i=i+1 Stop if mk (i) − mk (i − 1) < mk = mk (i) wk = wk (i) STEP 3: Quantization 21: m ˆ k = arg maxci ∈C H 2 ˘ |cH i Hk Hk mk |
22: 23: 24: 25: 26: 27: 28:
cH i (
PK
l=1,l=k
¯
H 2 HH Hk ml mH HH Hk )ci +cH i Hk Hk ci σ k l k
w ˆ k = Hk m ˆk STEP 4: Mitigation of Inter-user Interference for k = 1 :HK hk = w ˆh k Hk iT T T ˜ k = hT . . . hT H hk+1 . . . hK 1 k−1 ˆ (1) ˜H ˜k V ˜k = U ˜ kD ˜ H ˜ (0) v (0)
k
˜k mk = v end Final TX/RX Beamformers 29 : TX BF ⇒ mk 30 : RX BF ⇒ w ˆk
k
IV. ACHIEVABLE S UM R ATE L OSS In this section, achievable rate bounds are derived to compare the achievable sum rate for the proposed coordinated beamforming with/without quantized codebooks assuming
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equal power allocation across users for the Rayleigh fading channel. A. Sum Rate Upper Bound with Unquantized Beamformers After nulling inter-user interference, the transmitter needs to transmit the data stream through the best eigenmode of the resulting equivalent channel. Hence the coordinated beamforming is upper bounded by miltiple interference free K links with Nr × Nt antennas (i.e., no inter-user interference and achieving the largest singular value of the channel) [15]. The upper bound of the ergodic achievable rate is given by
P¯ λ1 RCBF,upper,unquant = KE log2 1 + 2 (6) σ where P¯ is the normalized transmit power at each antenna. The characteristics of the maximum eigenvalues of HH k Hk are well discussed in [16]. Since we assume only beamforming scenario, we just need to know the best eigenvalue distribution to analyze the achievable rate. Thus the probability density function (pdf) of ordered eigenvalue of HH k Hk is 1 pλ1 (λ1 ) = s Πk=1 Γ(t − k + 1)Γ(s − k + 1) × |Ψc (λ1 )|tr(Ψ−1 c (λ1 )Φc (λ1 ))U (λ1 )
t+s−i−j −λ1
e
(N, N, N ) with 3bit and 6bit codebook, SNR= 10dB
where g(z, λ1 , · · · , λm ) =
(Nt − 1)! γ 2(Nt −1)
u1
···
uNt −1
du1 · · · duNt −1
z−λ
where = min(Nt , Nr ) and t = max(Nt , Nr ), Γ(z) =
∞ −ts z−1 e t dt, Re(z) > 0 and Φc (λ1 ), Ψc (x) are s × s 0 matrices whose entries are given by respectively, {Φc (λ1 )}i,j = (λ1 )
Fig. 2.
, i, j = 1, 2, · · · , s
{Ψc (x)}i,j = γ(t + s − i − j + 1, x), i, j = 1, 2, · · · , s. γ(·, ·) is the incomplete gamma function and is given by x γ(a, x) = ta−1 e−t dt, (7) 0
and U (·) is the unit step function. With above equations, (6) can be evaluated with a numerical integrations. B. Sum Rate Upper Bound with Quantized Beamfomers With a codebook M of size Nb (we use M instead of C for the codebook set for notational consistency with the algorithm), the distribution function of the receive SNR in a MIMO system with Nt transmit antennas and Nr receive antennas is lower bounded by Fϑ (z) such that H ˆH P rob(maxm ˆ k ∈M |m k Hk Hk mk | < z) ≥ Fϑ (z) where the random variable ϑ is the instantaneous quantized SNR, H ∼ ˆ k 2 ˆH ϑ = maxm ˆ k ∈M |m ˆ k ∈M Hk m k Hk Hk mk | = maxm with large Nb approximation. ⎧ 2 ⎪ , 2 ≤ i ≤ Nt − 1 (1) 0, λ1 ≥ z, λi ≥ z−λ1γ(1−γ) 2 ⎪
⎪ ⎪ ⎪ ⎨ (2) λ1 · · · λNt −1 pλ (λ1 , · · · , λNt −1 ) Fϑ (z) = ⎪ dλ 1 · · · dλ Nt −1 , λ1 ≤ z ⎪ ⎪ ⎪ ⎪ (3) λ1 · · · λNt −1 g(z, λ1 , · · · , λm ) ⎩ pλ (λ1 , · · · , λNt −1 )dλ1 · · · dλNt −1 , otherwise
where u1 ∈ [1 − γ 2 , λ1 −λNNt ], ui ∈ [0, min(1 − t P i−1 (z−λNt − i−1 j=1 (λj −λNt )uj u , )] for 2 ≤ i ≤ Nt − 1, j j=1 (λi −λNt ) (Nt −1)Nb 1 γ = 2 Nt (Nb −1) and pλ (λ1 , · · · , λNt −1 ) is the probability density function of the ordered eigenvalues of HH k Hk [17]. The upper bound on the ergodic achievable of coordinated beamforming with quantized codebook is given ¯ by RCBF,upper,quant = KE[log2 (1 + Pσ2ϑ )], where ϑ = ˆ k 2 .Then we can write, maxm ˆ k ∈M Hk m ∞ P¯ P rob log 1 + 2 ϑ > z dz RCBF,upper,quant =K σ 0 ∞ 2z − 1 =K P rob ϑ > ¯ 2 dz. P /σ 0 The ergodic achievable rate, therefore, is upper bounded by ∞ 2z − 1 RCBF,upper,quant ≤ K 1 − Fϑ ¯ 2 dz. P /σ 0 We define the achievable rate loss due to quatization as RCBF,loss = RCBF,upper,unquant − RCBF,upper,quant (8) and this achievable rate loss due to quantization will be compared with Monte Carlo simulations in Section V.
V. S IMULATION C OMPARISON To compare the best achievable rate with the proposed algorithm, we assume that the base station performs power allocation across the parallel channels created by the transmit and receive beamforming vectors. Let us assume that the
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VI. S UMMARY AND F UTURE W ORK In this paper, jointly optimized linear transmit beamforming and receive combining for the downlink of a multiuser MIMO systems was investigated. An algorithm for finding the beamforming and combing vectors was derived assuming that the combining vector is quantized so that it can be sent on a feedforward control channel to each user. The impacts of quantization loss of the limited feedforward messaging were discussed. Our approach is a practical solution to the multiuser joint beamforming and combining problem for time division duplexing systems where reciprocity provides channel state information at the transmitter. As the next step, we will consider the combination of limited feedback and limited feedforward quantization, to allow our approach to operate with only quantized channel state information at the transmitter. Fig. 3.
Capacity loss due to quantization for (2, 2, 2) scenario
ACKNOWLEDGMENT AWGN variance at the receiver is σ 2 . The maximum achievable sum rate is obtained through waterfilling by solving Rk =
K k=1
log2
Pk H 2 ˆ k Hk mk | [bps/Hz]. 1 + 2 |w σ
(9)
subject to a sum power constraint k Pk = P . We deviate from the unit power assumption since we compare with dirty paper coding, which also has a sum power constraint. Note that we still use the same algorithms. In this paper, we compare the achievable rate of the proposed scheme with single user open/closed loop MIMO [18], independent stream scheduler [19], and sum capacity by DPC [2] [3]. We show the achievable sum rate of our methods when the number of users in the service area of the base station is exactly equal to the number of transmit antennas for simplicity. Fig. 2 shows sum rates for DPC, CBF with/without codebook, and independent stream scheduler as a function of (N, N, N ), where each N means the number of transmit antennas, the number of receive antennas per user, and the number of users, respectively. It is very noticeable that the gap between the sum capacity achieved by DPC and the sum rate of the proposed linear solution is only 1.6 bps/Hz in (5, 5, 5) scenario. Note that the independent stream scheduler requires only partial channel state information while the others assume the perfect/full channel state information at the transmitter. Coordinated beamforming with a 3 bit or 6 bit Grassmannian codebook still shows the good performance compared to independent stream scheduler [19]. Fig. 3 illustrates that the derived quantization loss is well matched with simulation result. From our results, we realize that the performance of CBF with limited feedforward is very close to that of coordinated beamforming with unquatized receive beamformers, with the difference that the proposed scheme can be implemented in a real system with a TDD mode.
This work was supported by Samsung Electronics. R EFERENCES [1] G. Caire and S. Shamai (Shitz), “On the achievable throughput of a multi-antenna gaussian broadcast channel,” IEEE Trans. Info. Th., vol. 43, pp. 1691 – 1706, July 2003. [2] S. Vishwanath, N. Jindal, and A. Goldsmith, “Duality, achievable rates, and sum capacity of gaussian MIMO broadcast channels,” IEEE Trans. Info. Th., vol. 49, pp. 2658–2668, Aug. 2003. [3] H. Weingarten, Y. Steinberg, and S. Shamai (Shitz), “The capacity region of the gaussian multiple-input multiple-output broadcast channel,” IEEE Trans. Info. Th., vol. 52, pp. 3936–3964, Sep. 2006. [4] L. Choi and R. D. Murch, “A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach,” IEEE Trans. Wireless Comm., vol. 2, pp. 773–786, July 2003. [5] B. Farhang-Boroujeny, A. L. Swindlehurst, and M. Haardt, “Layering techniques for space-time communications in multi-user networks,” Proc. IEEE Veh. Technol. Conf., vol. 2, pp. 1339–1342, Oct. 2003. [6] Q. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Sig. Proc., vol. 52, pp. 462–471, Feb. 2004. [7] Z. Pan, K.-K. Wong, and T.-S. Ng, “Generalized multiuser orthogonal space-division multiplexing,” IEEE Trans. Wireless Comm., vol. 3, pp. 1969–1973, Nov. 2004. [8] B. M. Hochwald, C. B. Peel, and A. L. Swindlehurst, “A vectorperturbation technique for near capacity multiantenna multiuser communication - part I: channel inversion and regularization,” IEEE Trans. Comm., vol. 53, pp. 195–202, Jan. 2005. [9] M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Trans. on Veh. Technol., vol. 53, pp. 18–28, Jan. 2004. [10] Z. Shen, J. G. Andrews, R. W. Heath Jr., and B. L. Evans, “Low complexity user selection algorithms for multiuser MIMO systems with block diagonalization,” IEEE Trans. Sig. Proc., vol. 54, pp. 3658–3663, Sep. 2006. [11] N. Jindal, “MIMO broadcast channels with finite rate feedback,” IEEE Trans. Info. Th., vol. 52, pp. 5045–5059, Nov. 2006. [12] D. J. Love, R. W. Heath Jr., and T. Strohmer, “Grassmannian beamforming for multiple-input multiple-output wireless systems,” IEEE Trans. on Info. Theory, vol. 49, pp. 2735–2747, Oct. 2003. [13] R. Chen, R. W. Heath Jr., and J. G. Andrews, “Transmit selection diversity for unitary precoded multiuser spatial multiplexing systems with linear receivers,” to appear in IEEE Trans. Sig. Proc. [14] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “On beamforming with finite rate feedback in multiple-antenna systems,” IEEE Trans. Comm., vol. 49, pp. 2562–2579, Oct. 2003. [15] C.-B. Chae, R. W. Heath Jr., and D. Mazzarese, “Achievable sum rate bounds of zero-forcing based linear multi-user MIMO systems,” Proc. of Allerton Conf. on Comm. Control and Comp., Sep. 2006. [16] M. Kang and M.-S. Alouini, “Largest eigenvalue of complex Wishart matrices and performance analysis of MIMO MRC systems,” IEEE Jour. Select. Areas in Comm., vol. 21, pp. 418–26, Apr. 2003. [17] B. Mondal and R. W. Heath Jr., “Performance analysis of quantized beamforming MIMO systems,” to appear in IEEE Trans. Sig. Proc. [18] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications, Cambridge University Press, 2003. [19] R. W. Heath Jr., M. Airy, and A. J. Paulraj, “Multiuser diversity for MIMO wireless systems with linear receivers,” Proc. of Asilomar Conf. on Sign., Syst. and Computers, pp. 1194–1199, Nov. 2001.
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