Joint FEC Encoder and Linear Precoder Design for MIMO Systems with Antenna Correlation

Transcription

1 Joint FEC Encoder and Linear Precoder Design for MIMO Systems with Antenna Correlation Chongbin Xu, Peng Wang, Zhonghao Zhang, and Li Ping City University of Hong Kong 1

2 Outline Background Mutual Information Analysis Practical Transmitter Design Conclusions 2

3 Massive MIMO Massive multiple-input multiple-output (MIMO) has attracted extensive research attention. [1]-[2] For massive MIMO, we refer to MIMO systems with an extremely large number of antennas at the base station (BS). The use of the extremely large number of antennas leads to the so called favorable propagation property, i.e., This property results in h ih h j /N 0 for i j. Significantly improved system throughput, and Simplified transmission strategies. 3

4 Practical Concern However, in practice, the favorable propagation property may not hold. The multiple antennas at one user can be correlated due to the limited physical size of the mobile terminal; and It is also reasonable to expect that the MIMO size will become large but may not be massive in the near future. In this presentation, we will consider the uplink transmission of not so large MIMO systems with antenna correlation, which may have useful applications in the evolution path towards massive MIMO. 4

5 Existing Works Existing works on practical transmitter design for MIMO systems are mostly focused on two extremes, i.e., small MIMO that relatively expensive techniques can be afforded [3]-[5] massive MIMO that the favorable propagation property allows low-cost techniques. The system setting considered in this presentation is somewhere between the two extremes. For such a setting, many conventional design techniques become too expensive, but the conditions for massive MIMO cannot yet be fully satisfied. 5

6 Our Research Our focus is on the transmitter design techniques that can provide good comprise between performance and complexity 6

7 Outline Background Mutual Information Analysis Practical Transmitter Design Conclusions 7

8 System Model MIMO system with N R receive antennas and N T transmit antennas: r = Hy+ η Kronecker model to characterize antenna correlation: [6]-[7] For uplink, we assume that N R N T and there is sufficient spacing among receive antennas and so C R does not contain dominant eigen-values. H = C H C 1/2 1/2 R w T 8

9 Asymptotic Analysis For the N R N T matrix H w, when N R, Based on this property, we can show that H lim log2 det( IN + HQH ) N R R R T ( 1/2 1/2 H 1/2 1/2 I ) N Q C H C C H C = lim log det + ( ) ( ) N H C H tr{ C } I = N I. H w R w R N R 2 T R W T R W T 1/2 1/2 2 IN CT QCT NR = log det( + ). where Q is the transmission covariance matrix. T N T 9

10 Asymptotic Analysis (Continued) lim log det( I + HQH ) = log det( I + C QC N ) N R H 1/2 1/2 2 N 2 N T T R R From the above expression, we can seen that the knowledge of C T is sufficient for the transmitter design. The capacity of the system with full channel state information at the transmitter (CSIT) can be approached by a simple statistical water-filling (SWF) scheme [8]-[9], i.e., water-filling over the channel characterized by C T. T 10

11 Numerical Results average rate in bits/symbols Performance of systems with different receive antenna configurations. N T = 8, ρ T = 0.8, and ρ R =

12 Theoretical Guidelines The above results show that it is possible to design transmitter based on only the knowledge of C T. This greatly reduces the CSIT acquisition cost, especially for relatively large MIMO. Based on this observation, a simple SWF scheme can obtain near-optimal performance. Now the question is how to implement SWF in practical systems? 12

13 Outline Background Mutual Information Analysis Practical Transmitter Design Conclusions 13

14 Possible Problems Demodulation Complexity: With SWF, different rates are transmitted on different eigen-directions of C T. For this purpose, adaptive modulation is a standard method. However, the related demodulation complexity can become a problem. Design Complexity: For MIMO systems, the reliabilities of different transmitted symbols are generally different. This makes some common system design techniques complicated, such as extrinsic information transfer (EXIT) chart technique [10]-[11]. 14

15 Main Ideas To reduce the demodulation complexity, we will consider systems with uniform constellation. To facilitate the system design, we will consider a Hadamard precoding structure [12]-[14], which can lead to uniform reliability of all symbols. c uniform constellation ENC x V Had 1/2 Q y H r LMMSE DEC ĉ linear precoder Hadamard matrix channel η iterative receiver 15

16 Hadamard Matrix Roughly, the Hadamard matrix performs a spreading operation. It spreads each symbol in x into all channel realizations and all directions, hence leads to a uniform reliability for all symbols in x. Then the LMMSE equalizer can be characterized by a single variable transfer function. c ENC x V Had 1/2 Q y H linear precoder channel η iterative receiver Hadamard matrix single variable function ρ = φ( v) r LMMSE DEC ĉ 16

17 Transfer Function of LMMSE Equalizer Proposition 1: For N R, the transfer function of the LMMSE equalizer is given by ω ( v) φ( v) = 1 vω ( v) where ω v 1 n T n () =. 2 N T n vλn( CT) λn( Q) + σ / N R This shows that φ(v) is asymptotically determined by C T. This is consistent with the mutual information analysis, except here a practical iterative LMMSE receiver is considered. λ ( C ) λ ( Q) 17

18 Numerical Example φ( v) curves of SWF from top to bottom N =, 16, 8, 4 R The φ(v) curves of SWF for different N R values. N T = 8, ρ T = 0.8, and ρ R = 0.5, and E b /N 0 = 10log10(N R ). 18

19 EXIT Chart Type Analysis Similarly to the LMMSE equalizer, the FEC decoder can also be characterized by a transfer function v = ψ(ρ). The performance of the iterative receiver is determined by a fixed point of the two functions ρ = φ(v) and v = ψ(ρ). LMMSE DEC ρ = φ() v v = ψ( ρ) SINR of LMMSE outputs variance of decoding outputs 19

20 Joint Encoder and Precoder Design Based on the above analysis, the FEC encoder and linear precoder can be jointly designed according to the curve matching principles [15]-[16]. 1 φ( v) of SWF precoder ψρ ( ) of irregular code variance v

21 Simulation Results Simulation performance of the proposed scheme. N T = 8, ρ T = 0.8, and ρ R = 0.5, and rate = 4 bits/symbol. 21

22 Further Consideration In the above discussion, the LDPC code is optimized for every realization of C T. This may incur considerable design and implementation complexity. An alternative is to fix coding for all realizations of C T and only employ adaptive precoding for different C T. Numerical results show that the above approach is effective within a certain statistical range of C T. 22

23 Simulation Results The code is optimized at ρ T = 0.8 and the precoder is adaptive for different ρ T. N T = 8, ρ R = 0.5, rate = 4 bits/symbol. 23

24 Outline Background Mutual Information Analysis Practical Transmitter Design Conclusions 24

25 Conclusions We consider uplink transmission in not so large MIMO systems with antenna correlation. We focus on transmitter design that can provide good comprise between performance and complexity. We develop a joint FEC encoder and linear precoder design that has the following properties. only requiring the knowledge of C T and so low CSIT acquisition cost; uniform constellation and so modest demodulation complexity; closed-form transfer function for LMMSE equalizer and so efficient system design. Good performance is demonstrated by simulation results. 25

26 Questions and Comments Thank you! 26

27 References [1] T. L. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., vol. 9, no. 11, pp , Nov [2] 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, pp , Jan [3] M. Vu, and A. Paulraj, MIMO wireless linear precoding, IEEE Signal Process. Mag., vol. 24, no. 5, pp , Sep [4] C. Xiao, Y. R. Zheng, and Z. Ding, Globally optimal linear precoders for finite alphabet signals over complex vector Gaussian channels, IEEE Trans. Signal Process., vol. 59, no. 7, pp , Jul [5] W. Zeng, C. Xiao, M. Wang, and J. Lu, Linear precoding for finite-alphabet inputs over MIMO fading channels with statistical CSI, IEEE Trans. Signal Process., vol. 60, no. 6, pp , Jul [6] C. N. Chuah, D. N. C. Tse, and J. M. Kahn, Capacity scaling in MIMO wireless systems under correlated fading, IEEE Trans. Inf. Theory, vol. 48, pp , Mar

28 References [7] J. P. Kermoal, L. Schumacher, K. I. Pedersen, P. E. Mogensen, and F. Fredriksen, A stochastic MIMO radio channel model with experimental validation, IEEE J. Sel. Areas Commun., vol. 20, no. 6, pp , Aug [8] E. Yoon, J. Hansen, and A. Paulraj, Space-frequency precoding with space-tap correlation information at the transmitter, IEEE Trans. Commun., vol. 55, no. 9, pp , Sep [9] X. Li, S. Jin, X. Gao, and K. K. Wong, Near-optimal power allocation for MIMO channels with mean or covariance feedback, IEEE Trans.Commun., vol. 58, no. 1, pp , Jan [10] S. ten Brink, Convergence behavior of iteratively decoded parallel concatenated codes, IEEE Trans. Commun., vol. 49, no. 10, pp , Oct [11] S. ten Brink, G. Kramer, and A. Ashikhmin, Design of low density parity-check codes for modulation and detection, IEEE Trans. Commun., vol. 52, no. 4, pp , Apr [12] P. Lusina, E. M. Gabidulin, and M. Bossert, Efficient decoding of space-time Hadamard codes using the Hadamard transform, in Proc. ISIT2001, Washington DC, Jun ,

29 References [13] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, Joint Tx-Rx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization, IEEE Trans. Signal Process., vol. 51, no. 9, pp , Sep [14] X. Yuan, C. Xu, Li Ping, and X. Lin, Precoder design for multiuser MIMO ISI channels based on iterative LMMSE detection, IEEE J. Sel. Topics Signal Process., vol. 3, no. 6, pp , Dec [15] K. R. Narayanan, D. N. Doan, and R. V. Tamma, Design and analysis of LDPC codes for turbo equalization with optimal and suboptimal soft output equalizers, in Proc. Allerton Conf. Commun., Control, and Computing, Monticello, IL, pp , Oct [16] X. Yuan, and Li Ping, Achievable rates of coded linear systems with iterative MMSE detection, in Proc. IEEE Globecom 09, Honolulu, HI, USA, Nov.30 -Dec

30 System Model Kronecker model: H = C H C 1/2 1/2 R w T For simplicity, we further adopt the exponential model in simulation C C R T ( mn, ) ( mn, ) = = ρ ρ m n jm ( n) θ R m n jm ( n) θ T e where ρ R and ρ T are, respectively, receive and transmit correlation factors, and θ R and θ T are uniformly distributed over [0, 2π). ( H ) H tr{ CR}, m= n; lim HwCRHw = lim ( H, w ) m,: CR( Hw ) mn :, n R NR 0, m n. N e R T 30

31 Complexity The complexity of the proposed scheme is considerably lower than the conventional adaptive modulation techniques. The key here is that the worse-case complexity can be very high if variable constellation sizes are involved. To see this, denote the number of bits carried by the symbol on the n-th eigen-direction by Q n, and its average value by Q avrg. Note that Q n can vary in the range of [0, N T Q avrg ], which increases with the MIMO size. Then the demodulation Q complexity O(2 n ) for some large Q n values can be a problem even if the MIMO size is only modestly large. 31

32 MRC Performance MRC over the true channel H MRC over artificial parallel channel N BS = 64 N BS = 32 N BS = E b /N 0 in db Performance of MRC receiver with equal power allocation. N T = 8, ρ T = 0.8, ρ R = 0.5, and rate = 4 bits/symbol. 32

33 Multi-User Results N T = 8, N R = 64, ρ T = 0.8, ρ R = 0.5. total rate is 4 bits/symbol for K = 1, and 16 bits/symbol for K = 4. 33

34 Multi-User Results N T = 8, N R = 64, ρ R = 0.5; γ 1 = γ 2 = 0.5, γ 3 = γ 4 = 2; ρ T, 1 = ρ T,3 = 0.6, ρ T, 2 = ρ T, 4 = 0.8; and target BER = 10-4 in precoder design. 34