Hermitian Precoding for Distributed MIMO Systems with Individual Channel State Information

Jianwen Zhang, Xiaojun Yuan and Li Ping

Department of Electronic Engineering City University of Hong Kong Reference

. J. Zhang, X. Yuan, and Li Ping, “Hermitian prcoding for distributed MIMO systems with individual channel state information,” IEEE J. Sel. Areas Commun., vol. 31, no. 2, Feb. 2013.

2 Outline

. Background

. Hermitian Precoding . Optimization and Optimality . Numerical Results . Conclusion

3 Outline

. Background

. Hermitian Precoding . Optimization and Optimality . Numerical Results . Conclusion

4 Cellular system . A cellular network provides coverage of the entire area by dividing it into cells. . There are a number of fixed base-stations (BS), one for each cell. . A mobile communicates with the BS close to it.

BS3

BS1

BS2

5 Multiple Input Multiple Output (MIMO)

Single antenna Multiple antennas

More connections between the transmitter and the receiver in MIMO system. Thus more data can be transmitted.

6 Problem at cell edge

. Cell-edge users have poor signal strength since they are far away from the BS. . Base station cooperation can improve the signal strength.

BS 2

BS 1

7 Base-station cooperation

Better signal quality and higher transmission rate can be achieved through base station cooperation.

8 Channel state information at Transmitter (CSIT)

H BS 1 1

BS 2 H User 2

If full channel state information at transmitter (CSIT) (i.e., H1 and H2) is available for all BSs, we can regard the overall system as an overall MIMO one. The problem is well studied. R. Zhang, “Cooperative multi-cell block diagonalization with per-base- station power constraints,” IEEE J. Sel. Areas Commun., vol. 28, no. 9, pp. 1435-1445, 2010. 9 However, full CSIT is very costly …

We need an extra link between two BSs to share their CSIT.

10 Full CSIT is very costly …

Or CSIT is obtained through the receiver.

11 Full CSIT is very costly …

Both these methods reduce the power and bandwidth efficiency.

12 Full CSIT is very costly …

This problem becomes more severe when more BS cooperation are involved.

13 Individual CSIT

BS 1 H1

BS 2

. Individual CSIT: Each BS k only has individual CSIT (Hk), but has no knowledge about {Hkʹ, kʹ ≠ k}. . No channel information exchange between BSs is required. . Individual CSIT is particular suitable for TDD.

14 Time Division Duplex (TDD)

Individual CSIT can be obtained by channel estimation from the uplink channel in time-division duplex (TDD) systems. No feedback link is required.

15 Distributed MIMO

BS 2

BS 1

A distributed MIMO system refers to one with individual channel state information available at each BS.

16 Challenging issues

. What is the maximum rate for the distributed MIMO system with individual CSIT?

. How to achieve this maximum rate or what is the optimal transmission strategy?

17 Outline

. Background

. Hermitian Precoding . Optimization and Optimality . Numerical Results . Conclusion

18 Our main contributions

. What is the maximum rate for the distributed MIMO system with individual CSIT? . We provide an upper bound and a lower bound to approximate the maximum rate.

. How to achieve this maximum rate or what is the optimal transmission strategy? . We propose a local optimal transmission strategy which performs very close to the maximum rate.

19 System model

channel 1 precoder 1 received signal H F 2 1 1 message s  y Hkk F s n k1 H2 F2 n channel 2 precoder 2

H: channel matrix. n: channel noise vector. F: precoding matrix that defines the transmission strategy.

20 Objective

channel 1 precoder 1 received signal H F 2 1 1 s y Hkk F s n k1 H F n 2 2 channel 2 precoder 2

Our objective is to maximize transmission rate based on individual CSIT, but how?

21 System rate and eigenvalues

channel 1 precoder 1 received signal H F 2 1 1 s y Hkk F s n k1 H F n 2 2 channel 2 precoder 2 . System rate is given by: 1 E log 1 l 2  2 i 

where {li} are eigenvalues of SHkFk. . Large eigenvalues generally lead to better performance.

. The problem becomes how to design Fk to increase the eigenvalues of SHkFk? 22 Basic design strategy

channel 1 precoder 1 received signal H F 2 1 1 s y Hkk F s n k1 H F n 2 2 channel 2 precoder 2 . From matrix theory, if A, B are positive semi-definite matrices, the addition A+B results in increased eigenvalues, i.e.,

lii ABAB  l   l1   . The above inequality does not hold if A and B are not Hermitian positive semi-definite matrices. In particular, we may have A+B = 0, which means that cooperation may lead to worse performance.

. This fact motivates our design strategy to make {HkFk} Hermitian positive semi-definite. 23 Details of Hermitian precoding

channel 1 precoder 1 received signal H F 2 1 1 s y Hkk F s n k1 H F n 2 2 channel 2 precoder 2

H . Let Hk = UkDkVk . . A Hermitian precoder is given by H Fk = VkWkUk .

. Let Wk be non-negative diagonal. Then

H HkFk = UkDkWkUk is Hermitian positive semi-definite.

24 . This, intuitively, insures large eigenvalues for SHkFk . Hermitian precoding

channel 1 precoder 1 received signal H F 2 1 1 s y Hkk F s n k1 H F n 2 2 channel 2 precoder 2

H H Hk = UkDkVk Fk = VkWkUk

Note that, given Hk, the structure of Fk is fixed except Wk.

25 Outline

. Background

. Hermitian Precoding . Optimization and Optimality . Numerical Results . Conclusion

Optimality of Hermitian precoding

The discussions above show that, intuitively, the Hermitian precoder structure can provide improved performance. However, rigorous proof for the global optimality of Hermitian precoding scheme is difficult, since the problem is not convex. In the reference below, we prove the local optimality for the Hermitian precoder.

J. Zhang, X. Yuan, and Li Ping, “Hermitian prcoding for distributed MIMO systems with individual channel state information,” IEEE J. Sel. Areas Commun., vol. 31, no. 2, Feb. 2013.

Optimality of Hermitian precoding

Theorem 1:

The Hermitian precoder structure is locally optimal for the rate maximization problem in a parallel relay network with individual CSIT and individual power constraint.

channel 1 precoder 1 received signal H F 2 1 1 s y Hkk F s n k1 H F n 2 2 channel 2 precoder 2

H H Hk = UkDkVk Fk = VkWkUk

Mathematical details of Theorem 1

Theorem 1:

H A locally optimal solution for {Wk} and {Σk Σk } to the following problem are real diagonal matrices.

KK2 1 22  max E logdetIHFHHHGHHH2 kk  kkkfg  (  )  kk  kkk   (  ) k FGkk,    k k k k s.t. tr{FFGGHH } P k k k k k

HH with FHVWUGHV kkkfg   kkk and kkk    kkkΣ U

29 Proof of Theorem 1 Proof: We first consider the case of K = 2. Since all transmitters experience the same channel distribution, these two transmitters are symmetric. Hence, we take transmitter 1 as an example. The objective function of the optimization problem can be equivalently rewritten as below:

max  (Λ11 ,W ) Λ ,W  : trΛ P , WWH Λ 1 1 1 1 1 1 1 with Λ W H 1 11D 1   Λ1,WIDHFH 1  E logdet  1 2 22  1  2 WIFGH1  FHHH 1 2 2 22

HHH where Λ1VFFGGV 1 1 1 1 1 1

30 Proof of Theorem 1 The following lemmas are necessary for the proof of theorem 1.

Lemma 1: max  Λ1 ,W 1 max  Λ 1 ,W 1  Λ1: trΛ 1PP 1 Λ 1 :trΛ 1 1 , HH WWW1: 1 1Λ 1 WWW:     Λ  1 1diag 1 diag 1 diag

Lemma 2: For any positive semi-definite matrix Λ1 and any matrix W1, we have  Λ ,,W    Λ  W  1 1 1diag 1 diag  Lemma 3: The optimal solution to the problem below is achieved at real- valued Λ1 and W1. max  Λ ,W H  11 Λ1,W 1 :trΛ 1P 1 , WW 1 1Λ 1 Λ11 and W are diagonal We skip the proof of Lemma 1 – Lemma 3 here.

Proof of Theorem 1

From Lemma 1 – Lemma 3, we have: max   Λ ,W  H 11 Λ1,W 1 : trΛ 1P 1 ,WW 1 1Λ 1 using Lemma 1  max   Λ11 ,W  Λ ,W  :trΛ P , WW  H  Λ  1 1 1 1 1diag 1 diag 1 diag using Lemma 2  max  Λ , W H  11diag  diag  Λ ,W :trΛ P , WW Λ  1 1  1diag 1 1 diag 1 diag 1  diag

using Lemma 3  max  Λ ,W H  11 Λ1,W 1 :trΛ 1P 1 , WW 1 1Λ 1 Λ11 and W are real diagonal

32 Hermitian precoding

channel 1 precoder 1 received signal H F 2 1 1 s y Hkk F s n k1 H F n 2 2 channel 2 precoder 2

H H Hk = UkDkVk Fk = VkWkUk

Note that, given Hk, the structure of Fk is fixed except Wk. Performance can be improved by optimizing Wk.

Optimization for Wk Theorem 2:

w1   Let W  wi    wN

The optimal wi is the root of following equation:

2 3 2 2 2 ldi w i20 l d i w i  l  l v  d i w i   d i 

This equation is easy to solve since it is an univariate cubic equation of wi. Thus the complexity is low.

Mathematical details of Theorem 2

Theorem 2:

Denote by wi the diagonal elements of Wk and di the diagonal elements of Dk. The optimal wi is the root of following equation: 2 3 2 2 2 ldi w i20 l d i w i  l  l v  d i w i   d i 

M wP2  where λ ≥ 0 is chosen to satisfy the power constraint  i=1 ik , μ and ν are given as: 1 K   E trDWkk M k2 KK2 1 2  E tr DWDk k kΣ k M  kk22

Proof of Theorem 2

Proof: Due to the symmetry of the transmitters, it suffices to study the optimization problem with k =1. The objective of the optimization problem is given by:

KK2 1 22 E logdetIUDWUUDWUUDHHHH  Σ UUD  Σ UH  2 1 1 1 1k k k k 1 1 1 1 k k k k 1 kk22

KK2 1 2 2 =E logdetIDWUUDWUUD HHHH Σ  UUD Σ UUD  2 1 1 1k k k k 1 1 1 1 k k k k 1 1 kk22

KK2 1 2 2 =E logdetIDWUDWUD HH Σ  UDΣ UD 2 1 1k k k k 1 1 k k k k 1  kk22  The above equations follows from the properties 2 and 3 in Section II-B. Then the related optimization problem is given by:

KK2 1 2 2 max E logdetIDWUDWUD HH Σ  UDΣ UD 2 1 1k k k k 1 1 k k k k 1 W11,Σ  :  22  kk22 tr{W1Σ 1 } P 1  36

Proof of Theorem 2

The above problem is difficult to solve due to its non-convexity nature. Thus, instead of directly solving it, we resort to a widely used upper-bound technique (cf., [16]). From the Jensen’s inequality (as logdet(∙) is concave), the objective function is upper-bounded by

KK2 1 2 2  W ,Σ   logdetIDWUDWUD  E  HH Σ  UDΣ UD 1 1 2 1 1k k k k 1 1 k k k k 1 kk22 1 HHH logdetID 2  1ΛDDWWDI 1  1 1   1 1     The optimization problem is rewritten by: M 1 2 max log 12 di  2  d i w i    M 2   P1 , wi  , i  1,..., M i1  i =1

where wi, di, and Λi are the ith diagonal element of W1, D1, and Λ, respectively. It is seen that the objective in (19b) is an increasing function 2 2 of wi. The optimal wi is w ii   . From (17), S 1, ii = Λi − wi = 0, where Σ1i,i stands for the (i, i)th element of Σ1. Therefore, for the problem in (15), the optimal Σ1 is Σ1= 0. 37

Proof of Theorem 2

With this result, the optimization problem of W1 is given as: M 1 max log 1d22 w  2 d w  M  2  i i i i  wP2  ,  i =1 i 1 i1  We next solve it using the Karush-Kuhn-Tucker (KKT) conditions. The corresponding KKT conditions are given by 2 3 2 2 2 ldi w i20 l d i w i  l  l v  d i w i   d i 

M ll(wP2  )  0 and  0 i=1 i 1

M w2  P and w 0 i=1 ii1 The above KKT conditions can be numerically solved, which yields the optimal W1.

38 Outline

. Background

. Hermitian Precoding . Optimization and Optimality . Numerical Results . Conclusion

39 Numerical results

9 Full CSIT in [1] 8 Hermitian PrecodingFull CSIT in [1] ) No-CSIT in [2] z h / 7 SVD-WF in [3] c . Close to the full CSIT case. e s / t i

b Hermitian ( 6 .

e Outperforms other reference t a R

e schemes. l 5 b a v e i . The performance of Hermitian h

c 4 A

e precoding is very close to that g a

r 3 e

v with full CSIT. A SVD-WF in [3] 2 . No-CSIT means no CSIT. 1 No-CSIT in [2] . SVD-WF means individual SVD 0 plus water-filling -10 -8 -6 -4 -2 0 2 4 6 8 10 SNR(dB) 2 M = 2, N = 2, K = 2, SNR = ΣPk/σ with Pk = P. [1] H.-F. Chong, M. Motani, and F. Nan, “Transmitter optimization for distributed Gaussian MIMO channels,” in Proc. Inf. Theory and Applications Workshop (ITA), 2010, San Diego, CA, Feb. 2010. [2] J. N. Laneman, and G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inf. Theory, vol. 49, no. 10, Oct. 2003. [3] T. M. Cover, and J. A. Thomas, Elements of Information Theory, John Wiley and Sons, 1991. 40 Numerical results

35 Full CSI in [1] . K: The number of cooperation Hermitian Precoding 30 BSs.

) SVD-WF in [3] z h / K = 8 c e s / t 25 i b (

. The performance of Hermitian

e K = 4 t a

R 20 e l precoding improves with K. b a v e K = 2 i h c 15 A e g a r e v 10 . The performance of the A reference schemes do not 5 improve when K increases.

0 This is because the reference -10 -8 -6 -4 -2 0 2 4 6 8 10 SNR (dB) scheme cannot guarantee that M = 4, N = 16 in Rayleigh-fading distributed the eigenvalues of SHkFk are 2 MIMO channels. SNR = ΣPk/σ with Pk = P. increased.

41 Numerical results

90 Full CSIT 80 Hermitian Precoding . M: The number of antennas

) SVD-WF

z at each user. h / 70 c e

s M = 16 / t i Full CSIT and . N: The number of antennas b (

60 e t

a Hermitian at each BS. R

e l 50 b

a . The gap between Hermitian v SVD-WF e i h c 40 precoding scheme and the A

e g a

r 30 full CSIT case diminishes e v A M = 4 with N increase. 20

10 M = 2

0 0 50 100 150 200 250 300 Antenna Number per Transmitter (N) 2 SNR = ΣPk/σ with Pk = P. K = 3.

Practical receiver

channel 1 precoder 1 H F FEC LMMSE 1 1 s FEC encoder Detector encoder H2 F2 n iterative receiver channel 2 precoder 2 transmitters

With Hermitian precoding, the received signal from different transmitters can no -longer be separated into orthogonal streams. Interference among different symbols is inevitable. Iterative detection can be used for interference cancelation in this case. Detail discussions can be found in X. Yuan, Q. Guo, X. Wang and Li Ping, “Evolution analysis of low-cost iterative equalization in coded linear systems with cyclic prefixes,” IEEE Journal of Select. Areas in Communications, February 2008. 43

Simulation with practical coding 0 10

SVD-WF . Hermitian precoding has

-1 lower transmission 10 )

R error.

E Hermitian Precoding F (

e t a R

r -2 o

r 10 r E

e m a r F

-3 10

Full CSIT Capacity -4 10 -4 -2 0 2 4 6 8 SNR (dB) Rayleigh fading channel with K = 2, N = 16, and M = 16. Regular (3, 6) LDPC codes with codeword length 8192 are used. QPSK used for modulation. SNR = ΣP /σ2 with P = P. k k 44 Outline

. Background

. Hermitian Precoding . Optimization and Optimality . Numerical Results . Conclusion

45 Conclusions

. A Hermitian precoding technique is proposed for distributed MIMO systems with individual CSIT.

. We have proved that Hermitian precoding is locally optimal.

. Hermitian precoding performs close to the full CSIT case.

. Hermitian precoding outperforms other reference schemes.

. We conjecture that Hermitian precoding is also globally optimal. However, its proof is still an open problem.

46 End of the presentation

Thank you. Intuition from single antenna case

For single antenna case, the received signal y = (h1f1+h2f2)s+n.

Im Optimal f1 is to compensate the phase of h1 and make h1f1 into positive. h1 Each term is real and positive h f +h f which lead to optimal received 1 1 2 2 Re signal power.

h Optimal f is to compensate the phase of h and 2 2 2 make h2f2 into positive.

In this case, the optimal strategy for fk is to make hkfk into real and positive. MIMO ?

48 Time Division Duplex (TDD)

Uplink Downlink

H H

. In TDD systems, the channels of the uplink and the downlink are assumed to be symmetric. . Thus the CSI of the downlink can be obtained by channel estimation from the uplink channel. No feedback link is required.

49 Time Division Duplex (TDD)

. Step 1: Individual CSI can be obtained by channel estimation from the uplink channel in time-division duplex (TDD) systems. No feedback link is required.

50 Time Division Duplex (TDD)

[CSI]

. Step 1: Individual CSI can be obtained by channel estimation from the uplink channel in time-division duplex (TDD) systems. No feedback link is required.

. Step 2: CSI sharing among transmitters. This incurs bandwidth cost as well as delay.