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Tadilo Endeshaw, Batu Chalise and Luc Vandendorpe Université catholique de Louvain (Belgium) 12-Jun-14 1 Tadilo Endeshaw, Université catholique de Louvain, Belgium (CAMSAP 09)

MSE uplink-downlink duality of MIMO systems under imperfect CSI

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Full detail has been published in CAMSAP 2009 conference.

Text of MSE uplink-downlink duality of MIMO systems under imperfect CSI

  • 1. Tadilo Endeshaw, Batu Chalise and Luc Vandendorpe Universit catholique de Louvain (Belgium) 12-Jun-14 1Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09)

2. Presentation Outline Motivation of establishing MSE duality MSE duality under imperfect CSI As an application example of MSE duality under imperfect CSI, we examine Robust sum MSE minimization problem Proposed duality based iterative solution (alternating optimization) Simulation results Conclusions 12-Jun-14 2Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) 3. 12-Jun-14 Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) 3 K21 B,,B,BB T T K T 2 T 1 n,,n,nn 0,1~d NC 1S k k d C Motivation of MSE duality Consider the following downlink system model )(W K21 W,,W,Wblkdiag kk SM kW CkSN kB C 4. Motivation of MSE duality contd For the above downlink system model, the instantaneous mean square error (MSE) between and is given by Assume we are interested to solve the following problem 12-Jun-14 4Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) k ^ d kd 5. Direct treatment of the above problem has Complicated mathematical structure. Difficult to examine. Now, let us also see the following uplink system model 12-Jun-14 5Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) Motivation of MSE duality contd K21 T,,T,TT )(V K21 V,,V,Vblkdiag K21 H,,H,HH 6. 12-Jun-14 Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) 6 Motivation of MSE duality contd The above uplink problem has - Simple mathematical structure. - Global optimal solution. 7. For any given , if we can get proper scaling factors , such that (or any other combination) we conclude that, global optimal solution of the downlink problem is guaranteed. 12-Jun-14 7Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) Motivation of MSE duality contd 8. Such approach of solving the downlink problem is called duality based approach Thus, duality based approach of solving the downlink problem has two benefits Simple mathematical structure. Exploit the hidden convexity of the downlink problem. Existing work on duality based approach for solving the downlink problem assume that perfect CSI is available at the BS and MSs. 12-Jun-14 8Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) Motivation of MSE duality contd 9. MSE duality under imperfect CSI In this work we establish three kinds of MSE dualities. Namely: Sum MSE duality User wise MSE duality and Symbol wise MSE duality when imperfect CSI is available at the BS and MSs. Then, as an application example we examine the robust sum MSE minimization problem Utilize Bayesian robust design approach 12-Jun-14 9Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) 10. Channel modeling Considering antenna correlation at the BS, we model the Rayleigh fading channel as When MMSE channel estimation is employed at the MSs, can be expressed as where is the estimated channel and - We establish MSE duality for any - Then, we solve the following robust design problem. where is the kth user AMSE. 12-Jun-14 10Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) 11. AMSE transfer from uplink to downlink Sum AMSE transfer: The sum AMSE of the uplink and downlink channels are given by If we choose , with we can achieve 12-Jun-14 11Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) 12. AMSE transfer from uplink to downlink Like the above transformations, we can also transfer the kth user and lth symbol AMSEs from downlink channel to uplink channel. By similar approach, we can transfer the AMSE from downlink to uplink channel. 12-Jun-14 12 Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) Application example 13. Application example contd 12-Jun-14 13Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) For convenience, consider the problem in the following 2 cases Case 1: When Case 2: For any Case 1: In such a case, the robust sum MSE minimization problem in the uplink channel can be formulated as a semi-definite programming (SDP) problem for which Global optimum is guaranteed. Consequently, global optimum of the original downlink problem is guaranteed by using our sum AMSE transfer. 14. 12-Jun-14 14Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) Case 2: The robust problem cannot be formulated as an SDP problem. Thus, the solution method discussed for Case 1 cannot be applied here. Hence, we propose the alternating optimization technique. To do this we decompose the precoders and decoders as Thus, the new equivalent uplink and downlink system models become Application example contd 15. 12-Jun-14 Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) 15 Application example contd 16. By collecting the powers and filter matrices as where are the filters for the lth symbol with , the AMSE of the lth symbol in the uplink channel can be written as 12-Jun-14 16Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) Application example contd 17. where 12-Jun-14 17Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) and f(i) is the smallest k, s.t, . For fixed , the power allocation part of the robust sum MSE minimization problem is expressed as Application example contd 18. 12-Jun-14 18Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) Since is a Posynomial, the above optimization problem is a Geometric programming (GP), for which - Global optimal solution is guaranteed. - Solved with a worst-case Polynomial time complexity. Application example contd 19. 12-Jun-14 19Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) Thus, our alternating optimization is performed as follows Uplink channel - First we get optimal Q by solving the GP problem. - With optimal Q of the GP, are updated by MMSE receiver Downlink channel - Now, we first ensure the same performance as the uplink channel by using the sum AMSE transfer (i.e., uplink to downlink channel). This is achieved by choosing and Application example contd 20. 12-Jun-14 20Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) - With the optimal , are updated by MMSE receiver. Uplink channel - First we ensure - With the optimal , Update by MMSE receiver. Application example contd 21. Simulation result (For case I) Comparison of GM and Alg I, K=2, N=4 and =2 K : # of users N : # of BS antennas : user ks # of antennas Alg I: The proposed alternating optimization 12-Jun-14 21Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) kM kM We model Rc as 22. 12-Jun-14 Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) 22 Simulation result (For case II) Comparison of robust and naive designs K=2, N=4 and = 2kM 23. Conclusions In this work we establish 3 types of MSE duality under imperfect CSI. As an application example robust sum MSE minimization Our robust design has better performance than the non-robust/naive design. Large antenna correlation factor further increases the sum AMSE of the downlink system. 12-Jun-14 23 Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09) 24. Thank You! 12-Jun-14 24Tadilo Endeshaw, Universit catholique de Louvain, Belgium (CAMSAP 09)