Transceiver design for single-cell and multi-cell downlink multiuser MIMO systems
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The document outlines a presentation on transceiver design for single-cell and multi-cell downlink multiuser MIMO systems. It discusses MSE uplink-downlink duality under imperfect CSI, showing that the sum MSE, user MSE, and symbol MSE are dual between the uplink and downlink channels. It demonstrates how to ensure the uplink and downlink MSE values are equal to each other by appropriately setting the transmit covariance matrices. The presentation also covers transceiver design algorithms for coordinated base station systems and generalized duality for multiuser MIMO systems.
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Transceiver design for single-cell and multi-cell downlink multiuser MIMO systems
- 1. Transceiver design for single-cell and multi-cell downlink multiuser MIMO systems Tadilo Endeshaw Bogale University Catholique de Louvain (UCL), ICTEAM Dec. 2013
- 2. Presentation Outline Presentation Outline 1 MSE uplink-downlink duality under imperfect CSI MSE uplink-downlink duality under imperfect CSI Application of AMSE duality Simulation Results Drawbacks and Looking ahead 2 Transceiver design for Coordinated BS Systems Block diagram and Problem formulation Proposed Algorithms Simulation Results Drawbacks and Looking ahead 3 Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements Simulation Results 4 Thesis Conclusions 5 Future Research Directions Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 2 / 24
- 3. MSE duality MSE uplink-downlink duality under imperfect CSI MSE uplink-downlink duality under imperfect CSI (a) d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK (b) H1 H2 HK n TH V1 V2 VK d d1 d2 dK Assumption: CSI model HH k = HH k + R 1/2 mk EH wk R 1/2 bk Objectives: Exploit MSE duality (sum MSE, user MSE and symbol MSE duality) between UL and DL channels Apply duality to solve transceiver design problems Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 3 / 24
- 4. MSE duality MSE uplink-downlink duality under imperfect CSI Sum MSE uplink-downlink duality (a) d1 d2 dK HH 2 HH K αK P −1 2 2 P −1 2 1 P −1 2 K n1 nK n2 dK d2 d1 = d GP 1 2 α1HH 1 α2 UH 1 UH 2 UH K (b) d1 GH d2 dK Q 1 2 1 Q 1 2 2 Q 1 2 K H1 H2 HK n dQ−1 2α U1 U2 UK ξ DL k = tr{ISk + P −1/2 k αk UH k ΓDL k Uk αk P −1/2 k −2ℜ{P 1/2 k GH k Hk Uk αk P −1/2 k }} ΓDL k = σ2 ek tr{Rbk GPGH }Rmk + HH k GPGH Hk + σ2 IMk ξ UL k = tr{ISk + Q −1/2 k αk GH k ΓcGk αk Q −1/2 k −2ℜ{Q −1/2 k αk GH k Hk Uk Q 1/2 k }} ΓUL c = K i=1(σ2 ei tr{Rmi Ui Qi UH i }Rbi + Hi Ui Qi UH i HH i ) + σ2 IN Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 4 / 24
- 5. MSE duality MSE uplink-downlink duality under imperfect CSI Sum MSE uplink-downlink duality (a) d1 d2 dK HH 2 HH K αK P −1 2 2 P −1 2 1 P −1 2 K n1 nK n2 dK d2 d1 = d GP 1 2 α1HH 1 α2 UH 1 UH 2 UH K (b) d1 GH d2 dK Q 1 2 1 Q 1 2 2 Q 1 2 K H1 H2 HK n dQ−1 2α U1 U2 UK ξ DL k = tr{ISk + P −1/2 k αk UH k ΓDL k Uk αk P −1/2 k −2ℜ{P 1/2 k GH k Hk Uk αk P −1/2 k }} ΓDL k = σ2 ek tr{Rbk GPGH }Rmk + HH k GPGH Hk + σ2 IMk ξ UL k = tr{ISk + Q −1/2 k αk GH k ΓcGk αk Q −1/2 k −2ℜ{Q −1/2 k αk GH k Hk Uk Q 1/2 k }} ΓUL c = K i=1(σ2 ei tr{Rmi Ui Qi UH i }Rbi + Hi Ui Qi UH i HH i ) + σ2 IN Given ξDL K k=1 ξ DL k We can ensure K k=1 ξ UL k = ξDL by setting Qk = ˜βα2 k P−1 k with ˜β = K k=1 tr{Pk } K k=1 tr{P−1 k αk } K k=1 tr{Qk } = K k=1 tr{Pk }(Also met) Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 4 / 24
- 6. MSE duality MSE uplink-downlink duality under imperfect CSI Sum MSE uplink-downlink duality (a) d1 d2 dK HH 2 HH K αK P −1 2 2 P −1 2 1 P −1 2 K n1 nK n2 dK d2 d1 = d GP 1 2 α1HH 1 α2 UH 1 UH 2 UH K (b) d1 GH d2 dK Q 1 2 1 Q 1 2 2 Q 1 2 K H1 H2 HK n dQ−1 2α U1 U2 UK ξ DL k = tr{ISk + P −1/2 k αk UH k ΓDL k Uk αk P −1/2 k −2ℜ{P 1/2 k GH k Hk Uk αk P −1/2 k }} ΓDL k = σ2 ek tr{Rbk GPGH }Rmk + HH k GPGH Hk + σ2 IMk ξ UL k = tr{ISk + Q −1/2 k αk GH k ΓcGk αk Q −1/2 k −2ℜ{Q −1/2 k αk GH k Hk Uk Q 1/2 k }} ΓUL c = K i=1(σ2 ei tr{Rmi Ui Qi UH i }Rbi + Hi Ui Qi UH i HH i ) + σ2 IN Given ξDL K k=1 ξ DL k We can ensure K k=1 ξ UL k = ξDL by setting Qk = ˜βα2 k P−1 k with ˜β = K k=1 tr{Pk } K k=1 tr{P−1 k αk } K k=1 tr{Qk } = K k=1 tr{Pk }(Also met) Given ξUL K k=1 ξ UL k We ensure K k=1 ξ DL k = ξUL by setting Pk = βα2 k Q−1 k with β = K k=1 tr{Qk } K k=1 tr{Q−1 k αk } K k=1 tr{Pk } = K k=1 tr{Qk }(Also met) Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 4 / 24
- 7. MSE duality MSE uplink-downlink duality under imperfect CSI User MSE uplink-downlink duality (a) d1 d2 dK HH 2 HH K αK P −1 2 2 P −1 2 1 P −1 2 K n1 nK n2 dK d2 d1 = d GP 1 2 α1HH 1 α2 UH 1 UH 2 UH K (b) d1 GH d2 dK Q 1 2 1 Q 1 2 2 Q 1 2 K H1 H2 HK n dQ−1 2α U1 U2 UK ξ DL k = tr{ISk + P −1/2 k αk UH k ΓDL k Uk αk P −1/2 k −2ℜ{P 1/2 k GH k Hk Uk αk P −1/2 k }} ΓDL k = σ2 ek tr{Rbk GPGH }Rmk + HH k GPGH Hk + σ2 IMk ξ UL k = tr{ISk + Q −1/2 k αk GH k ΓcGk αk Q −1/2 k −2ℜ{Q −1/2 k αk GH k Hk Uk Q 1/2 k }} ΓUL c = K i=1(σ2 ei tr{Rmi Ui Qi UH i }Rbi + Hi Ui Qi UH i HH i ) + σ2 IN Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 5 / 24
- 8. MSE duality MSE uplink-downlink duality under imperfect CSI User MSE uplink-downlink duality (a) d1 d2 dK HH 2 HH K αK P −1 2 2 P −1 2 1 P −1 2 K n1 nK n2 dK d2 d1 = d GP 1 2 α1HH 1 α2 UH 1 UH 2 UH K (b) d1 GH d2 dK Q 1 2 1 Q 1 2 2 Q 1 2 K H1 H2 HK n dQ−1 2α U1 U2 UK ξ DL k = tr{ISk + P −1/2 k αk UH k ΓDL k Uk αk P −1/2 k −2ℜ{P 1/2 k GH k Hk Uk αk P −1/2 k }} ΓDL k = σ2 ek tr{Rbk GPGH }Rmk + HH k GPGH Hk + σ2 IMk ξ UL k = tr{ISk + Q −1/2 k αk GH k ΓcGk αk Q −1/2 k −2ℜ{Q −1/2 k αk GH k Hk Uk Q 1/2 k }} ΓUL c = K i=1(σ2 ei tr{Rmi Ui Qi UH i }Rbi + Hi Ui Qi UH i HH i ) + σ2 IN Given ξ DL k , ξ UL k = ξ DL k , K k=1 tr{Qk } = K k=1 tr{Pk }(ensured) by Qk = ˜βk α2 k P−1 k where ˜T · [˜β1, . . . , ˜βK ]T = σ2 [tr{P1}, . . . , tr{PK }]T , ˜T is constant Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 5 / 24
- 9. MSE duality MSE uplink-downlink duality under imperfect CSI User MSE uplink-downlink duality (a) d1 d2 dK HH 2 HH K αK P −1 2 2 P −1 2 1 P −1 2 K n1 nK n2 dK d2 d1 = d GP 1 2 α1HH 1 α2 UH 1 UH 2 UH K (b) d1 GH d2 dK Q 1 2 1 Q 1 2 2 Q 1 2 K H1 H2 HK n dQ−1 2α U1 U2 UK ξ DL k = tr{ISk + P −1/2 k αk UH k ΓDL k Uk αk P −1/2 k −2ℜ{P 1/2 k GH k Hk Uk αk P −1/2 k }} ΓDL k = σ2 ek tr{Rbk GPGH }Rmk + HH k GPGH Hk + σ2 IMk ξ UL k = tr{ISk + Q −1/2 k αk GH k ΓcGk αk Q −1/2 k −2ℜ{Q −1/2 k αk GH k Hk Uk Q 1/2 k }} ΓUL c = K i=1(σ2 ei tr{Rmi Ui Qi UH i }Rbi + Hi Ui Qi UH i HH i ) + σ2 IN Given ξ DL k , ξ UL k = ξ DL k , K k=1 tr{Qk } = K k=1 tr{Pk }(ensured) by Qk = ˜βk α2 k P−1 k where ˜T · [˜β1, . . . , ˜βK ]T = σ2 [tr{P1}, . . . , tr{PK }]T , ˜T is constant Given ξ UL k , ξ DL k = ξ UL k , K k=1 tr{Pk } = K k=1 tr{Qk }(ensured) by Pk = βk α2 k Q−1 k where T · [β1, . . . , βK ]T = σ2 [tr{Q1}, . . . , tr{QK }]T , T is constant Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 5 / 24
- 10. MSE duality Application of AMSE duality Robust Weighted Sum MSE Minimization (a) d1 d2 dK HH 2 HH K αK P −1 2 2 P −1 2 1 P −1 2 K n1 nK n2 dK d2 d1 = d GP 1 2 α1HH 1 α2 UH 1 UH 2 UH K (b) d1 GH d2 dK Q 1 2 1 Q 1 2 2 Q 1 2 K H1 H2 HK n dQ−1 2α U1 U2 UK minGk ,Uk ,αk ,Pk K k=1 τk ξ DL k s.t K k=1 tr{Pk } ≤ Pmax Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 6 / 24
- 11. MSE duality Application of AMSE duality Robust Weighted Sum MSE Minimization (a) d1 d2 dK HH 2 HH K αK P −1 2 2 P −1 2 1 P −1 2 K n1 nK n2 dK d2 d1 = d GP 1 2 α1HH 1 α2 UH 1 UH 2 UH K (b) d1 GH d2 dK Q 1 2 1 Q 1 2 2 Q 1 2 K H1 H2 HK n dQ−1 2α U1 U2 UK minGk ,Uk ,αk ,Pk K k=1 τk ξ DL k s.t K k=1 tr{Pk } ≤ Pmax Case I : τk = 1, Rmk = I, Rbk = Rb, σ2 ek = σ2 e ⋄ Define Uk = Uk Qk UH k ⋄ Optimize Uk (SDP problem)⋆ ⋄ Get Uk and Qk from Uk ⋄ Update Rx by MMSE and get Gk , αk from Rx Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 6 / 24
- 12. MSE duality Application of AMSE duality Robust Weighted Sum MSE Minimization (a) d1 d2 dK HH 2 HH K αK P −1 2 2 P −1 2 1 P −1 2 K n1 nK n2 dK d2 d1 = d GP 1 2 α1HH 1 α2 UH 1 UH 2 UH K (b) d1 GH d2 dK Q 1 2 1 Q 1 2 2 Q 1 2 K H1 H2 HK n dQ−1 2α U1 U2 UK minGk ,Uk ,αk ,Pk K k=1 τk ξ DL k s.t K k=1 tr{Pk } ≤ Pmax Case I : τk = 1, Rmk = I, Rbk = Rb, σ2 ek = σ2 e ⋄ Define Uk = Uk Qk UH k ⋄ Optimize Uk (SDP problem)⋆ ⋄ Get Uk and Qk from Uk ⋄ Update Rx by MMSE and get Gk , αk from Rx ⋄ Transfer to DL as Pk = βα2 k Q−1 k where β = K k=1 tr{Qk } K k=1 tr{Q−1 k αk } ⋄ Update Rx by MMSE ⋄ Get Uk and αk from Rx Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 6 / 24
- 13. MSE duality Application of AMSE duality Robust Weighted Sum MSE Minimization (a) d1 d2 dK HH 2 HH K αK P −1 2 2 P −1 2 1 P −1 2 K n1 nK n2 dK d2 d1 = d GP 1 2 α1HH 1 α2 UH 1 UH 2 UH K (b) d1 GH d2 dK Q 1 2 1 Q 1 2 2 Q 1 2 K H1 H2 HK n dQ−1 2α U1 U2 UK minGk ,Uk ,αk ,Pk K k=1 τk ξ DL k s.t K k=1 tr{Pk } ≤ Pmax Case II : General τk , Rmk , Rbk , σ2 ek ⋄ Initialize Uk , Qk and get Gk , αk from MMSE Rx ⋄ Decompose Qk = qk ˜Qk , tr{ ˜Qk } = 1 ⋄ Optimize qk (GP problem)⋆ ⋄ Get Qk from qk and ˜Qk ⋄ Update Rx by MMSE and get Gk , αk from Rx Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 6 / 24
- 14. MSE duality Application of AMSE duality Robust Weighted Sum MSE Minimization (a) d1 d2 dK HH 2 HH K αK P −1 2 2 P −1 2 1 P −1 2 K n1 nK n2 dK d2 d1 = d GP 1 2 α1HH 1 α2 UH 1 UH 2 UH K (b) d1 GH d2 dK Q 1 2 1 Q 1 2 2 Q 1 2 K H1 H2 HK n dQ−1 2α U1 U2 UK minGk ,Uk ,αk ,Pk K k=1 τk ξ DL k s.t K k=1 tr{Pk } ≤ Pmax Case II : General τk , Rmk , Rbk , σ2 ek ⋄ Initialize Uk , Qk and get Gk , αk from MMSE Rx ⋄ Decompose Qk = qk ˜Qk , tr{ ˜Qk } = 1 ⋄ Optimize qk (GP problem)⋆ ⋄ Get Qk from qk and ˜Qk ⋄ Update Rx by MMSE and get Gk , αk from Rx ⋄ Transfer to DL as Pk = βk α2 k Q−1 k where T · [β1, . . . , βK ]T = σ2 [tr{Q1}, . . . , tr{QK }]T T Constant ⋄ Update Rx by MMSE ⋄ Get Uk and αk from Rx ⋄ Switch to UL and iterate Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 6 / 24
- 15. MSE duality Simulation Results Simulation Results 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 1.2 1.4 SNR (dB) (a) AveragesumMSE GM (Na) GM (Ro) GM (Pe) Alg I (Na) Alg I (Ro) Alg I (Pe) 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 SNR (dB) (b) AveragesumMSE Na (ρ b = 0.25) Ro (ρ b = 0.25) Pe (ρ b = 0.25) Na (ρ b = 0.75) Ro (ρ b = 0.75) Pe (ρ b = 0.75) 10 15 20 25 30 35 0 0.5 1 1.5 SNR (dB) (c) AveragesumMSE Na (ρ b = 0.25, ρ m = 0.25) Ro (ρ b = 0.25, ρ m = 0.25) Pe (ρ b = 0.25, ρ m = 0.25) Na (ρ b = 0.25, ρ m = 0.75) Ro (ρ b = 0.25, ρ m = 0.75) Pe (ρ b = 0.25, ρ m = 0.75) Settings N = 4, K = 2, Mk = 2, Pmax = 10mw, τk = 1 Observations ⋄ Robust outperforms nonrobust ⋄ Perfect CSI gives the best AMSE ⋄ Large antenna correlation further increases sum AMSE Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 7 / 24
- 16. MSE duality Drawbacks and Looking ahead Drawbacks and Looking ahead Drawbacks The duality solve only total BS power based problems The duality FAIL to solve Practically relevant per BS antenna power based problems Looking Ahead No clue to resolve the drawback!! Switch to distributed transceiver design for Coordinated BS systems Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 8 / 24
- 17. Transceiver design for Coordinated BS Systems Block diagram and Problem formulation Coordinated BS Block Diagram Assumptions: The lth BS precods the overall data d = [d1, · · · , dK ] by Bl The kth MS uses the receiver Wk to recover its data dk Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 9 / 24
- 18. Transceiver design for Coordinated BS Systems Block diagram and Problem formulation Coordinated BS Block Diagram Assumptions: The lth BS precods the overall data d = [d1, · · · , dK ] by Bl The kth MS uses the receiver Wk to recover its data dk ˆdk = WH k ( L l=1 HH lk Bl d + nk ) = WH k (HH k Bd + nk ) where HH k = [HH 1k , · · · , HH Lk ] B = [B1; · · · ; BL] ⋄ Interpreted as a gaint MIMO ⋄ Treated like conventional MIMO BUT with per BS power constraint Or per BS antenna power constraint Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 9 / 24
- 19. Transceiver design for Coordinated BS Systems Block diagram and Problem formulation System Model and Problem Statement d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK max{Bk ,Wk }K k=1 K k=1 Sk i=1 ωki Rki s.t [ K k=1 Bk BH k ]n,n ≤ Pn, ∀n Rki = log2 (ξ−1 ki ) ξki = wH ki (HH k BBH Hk + σ2 k I)wki −2ℜ{wH ki HH k bki } + 1 Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 10 / 24
- 20. Transceiver design for Coordinated BS Systems Block diagram and Problem formulation System Model and Problem Statement d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK max{Bk ,Wk }K k=1 K k=1 Sk i=1 ωki Rki s.t [ K k=1 Bk BH k ]n,n ≤ Pn, ∀n Rki = log2 (ξ−1 ki ) ξki = wH ki (HH k BBH Hk + σ2 k I)wki −2ℜ{wH ki HH k bki } + 1 Reexpressed as min{bs,ws}S w=1 S s=1 ξωs s s.t [ S s=1 bsbH s ]n,n ≤ Pn, ∀n ξs = wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws −2ℜ{wH s ˜HH s bs} + 1 ⋄ Non linear and non convex Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 10 / 24
- 21. Transceiver design for Coordinated BS Systems Block diagram and Problem formulation System Model and Problem Statement d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK max{Bk ,Wk }K k=1 K k=1 Sk i=1 ωki Rki s.t [ K k=1 Bk BH k ]n,n ≤ Pn, ∀n Rki = log2 (ξ−1 ki ) ξki = wH ki (HH k BBH Hk + σ2 k I)wki −2ℜ{wH ki HH k bki } + 1 Reexpressed as min{bs,ws}S w=1 S s=1 ξωs s s.t [ S s=1 bsbH s ]n,n ≤ Pn, ∀n ξs = wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws −2ℜ{wH s ˜HH s bs} + 1 ⋄ Non linear and non convex Existing iterative algorithm[1] ⋄ Solve this problem as it is ⋄ Complexity per iteration : O( (N + S)(2NS + 1)2 (2S2 + 2NS + S)) +O(K ˜M2.376 ) + CGP [1] Shi, S., Schubert, M., and Boche, H. ”Per-antenna power constrained rate optimization for multiuser MIMO systems”, Proc. WSA, Belrin, Germany, Feb., 2008. Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 10 / 24
- 22. Transceiver design for Coordinated BS Systems Proposed Algorithms Problem Reformulation min{bs,ws}S s=1 S s=1 ξωs s , s.t [ S s=1 bsbH s ]n,n ≤ Pn, ∀n ξs = wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1 Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 11 / 24
- 23. Transceiver design for Coordinated BS Systems Proposed Algorithms Problem Reformulation min{bs,ws}S s=1 S s=1 ξωs s , s.t [ S s=1 bsbH s ]n,n ≤ Pn, ∀n ξs = wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1 Key facts ⋄ S s=1 fs, fs > 0 ≡ min{νs}S s=1 1 S S s=1 fsνs S s.t S s=1 νs = 1, νs ≥ 0 ⋄ abω , a, b > 0, 0 < ω < 1 ≡ minτ>0 κ(aγ τ + bτµ ) γ = 1 1−ω , µ = 1 ω − 1, κ = ωµ(1−ω) Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 11 / 24
- 24. Transceiver design for Coordinated BS Systems Proposed Algorithms Problem Reformulation min{bs,ws}S s=1 S s=1 ξωs s , s.t [ S s=1 bsbH s ]n,n ≤ Pn, ∀n ξs = wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1 Key facts ⋄ S s=1 fs, fs > 0 ≡ min{νs}S s=1 1 S S s=1 fsνs S s.t S s=1 νs = 1, νs ≥ 0 ⋄ abω , a, b > 0, 0 < ω < 1 ≡ minτ>0 κ(aγ τ + bτµ ) γ = 1 1−ω , µ = 1 ω − 1, κ = ωµ(1−ω) Reformulate WSR max problem as minτs,νs,bs,ws S s=1 κs[νγs s τs + τµs s (wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1)] s.t [ S s=1 bsbH s ]n,n ≤ pn, S s=1 νs = 1, νs > 0, τs > 0 ∀s, n Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 11 / 24
- 25. Transceiver design for Coordinated BS Systems Proposed Algorithms Proposed Centralized Algorithm Reformulated WSR max problem minτs,νs,bs,ws S s=1 κs[νγs s τs + τµs s (wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1)] s.t [ S s=1 bsbH s ]n,n ≤ pn, S s=1 νs = 1, νs > 0, τs > 0 ∀s, n Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 12 / 24
- 26. Transceiver design for Coordinated BS Systems Proposed Algorithms Proposed Centralized Algorithm Reformulated WSR max problem minτs,νs,bs,ws S s=1 κs[νγs s τs + τµs s (wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1)] s.t [ S s=1 bsbH s ]n,n ≤ pn, S s=1 νs = 1, νs > 0, τs > 0 ∀s, n ⋄ For fixed B : Optimize ws, νs, τs (closed form solution) Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 12 / 24
- 27. Transceiver design for Coordinated BS Systems Proposed Algorithms Proposed Centralized Algorithm Reformulated WSR max problem minτs,νs,bs,ws S s=1 κs[νγs s τs + τµs s (wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1)] s.t [ S s=1 bsbH s ]n,n ≤ pn, S s=1 νs = 1, νs > 0, τs > 0 ∀s, n ⋄ For fixed B : Optimize ws, νs, τs (closed form solution) ⋄ For fixed ws, νs, τs : Optimize bs (SDP problem) Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 12 / 24
- 28. Transceiver design for Coordinated BS Systems Proposed Algorithms Proposed Centralized Algorithm Reformulated WSR max problem minτs,νs,bs,ws S s=1 κs[νγs s τs + τµs s (wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1)] s.t [ S s=1 bsbH s ]n,n ≤ pn, S s=1 νs = 1, νs > 0, τs > 0 ∀s, n Repeat ⋄ For fixed B : Optimize ws, νs, τs (closed form solution) ⋄ For fixed ws, νs, τs : Optimize bs (SDP problem) Until Convergence Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 12 / 24
- 29. Transceiver design for Coordinated BS Systems Proposed Algorithms Proposed Centralized Algorithm Reformulated WSR max problem minτs,νs,bs,ws S s=1 κs[νγs s τs + τµs s (wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1)] s.t [ S s=1 bsbH s ]n,n ≤ pn, S s=1 νs = 1, νs > 0, τs > 0 ∀s, n Repeat ⋄ For fixed B : Optimize ws, νs, τs (closed form solution) ⋄ For fixed ws, νs, τs : Optimize bs (SDP problem) Until Convergence Computational complexity Exist : O(K ˜M2.376 ) + O( (N + S)(2NS + 1)2 (2S2 + 2NS + S)) + CGP per ite Prop : O(K ˜M2.376 ) + O( √ N + 1(2NS + 1)2 (2S2 + 2NS)) per ite (Better!) Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 12 / 24
- 30. Transceiver design for Coordinated BS Systems Proposed Algorithms Proposed Distributed Algorithm Reformulated WSR max problem minτs,νs,bs,ws S s=1 κs[νγs s τs + τµs s (wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1)] s.t [ S s=1 bsbH s ]n,n ≤ pn, S s=1 νs = 1, νs > 0, τs > 0 ∀s, n Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 13 / 24
- 31. Transceiver design for Coordinated BS Systems Proposed Algorithms Proposed Distributed Algorithm Reformulated WSR max problem minτs,νs,bs,ws S s=1 κs[νγs s τs + τµs s (wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1)] s.t [ S s=1 bsbH s ]n,n ≤ pn, S s=1 νs = 1, νs > 0, τs > 0 ∀s, n ⋄ For fixed B : Same as centralized Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 13 / 24
- 32. Transceiver design for Coordinated BS Systems Proposed Algorithms Proposed Distributed Algorithm Reformulated WSR max problem minτs,νs,bs,ws S s=1 κs[νγs s τs + τµs s (wH s (˜HH s BBH ˜Hs + ˜σ2 s I)ws − 2ℜ{wH s ˜HH s bs} + 1)] s.t [ S s=1 bsbH s ]n,n ≤ pn, S s=1 νs = 1, νs > 0, τs > 0 ∀s, n ⋄ For fixed B : Same as centralized For fixed ws, νs, τs ⋄ Formulate bs optimization as SDP ⋄ Get dual of SDP : ({λn ≥ 0}N n=1 are dual variables) ⋄ Apply MFM and get λi iteratively by ⋄ λ⋆ i = |¯gi |/ √ pi , g⋆ i = λ(RRH + λ)−1 fi where ¯gi is ith row of [g1, g2, · · · , gN ] (i.e., needs inner iteration) R, fi are constants ⋄ Compute optimal bs by employing {λ⋆ i }N i=1 Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 13 / 24
- 33. Transceiver design for Coordinated BS Systems Proposed Algorithms Proposed Distributed Algorithm ⋄ For fixed B : Same as centralized For fixed ws, νs, τs ⋄ Formulate bs optimization as SDP ⋄ Get dual of SDP : ({λn ≥ 0}N n=1 are dual variables) ⋄ Apply MFM and get λi iteratively by ⋄ λ⋆ i = |¯gi |/ √ pi , g⋆ i = λ(RRH + λ)−1 fi where ¯gi is ith row of [g1, g2, · · · , gN ] (i.e., needs inner iteration) R, fi are constants ⋄ Compute optimal bs by employing {λ⋆ i }N i=1 Computational complexity Exist : O(K ˜M2.376 ) + O( (N + S)(2NS + 1)2 (2S2 + 2NS + S)) + CGP per ite Pro (cent) : O(K ˜M2.376 ) + O( √ N + 1(2NS + 1)2 (2S2 + 2NS)) per ite Pro (dist) : O(K ˜M2.376 ) + inner ite × O(N2.376 ) per ite (Much better!) Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 13 / 24
- 34. Transceiver design for Coordinated BS Systems Simulation Results Simulation Results for inner iteration Large scale network: L = 25, K = 50, Mk = 2 at SNR = 10dB 2 4 6 8 10 12 14 16 18 20 0 20 40 60 80 100 120 140 Number of iterations Objectivefunction Small number of inner iteration is required Indeed distributed needs less computation than centralized Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 14 / 24
- 35. Transceiver design for Coordinated BS Systems Simulation Results Comparison of Proposed and Existing Algorithms Set N = 4, L = 2, K = 4, Mk = 2, ω = [.6, .4, .5, .8, .25, .8, .46, .28] 5 10 15 20 25 4 5 6 7 8 9 10 11 12 Number of iterations Weightedsumrate(bps/Hz) SNR=10dB Proposed centralized algorithm Proposed distributed algorithm Existing algorithm [1] 0 5 10 15 20 4 6 8 10 12 14 16 18 20 22 SNR (dB) Weightedsumrate(bps/Hz) Proposed centralized algorithm Proposed distributed algorithm Existing algorithm [1] Proposed algorithms have faster convergence than existing Proposed algorithms have slightly higher WSR than existing Distributed algorithm achieves the same WSR as centralized [1] Shi, S., Schubert, M., and Boche, H. ”Per-antenna power constrained rate optimization for multiuser MIMO systems”, Proc. WSA, Belrin, Germany, Feb., 2008. Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 15 / 24
- 36. Transceiver design for Coordinated BS Systems Drawbacks and Looking ahead Drawbacks and Looking ahead Drawbacks: The proposed distributed algorithm is problem dependent (i.e., for each problem we need to formulate its Lagrangian dual problem). Looking ahead The WSR max problem can be analyzed like in a conventional multiuser MIMO system with per antenna power constraint. A clear relation between WSR and WSMSE is exploited. Key observation of MSE duality: The role of transmitters and receivers are interchanged. Exploiting MSE duality for generalized power constraint should help to get problem independent distributed algorithm for many classes of transceiver design problems Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 16 / 24
- 37. Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements System Model and Problem Statements d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK Objectives To solve P1 and P2 by MSE duality approach To show the benefits of the MSE duality solution approach To show the extension of the duality for solving other transceiver design problems P1 : minBk ,Wk K k=1 Sk i=1 ηki ξDL ki s.t [ K k=1 Bk BH k ]n,n ≤ ˘pn, bH ki bki ≤ ˘pki , ∀n, k, i P2 : min{Bk ,Wk }K k=1 max ρki ξDL ki s.t [ K k=1 Bk BH k ]n,n ≤ ˘pn, bH ki bki ≤ ˘pki , ∀n, k, i Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 17 / 24
- 38. Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements Existing MSE Uplink-downlink Duality (Revisited) (a) d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK (b) H1 H2 HK n TH V1 V2 VK d d1 d2 dK The duality can maintain ξDL ki = ξUL ki The duality cannot ensure [ K k=1 Bk BH k ]n,n ≤ ˘pn and bH kibki ≤ ˘pki P1 : min{Bk ,Wk }K k=1 K k=1 Sk i=1 ηki ξDL ki s.t [ K k=1 Bk BH k ]n,n ≤ ˘pn, bH ki bki ≤ ˘pki P2 : min{Bk ,Wk }K k=1 max ρki ξDL ki s.t [ K k=1 Bk BH k ]n,n ≤ ˘pn, bH ki bki ≤ ˘pki Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 18 / 24
- 39. Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements New MSE Downlink-Interference Duality d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK V1 V2 nI 1S1 nI K1 nI KSK nI 11 tH KSK tH K1 tH 1S1 tH 11 H111 H11S1 H21S1 H1KSK H1K1 HKK1 HK1S1 H2K1 HKKSK ˆdK1 ˆd11 VK HK11 H2KSK H211 d1 ˆdKSK ˆd1S1 d2 dK P1 : min{Bk ,Wk }K k=1 K k=1 Sk i=1 ηki ξDL ki s.t [ K k=1 Bk BH k ]n,n ≤ ˘pn, bH ki bki ≤ ˘pki Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
- 40. Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements New MSE Downlink-Interference Duality d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK V1 V2 nI 1S1 nI K1 nI KSK nI 11 tH KSK tH K1 tH 1S1 tH 11 H111 H11S1 H21S1 H1KSK H1K1 HKK1 HK1S1 H2K1 HKKSK ˆdK1 ˆd11 VK HK11 H2KSK H211 d1 ˆdKSK ˆd1S1 d2 dK P1 : min{Bk ,Wk }K k=1 K k=1 Sk i=1 ηki ξDL ki s.t [ K k=1 Bk BH k ]n,n ≤ ˘pn, bH ki bki ≤ ˘pki ⋄ Initialize Bk and update Wk by MMSE Repeat Transformation (DL to Interference) ⋄ Set dI ki ∼ (0, ηki ), nI ki ∼ (0, Ψ + µki I), Ψ = diag(ψn) ⋄ Get ψn, µki iteratively Key we show that ψn, µki > 0 always exist! ⋄ Set vki = wki and update tki by MMSE Transformation (Interference to DL) ⋄ Set bki = βtki , β2 = K i=1 Si j=1 ηij wH ij Ri wij K i=1 Si j=1 tH ij (Ψ+µij I)tij ⋄ Update Wk by MMSE Until convergence Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
- 41. Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements New MSE Downlink-Interference Duality d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK V1 V2 nI 1S1 nI K1 nI KSK nI 11 tH KSK tH K1 tH 1S1 tH 11 H111 H11S1 H21S1 H1KSK H1K1 HKK1 HK1S1 H2K1 HKKSK ˆdK1 ˆd11 VK HK11 H2KSK H211 d1 ˆdKSK ˆd1S1 d2 dK P1 : min{Bk ,Wk }K k=1 K k=1 Sk i=1 ηki ξDL ki s.t [ K k=1 Bk BH k ]n,n ≤ ˘pn, bH ki bki ≤ ˘pki ⋄ Initialize Bk and update Wk by MMSE Repeat Transformation (DL to Interference) ⋄ Set dI ki ∼ (0, ηki ), nI ki ∼ (0, Ψ + µki I), Ψ = diag(ψn) ⋄ Get ψn, µki iteratively Key we show that ψn, µki > 0 always exist! ⋄ Set vki = wki and update tki by MMSE Transformation (Interference to DL) ⋄ Set bki = βtki , wki = vki β , β2 = K i=1 Si j=1 ηij wH ij Ri wij K i=1 Si j=1 tH ij (Ψ+µij I)tij ⋄ Decompose Bk = Gk P 1/2 k , Wk = Gk P −1/2 k αk ⋄ Optimize Pk (increases convergence speed) ⋄ Again update Bk = Gk P 1/2 k and Wk by MMSE Until convergence Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
- 42. Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements New MSE Downlink-Interference Duality d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK V1 V2 nI 1S1 nI K1 nI KSK nI 11 tH KSK tH K1 tH 1S1 tH 11 H111 H11S1 H21S1 H1KSK H1K1 HKK1 HK1S1 H2K1 HKKSK ˆdK1 ˆd11 VK HK11 H2KSK H211 d1 ˆdKSK ˆd1S1 d2 dK P2 : min{Bk ,Wk }K k=1 max ρki ξDL ki s.t [ K k=1 Bk BH k ]n,n ≤ ˘pn, bH ki bki ≤ ˘pki Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
- 43. Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements New MSE Downlink-Interference Duality d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK V1 V2 nI 1S1 nI K1 nI KSK nI 11 tH KSK tH K1 tH 1S1 tH 11 H111 H11S1 H21S1 H1KSK H1K1 HKK1 HK1S1 H2K1 HKKSK ˆdK1 ˆd11 VK HK11 H2KSK H211 d1 ˆdKSK ˆd1S1 d2 dK P2 : min{Bk ,Wk }K k=1 max ρki ξDL ki s.t [ K k=1 Bk BH k ]n,n ≤ ˘pn, bH ki bki ≤ ˘pki ⋄ Initialize Bk and update Wk by MMSE Repeat Transformation (DL to Interference) ⋄ Set dI ki ∼ (0, 1), nI ki ∼ (0, Ψ + µki I), Ψ = diag(ψn) ⋄ Get ψn, µki iteratively ⋄ Get ¯β2 [ ¯β2 11, · · · ¯β2 KSK ] as ¯β2 = ¯Zx x = [ψ1, · · · , ψN , µ11, · · · , µKSK ], ¯Z is constant ⋄ Set vki = ¯βki wki and update tki by MMSE Transformation (Interference to DL) ⋄ Get β2 [β2 11, · · · β2 KSK ] as β2 = Zx, Z is constant Key we show that ψn, µki , ¯β2 ki , β2 ki > 0 always exist! ⋄ Set bki = βki tki and update Wk by MMSE Until convergence Unbalanced weighted MSE Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
- 44. Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements New MSE Downlink-Interference Duality d2 d1 HH 1 HH 2 HH K WH 2 n1 nK n2 WH K WH 1 d1 d2 dK = d B dK V1 V2 nI 1S1 nI K1 nI KSK nI 11 tH KSK tH K1 tH 1S1 tH 11 H111 H11S1 H21S1 H1KSK H1K1 HKK1 HK1S1 H2K1 HKKSK ˆdK1 ˆd11 VK HK11 H2KSK H211 d1 ˆdKSK ˆd1S1 d2 dK P2 : min{Bk ,Wk }K k=1 max ρki ξDL ki s.t [ K k=1 Bk BH k ]n,n ≤ ˘pn, bH ki bki ≤ ˘pki ⋄ Initialize Bk and update Wk by MMSE Repeat Transformation (DL to Interference) ⋄ Set dI ki ∼ (0, 1), nI ki ∼ (0, Ψ + µki I), Ψ = diag(ψn) ⋄ Get ψn, µki iteratively ⋄ Get ¯β2 [ ¯β2 11, · · · ¯β2 KSK ] as ¯β2 = ¯Zx x = [ψ1, · · · , ψN , µ11, · · · , µKSK ], ¯Z is constant ⋄ Set vki = ¯βki wki and update tki by MMSE Transformation (Interference to DL) ⋄ Get β2 [β2 11, · · · β2 KSK ] as β2 = Zx, Z is constant Key we show that ψn, µki , ¯β2 ki , β2 ki > 0 always exist! ⋄ Set bki = βki tki , wki = vki /βki and decompose Bk = Gk P 1/2 k , Wk = Gk P −1/2 k αk ⋄ Optimize Pk , −Ensures balanced weighted MSE −Increases convergence speed ⋄ Again update Bk = Gk P 1/2 k and Wk by MMSE Until convergence Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
- 45. Transceiver design for multiuser MIMO systems: Generalized duality Simulation Results Simulation Results 10 15 20 25 30 35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR (dB) WeightedsumMSE Proposed Duality Algorithm in [1] −25 −20 −15 −10 −5 0 7.5 8 8.5 9 9.5 10 σav 2 (dB) TotalBSpower Proposed Duality Algorithm in [1] 10 15 20 25 30 35 0 0.05 0.1 0.15 0.2 0.25 SNR (dB) MaximumsymbolMSE Proposed Duality Algorithm in [1] −25 −20 −15 −10 −5 0 7 7.5 8 8.5 9 9.5 10 σ av 2 (dB) TotalBSpower Proposed Duality Algorithm in [1] Settings N = 4, K = 2, Mk = 2, ˘pki = 2.5mw, ˘pn = 2.5mw, ηki = ρki = 1 [1] Shi, S., Schubert, M., Vucic, N., and Boche, H. ”MMSE Optimization with Per-Base-Station Power Constraints for Network MIMO Systems”, Proc. IEEE ICC, Beijing, China, May, 2008. P1 P2 Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 20 / 24
- 46. Transceiver design for multiuser MIMO systems: Generalized duality Simulation Results Simulation Results 10 15 20 25 30 35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR (dB) WeightedsumMSE Proposed Duality Algorithm in [1] −25 −20 −15 −10 −5 0 7.5 8 8.5 9 9.5 10 σav 2 (dB) TotalBSpower Proposed Duality Algorithm in [1] 10 15 20 25 30 35 0 0.05 0.1 0.15 0.2 0.25 SNR (dB) MaximumsymbolMSE Proposed Duality Algorithm in [1] −25 −20 −15 −10 −5 0 7 7.5 8 8.5 9 9.5 10 σ av 2 (dB) TotalBSpower Proposed Duality Algorithm in [1] Settings N = 4, K = 2, Mk = 2, ˘pki = 2.5mw, ˘pn = 2.5mw, ηki = ρki = 1 [1] Shi, S., Schubert, M., Vucic, N., and Boche, H. ”MMSE Optimization with Per-Base-Station Power Constraints for Network MIMO Systems”, Proc. IEEE ICC, Beijing, China, May, 2008. P1 P2 Complexity (P1) Proposed duality O(N2.376 ) + O(KM2.376 ) + CGP (≡ Linear programming) Algorithm in [1] O( (N + KM + 1)(2MKN + 1)2 (2(MK)2 + 4NMK)) + O(KM2.376 ) Complexity (P2) Proposed duality O(N2.376 ) + O(KM2.376 ) + CGP (≡ Linear programming) Algorithm in [1] O( (N + KM + 1)(2MKN + 1)2 (2(MK)2 + 4NMK)) + O(KM2.376 ) Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 20 / 24
- 47. Thesis Conclusions Thesis Conclusions In this PhD work, we accomplish the following main tasks: We generalize the existing MSE duality to handle many practically relevant transceiver design problems. For stochastic robust design MSE-based problems, the duality can be extended straightforwardly to imperfect CSI scenario. For all of considered problems, the proposed duality algorithms require less total BS power (and complexity) compared to the existing solution approach which does not employ duality The relationship between WSMSE and WSR problems have been exploited. Consequently, the complicated nonlinear WSR problem can be examined by its equivalent linear WSMSE problem We also develop distributed transceiver design algorithms to solve weighted sum rate and MSE optimization problems Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 21 / 24
- 48. Future Research Directions Future Research Directions All of our algorithms are linear but suboptimal. So getting linear and optimal algorithm is still an open research topic (one approach could be to extend the well known Majorization theory to Multiuser MIMO setup). The proposed general duality is valid only for perfect CSI and imperfect CSI with stochastic robust design. The extension of the proposed duality to imperfect CSI with worst-case robust design is open for future research. In all of our distributive algorithms, we assume that the global channel knowledge is available at the central controller (or at all BSs) prior to optimization. Thus, developing distributed algorithm with local CSI knowledge is also an open research direction The robust rate and SINR-based problems (i.e, in stochastic design approach) have not been examined. Hence, solving such problems is open research topic. Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 22 / 24
- 49. Future Research Directions Selected List of Publications I T. E. Bogale, B. K. Chalise, and L. Vandendorpe, Robust transceiver optimization for downlink multiuser MIMO systems, IEEE Tran. Sig. Proc. 59 (2011), no. 1, 446 – 453. T. E. Bogale and L. Vandendorpe, MSE uplink-downlink duality of MIMO systems with arbitrary noise covariance matrices, 45th Annual conference on Information Sciences and Systems (CISS) (Baltimore, MD, USA), 23 – 25 Mar. 2011, pp. 1 – 6. T. E. Bogale and L. Vandendorpe, Weighted sum rate optimization for downlink multiuser MIMO coordinated base station systems: Centralized and distributed algorithms, IEEE Trans. Signal Process. 60 (2011), no. 4, 1876 – 1889. Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 23 / 24
- 50. Future Research Directions Selected List of Publications II , Weighted sum rate optimization for downlink multiuser MIMO systems with per antenna power constraint: Downlink-uplink duality approach, IEEE International Conference On Acuostics, Speech and Signal Processing (ICASSP) (Kyoto, Japan), 25 – 30 Mar. 2012, pp. 3245 – 3248. , Linear transceiver design for downlink multiuser MIMO systems: Downlink-interference duality approach, IEEE Trans. Sig. Process. 61 (2013), no. 19, 4686 – 4700. Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 24 / 24