transceiver design for single-cell and multi-cell downlink multiuser mimo systems
DESCRIPTION
Transceiver design for single-cell and multi-cell downlink multiuser MIMO systems PhD defense slideTRANSCRIPT
Transceiver design for single-cell and multi-celldownlink multiuser MIMO systems
Tadilo Endeshaw Bogale
University Catholique de Louvain (UCL), ICTEAM
Dec. 2013
Presentation Outline
Presentation Outline1 MSE uplink-downlink duality under imperfect CSI
MSE uplink-downlink duality under imperfect CSIApplication of AMSE dualitySimulation ResultsDrawbacks and Looking ahead
2 Transceiver design for Coordinated BS SystemsBlock diagram and Problem formulationProposed AlgorithmsSimulation ResultsDrawbacks and Looking ahead
3 Transceiver design for multiuser MIMO systems: Generalized dualitySystem Model and Problem StatementsSimulation Results
4 Thesis Conclusions5 Future Research Directions
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 2 / 24
MSE duality MSE uplink-downlink duality under imperfect CSI
MSE uplink-downlink duality under imperfect CSI
(a)
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
(b)
H1
H2
HK
n
TH
V1
V2
VK
d
d1
d2
dK
Assumption: CSI model HHk = HH
k + R1/2mk EH
wkR1/2bk
Objectives:Exploit MSE duality (sum MSE, user MSE and symbol MSE duality)between UL and DL channelsApply duality to solve transceiver design problems
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 3 / 24
MSE duality MSE uplink-downlink duality under imperfect CSI
Sum MSE uplink-downlink duality
(a)
d1
d2
dK
HH2
HHK αK
P−12
2
P−12
1
P−12
K
n1
nK
n2
dK
d2
d1
= d GP12
α1HH1
α2
UH1
UH2
UHK
(b)
d1
GHd2
dK
Q121
Q122
Q12K
H1
H2
HK
n
dQ−12α
U1
U2
UK
ξDLk = tr{ISk + P−1/2
k αk UHk Γ
DLk Ukαk P−1/2
k
−2ℜ{P1/2k GH
k Hk Ukαk P−1/2k }}
ΓDLk = σ2
ek tr{Rbk GPGH}Rmk+
HHk GPGHHk + σ2IMk
ξULk = tr{ISk + Q−1/2
k αk GHk ΓcGkαk Q−1/2
k
−2ℜ{Q−1/2k αk GH
k Hk Uk Q1/2k }}
ΓULc =
∑Ki=1(σ
2ei tr{RmiUiQiUH
i }Rbi+
HiUiQiUHi HH
i ) + σ2IN
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 4 / 24
MSE duality MSE uplink-downlink duality under imperfect CSI
Sum MSE uplink-downlink duality
(a)
d1
d2
dK
HH2
HHK αK
P−12
2
P−12
1
P−12
K
n1
nK
n2
dK
d2
d1
= d GP12
α1HH1
α2
UH1
UH2
UHK
(b)
d1
GHd2
dK
Q121
Q122
Q12K
H1
H2
HK
n
dQ−12α
U1
U2
UK
ξDLk = tr{ISk + P−1/2
k αk UHk Γ
DLk Ukαk P−1/2
k
−2ℜ{P1/2k GH
k Hk Ukαk P−1/2k }}
ΓDLk = σ2
ek tr{Rbk GPGH}Rmk+
HHk GPGHHk + σ2IMk
ξULk = tr{ISk + Q−1/2
k αk GHk ΓcGkαk Q−1/2
k
−2ℜ{Q−1/2k αk GH
k Hk Uk Q1/2k }}
ΓULc =
∑Ki=1(σ
2ei tr{RmiUiQiUH
i }Rbi+
HiUiQiUHi HH
i ) + σ2IN
GivenξDL ,∑K
k=1 ξDLk
We can ensure∑K
k=1 ξULk = ξDL
by settingQk = βα2k P−1
k
with β =∑K
k=1 tr{Pk}∑K
k=1 tr{P−1k αk}∑K
k=1 tr{Qk} =∑K
k=1 tr{Pk}(Also met)
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 4 / 24
MSE duality MSE uplink-downlink duality under imperfect CSI
Sum MSE uplink-downlink duality
(a)
d1
d2
dK
HH2
HHK αK
P−12
2
P−12
1
P−12
K
n1
nK
n2
dK
d2
d1
= d GP12
α1HH1
α2
UH1
UH2
UHK
(b)
d1
GHd2
dK
Q121
Q122
Q12K
H1
H2
HK
n
dQ−12α
U1
U2
UK
ξDLk = tr{ISk + P−1/2
k αk UHk Γ
DLk Ukαk P−1/2
k
−2ℜ{P1/2k GH
k Hk Ukαk P−1/2k }}
ΓDLk = σ2
ek tr{Rbk GPGH}Rmk+
HHk GPGHHk + σ2IMk
ξULk = tr{ISk + Q−1/2
k αk GHk ΓcGkαk Q−1/2
k
−2ℜ{Q−1/2k αk GH
k Hk Uk Q1/2k }}
ΓULc =
∑Ki=1(σ
2ei tr{RmiUiQiUH
i }Rbi+
HiUiQiUHi HH
i ) + σ2IN
GivenξDL ,∑K
k=1 ξDLk
We can ensure∑K
k=1 ξULk = ξDL
by settingQk = βα2k P−1
k
with β =∑K
k=1 tr{Pk}∑K
k=1 tr{P−1k αk}∑K
k=1 tr{Qk} =∑K
k=1 tr{Pk}(Also met)
GivenξUL ,∑K
k=1 ξULk
We ensure∑K
k=1 ξDLk = ξUL
by settingPk = βα2k Q−1
k
with β =∑K
k=1 tr{Qk}∑K
k=1 tr{Q−1k αk}∑K
k=1 tr{Pk} =∑K
k=1 tr{Qk}(Also met)
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 4 / 24
MSE duality MSE uplink-downlink duality under imperfect CSI
User MSE uplink-downlink duality
(a)
d1
d2
dK
HH2
HHK αK
P−12
2
P−12
1
P−12
K
n1
nK
n2
dK
d2
d1
= d GP12
α1HH1
α2
UH1
UH2
UHK
(b)
d1
GHd2
dK
Q121
Q122
Q12K
H1
H2
HK
n
dQ−12α
U1
U2
UK
ξDLk = tr{ISk + P−1/2
k αk UHk Γ
DLk Ukαk P−1/2
k
−2ℜ{P1/2k GH
k Hk Ukαk P−1/2k }}
ΓDLk = σ2
ek tr{Rbk GPGH}Rmk+
HHk GPGHHk + σ2IMk
ξULk = tr{ISk + Q−1/2
k αk GHk ΓcGkαk Q−1/2
k
−2ℜ{Q−1/2k αk GH
k Hk Uk Q1/2k }}
ΓULc =
∑Ki=1(σ
2ei tr{RmiUiQiUH
i }Rbi+
HiUiQiUHi HH
i ) + σ2IN
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 5 / 24
MSE duality MSE uplink-downlink duality under imperfect CSI
User MSE uplink-downlink duality
(a)
d1
d2
dK
HH2
HHK αK
P−12
2
P−12
1
P−12
K
n1
nK
n2
dK
d2
d1
= d GP12
α1HH1
α2
UH1
UH2
UHK
(b)
d1
GHd2
dK
Q121
Q122
Q12K
H1
H2
HK
n
dQ−12α
U1
U2
UK
ξDLk = tr{ISk + P−1/2
k αk UHk Γ
DLk Ukαk P−1/2
k
−2ℜ{P1/2k GH
k Hk Ukαk P−1/2k }}
ΓDLk = σ2
ek tr{Rbk GPGH}Rmk+
HHk GPGHHk + σ2IMk
ξULk = tr{ISk + Q−1/2
k αk GHk ΓcGkαk Q−1/2
k
−2ℜ{Q−1/2k αk GH
k Hk Uk Q1/2k }}
ΓULc =
∑Ki=1(σ
2ei tr{RmiUiQiUH
i }Rbi+
HiUiQiUHi HH
i ) + σ2IN
GivenξDLk , ξ
ULk = ξ
DLk ,
∑Kk=1 tr{Qk} =
∑Kk=1 tr{Pk}(ensured) by Qk = βkα
2k 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
MSE duality MSE uplink-downlink duality under imperfect CSI
User MSE uplink-downlink duality
(a)
d1
d2
dK
HH2
HHK αK
P−12
2
P−12
1
P−12
K
n1
nK
n2
dK
d2
d1
= d GP12
α1HH1
α2
UH1
UH2
UHK
(b)
d1
GHd2
dK
Q121
Q122
Q12K
H1
H2
HK
n
dQ−12α
U1
U2
UK
ξDLk = tr{ISk + P−1/2
k αk UHk Γ
DLk Ukαk P−1/2
k
−2ℜ{P1/2k GH
k Hk Ukαk P−1/2k }}
ΓDLk = σ2
ek tr{Rbk GPGH}Rmk+
HHk GPGHHk + σ2IMk
ξULk = tr{ISk + Q−1/2
k αk GHk ΓcGkαk Q−1/2
k
−2ℜ{Q−1/2k αk GH
k Hk Uk Q1/2k }}
ΓULc =
∑Ki=1(σ
2ei tr{RmiUiQiUH
i }Rbi+
HiUiQiUHi HH
i ) + σ2IN
GivenξDLk , ξ
ULk = ξ
DLk ,
∑Kk=1 tr{Qk} =
∑Kk=1 tr{Pk}(ensured) by Qk = βkα
2k P−1
k
where T · [β1, . . . , βK ]T = σ2 [tr{P1}, . . . , tr{PK}]
T, T is constant
GivenξULk , ξ
DLk = ξ
ULk ,
∑Kk=1 tr{Pk} =
∑Kk=1 tr{Qk}(ensured) by Pk = βkα
2k 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
MSE duality Application of AMSE duality
Robust Weighted Sum MSE Minimization
(a)
d1
d2
dK
HH2
HHK αK
P−12
2
P−12
1
P−12
K
n1
nK
n2
dK
d2
d1
= d GP12
α1HH1
α2
UH1
UH2
UHK
(b)
d1
GHd2
dK
Q121
Q122
Q12K
H1
H2
HK
n
dQ−12α
U1
U2
UK
minGk ,Uk ,αk ,Pk
∑Kk=1 τkξ
DLk
s.t∑K
k=1 tr{Pk} ≤ Pmax
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 6 / 24
MSE duality Application of AMSE duality
Robust Weighted Sum MSE Minimization
(a)
d1
d2
dK
HH2
HHK αK
P−12
2
P−12
1
P−12
K
n1
nK
n2
dK
d2
d1
= d GP12
α1HH1
α2
UH1
UH2
UHK
(b)
d1
GHd2
dK
Q121
Q122
Q12K
H1
H2
HK
n
dQ−12α
U1
U2
UK
minGk ,Uk ,αk ,Pk
∑Kk=1 τkξ
DLk
s.t∑K
k=1 tr{Pk} ≤ Pmax
Case I : τk = 1, Rmk = I,Rbk = Rb, σ2ek = σ2
e
⋄ DefineUk = Uk Qk UHk
⋄ OptimizeUk (SDP problem)⋆⋄ GetUk andQk from Uk
⋄ Update Rx by MMSE and getGk ,αk from Rx
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 6 / 24
MSE duality Application of AMSE duality
Robust Weighted Sum MSE Minimization
(a)
d1
d2
dK
HH2
HHK αK
P−12
2
P−12
1
P−12
K
n1
nK
n2
dK
d2
d1
= d GP12
α1HH1
α2
UH1
UH2
UHK
(b)
d1
GHd2
dK
Q121
Q122
Q12K
H1
H2
HK
n
dQ−12α
U1
U2
UK
minGk ,Uk ,αk ,Pk
∑Kk=1 τkξ
DLk
s.t∑K
k=1 tr{Pk} ≤ Pmax
Case I : τk = 1, Rmk = I,Rbk = Rb, σ2ek = σ2
e
⋄ DefineUk = Uk Qk UHk
⋄ OptimizeUk (SDP problem)⋆⋄ GetUk andQk from Uk
⋄ Update Rx by MMSE and getGk ,αk from Rx
⋄ Transfer to DL asPk = βα2k Q−1
k
whereβ =∑K
k=1 tr{Qk}∑K
k=1 tr{Q−1k αk}
⋄ Update Rx by MMSE⋄ GetUk andαk from Rx
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 6 / 24
MSE duality Application of AMSE duality
Robust Weighted Sum MSE Minimization
(a)
d1
d2
dK
HH2
HHK αK
P−12
2
P−12
1
P−12
K
n1
nK
n2
dK
d2
d1
= d GP12
α1HH1
α2
UH1
UH2
UHK
(b)
d1
GHd2
dK
Q121
Q122
Q12K
H1
H2
HK
n
dQ−12α
U1
U2
UK
minGk ,Uk ,αk ,Pk
∑Kk=1 τkξ
DLk
s.t∑K
k=1 tr{Pk} ≤ Pmax
Case II : Generalτk , Rmk ,Rbk , σ2ek
⋄ Initialize Uk ,Qk and getGk ,αk from MMSE Rx⋄ DecomposeQk = qk Qk , tr{Qk} = 1⋄ Optimizeqk (GP problem)⋆⋄ GetQk from qk andQk
⋄ Update Rx by MMSE and getGk ,αk from Rx
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 6 / 24
MSE duality Application of AMSE duality
Robust Weighted Sum MSE Minimization
(a)
d1
d2
dK
HH2
HHK αK
P−12
2
P−12
1
P−12
K
n1
nK
n2
dK
d2
d1
= d GP12
α1HH1
α2
UH1
UH2
UHK
(b)
d1
GHd2
dK
Q121
Q122
Q12K
H1
H2
HK
n
dQ−12α
U1
U2
UK
minGk ,Uk ,αk ,Pk
∑Kk=1 τkξ
DLk
s.t∑K
k=1 tr{Pk} ≤ Pmax
Case II : Generalτk , Rmk ,Rbk , σ2ek
⋄ Initialize Uk ,Qk and getGk ,αk from MMSE Rx⋄ DecomposeQk = qk Qk , tr{Qk} = 1⋄ Optimizeqk (GP problem)⋆⋄ GetQk from qk andQk
⋄ Update Rx by MMSE and getGk ,αk from Rx
⋄ Transfer to DL asPk = βkα2k Q−1
k
where T · [β1, . . . , βK ]T =
σ2 [tr{Q1}, . . . , tr{QK}]T
T Constant⋄ Update Rx by MMSE⋄ GetUk andαk from Rx⋄ Switch to UL and iterate
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 6 / 24
MSE duality Simulation Results
Simulation Results
10 15 20 25 30 350
0.2
0.4
0.6
0.8
1
1.2
1.4
SNR (dB)(a)
Ave
rage
sum
MS
E
GM (Na)GM (Ro)GM (Pe)Alg I (Na)Alg I (Ro)Alg I (Pe)
10 15 20 25 30 350
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
SNR (dB)(b)
Ave
rage
sum
MS
E
Na (ρ
b = 0.25)
Ro (ρb = 0.25)
Pe (ρb = 0.25)
Na (ρb = 0.75)
Ro (ρb = 0.75)
Pe (ρb = 0.75)
10 15 20 25 30 350
0.5
1
1.5
SNR (dB) (c)
Ave
rage
sum
MS
E
Na (ρb = 0.25, ρ
m= 0.25)
Ro (ρb = 0.25, ρ
m= 0.25)
Pe (ρb = 0.25, ρ
m= 0.25)
Na (ρb = 0.25, ρ
m= 0.75)
Ro (ρb = 0.25, ρ
m= 0.75)
Pe (ρb = 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
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 antennapower based problems
Looking Ahead
No clue to resolve the drawback!!
Switch to distributed transceiver design for Coordinated BSsystems
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 8 / 24
Transceiver design for Coordinated BS Systems Block diagram and Problem formulation
Coordinated BS Block Diagram
Assumptions :The l th BS precods the overall data d = [d1, · · · , dK ] by B l
The k th MS uses the receiver Wk to recover its data dk
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 9 / 24
Transceiver design for Coordinated BS Systems Block diagram and Problem formulation
Coordinated BS Block Diagram
Assumptions :The l th BS precods the overall data d = [d1, · · · , dK ] by B l
The k th MS uses the receiver Wk to recover its data dk
dk = WHk (∑L
l=1 HHlk B ld + nk )
= WHk (H
Hk Bd + nk )
whereHHk = [HH
1k , · · · ,HHLk ]
B = [B1; · · · ;BL]
⋄ Interpreted as a gaint MIMO⋄ Treated like conventional MIMOBUT with
per BS power constraint Orper BS antenna power constraint
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 9 / 24
Transceiver design for Coordinated BS Systems Block diagram and Problem formulation
System Model and Problem Statement
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
max{Bk ,Wk}Kk=1
∑Kk=1
∑Ski=1 ωkiRki
s.t [∑K
k=1 Bk BHk ]n,n ≤ Pn, ∀n
Rki = log2 (ξ−1ki )
ξki = wHki(H
Hk BBHHk + σ2
k I)wki
−2ℜ{wHkiH
Hk bki}+ 1
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 10 / 24
Transceiver design for Coordinated BS Systems Block diagram and Problem formulation
System Model and Problem Statement
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
max{Bk ,Wk}Kk=1
∑Kk=1
∑Ski=1 ωkiRki
s.t [∑K
k=1 Bk BHk ]n,n ≤ Pn, ∀n
Rki = log2 (ξ−1ki )
ξki = wHki(H
Hk BBHHk + σ2
k I)wki
−2ℜ{wHkiH
Hk bki}+ 1
Reexpressed as
min{bs,ws}Sw=1
∏Ss=1 ξ
ωss
s.t [∑S
s=1 bsbHs ]n,n ≤ Pn, ∀n
ξs = wHs (H
Hs BBHHs + σ2
s I)ws
−2ℜ{wHs HH
s bs}+ 1
⋄ Non linear and non convex
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 10 / 24
Transceiver design for Coordinated BS Systems Block diagram and Problem formulation
System Model and Problem Statement
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
max{Bk ,Wk}Kk=1
∑Kk=1
∑Ski=1 ωkiRki
s.t [∑K
k=1 Bk BHk ]n,n ≤ Pn, ∀n
Rki = log2 (ξ−1ki )
ξki = wHki(H
Hk BBHHk + σ2
k I)wki
−2ℜ{wHkiH
Hk bki}+ 1
Reexpressed as
min{bs,ws}Sw=1
∏Ss=1 ξ
ωss
s.t [∑S
s=1 bsbHs ]n,n ≤ Pn, ∀n
ξs = wHs (H
Hs BBHHs + σ2
s I)ws
−2ℜ{wHs 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 optimizationfor multiuser MIMO systems”, Proc. WSA, Belrin, Germany, Feb., 2008.
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 10 / 24
Transceiver design for Coordinated BS Systems Proposed Algorithms
Problem Reformulation
min{bs,ws}Ss=1
∏Ss=1 ξ
ωss , s.t [
∑Ss=1 bsbH
s ]n,n ≤ Pn, ∀n
ξs = wHs (H
Hs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 11 / 24
Transceiver design for Coordinated BS Systems Proposed Algorithms
Problem Reformulation
min{bs,ws}Ss=1
∏Ss=1 ξ
ωss , s.t [
∑Ss=1 bsbH
s ]n,n ≤ Pn, ∀n
ξs = wHs (H
Hs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1
Key facts
⋄ ∏Ss=1 fs, fs > 0 ≡
min{νs}Ss=1
(1S
∑Ss=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 = 11−ω
, µ = 1ω− 1, κ = ωµ(1−ω)
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 11 / 24
Transceiver design for Coordinated BS Systems Proposed Algorithms
Problem Reformulation
min{bs,ws}Ss=1
∏Ss=1 ξ
ωss , s.t [
∑Ss=1 bsbH
s ]n,n ≤ Pn, ∀n
ξs = wHs (H
Hs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1
Key facts
⋄ ∏Ss=1 fs, fs > 0 ≡
min{νs}Ss=1
(1S
∑Ss=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 = 11−ω
, µ = 1ω− 1, κ = ωµ(1−ω)
Reformulate WSR max problem as
minτs,νs,bs,ws
∑Ss=1 κs[
νγssτs
+ τµss (wH
s (HHs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1)]
s.t [∑S
s=1 bsbHs ]n,n ≤ pn,
∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 11 / 24
Transceiver design for Coordinated BS Systems Proposed Algorithms
Proposed Centralized Algorithm
Reformulated WSR max problem
minτs,νs,bs,ws
∑Ss=1 κs[
νγssτs
+ τµss (wH
s (HHs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1)]
s.t [∑S
s=1 bsbHs ]n,n ≤ pn,
∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 12 / 24
Transceiver design for Coordinated BS Systems Proposed Algorithms
Proposed Centralized Algorithm
Reformulated WSR max problem
minτs,νs,bs,ws
∑Ss=1 κs[
νγssτs
+ τµss (wH
s (HHs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1)]
s.t [∑S
s=1 bsbHs ]n,n ≤ pn,
∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n
⋄ For fixedB : Optimizews, νs, τs (closed form solution)
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 12 / 24
Transceiver design for Coordinated BS Systems Proposed Algorithms
Proposed Centralized Algorithm
Reformulated WSR max problem
minτs,νs,bs,ws
∑Ss=1 κs[
νγssτs
+ τµss (wH
s (HHs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1)]
s.t [∑S
s=1 bsbHs ]n,n ≤ pn,
∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n
⋄ For fixedB : Optimizews, νs, τs (closed form solution)⋄ For fixedws, νs, τs : Optimizebs (SDP problem)
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 12 / 24
Transceiver design for Coordinated BS Systems Proposed Algorithms
Proposed Centralized Algorithm
Reformulated WSR max problem
minτs,νs,bs,ws
∑Ss=1 κs[
νγssτs
+ τµss (wH
s (HHs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1)]
s.t [∑S
s=1 bsbHs ]n,n ≤ pn,
∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n
Repeat⋄ For fixedB : Optimizews, νs, τs (closed form solution)⋄ For fixedws, νs, τs : Optimizebs (SDP problem)
Until Convergence
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 12 / 24
Transceiver design for Coordinated BS Systems Proposed Algorithms
Proposed Centralized Algorithm
Reformulated WSR max problem
minτs,νs,bs,ws
∑Ss=1 κs[
νγssτs
+ τµss (wH
s (HHs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1)]
s.t [∑S
s=1 bsbHs ]n,n ≤ pn,
∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n
Repeat⋄ For fixedB : Optimizews, νs, τs (closed form solution)⋄ For fixedws, νs, τs : Optimizebs (SDP problem)
Until Convergence
Computational complexity
Exist : O(K M2.376) + O(√
(N + S)(2NS + 1)2(2S2 + 2NS + S)) + CGP per iteProp: 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
Transceiver design for Coordinated BS Systems Proposed Algorithms
Proposed Distributed Algorithm
Reformulated WSR max problem
minτs,νs,bs,ws
∑Ss=1 κs[
νγssτs
+ τµss (wH
s (HHs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1)]
s.t [∑S
s=1 bsbHs ]n,n ≤ pn,
∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 13 / 24
Transceiver design for Coordinated BS Systems Proposed Algorithms
Proposed Distributed Algorithm
Reformulated WSR max problem
minτs,νs,bs,ws
∑Ss=1 κs[
νγssτs
+ τµss (wH
s (HHs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1)]
s.t [∑S
s=1 bsbHs ]n,n ≤ pn,
∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n
⋄ For fixedB : Same as centralized
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 13 / 24
Transceiver design for Coordinated BS Systems Proposed Algorithms
Proposed Distributed Algorithm
Reformulated WSR max problem
minτs,νs,bs,ws
∑Ss=1 κs[
νγssτs
+ τµss (wH
s (HHs BBHHs + σ2
s I)ws − 2ℜ{wHs HH
s bs}+ 1)]
s.t [∑S
s=1 bsbHs ]n,n ≤ pn,
∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n
⋄ For fixedB : Same as centralized
For fixedws, νs, τs
⋄ Formulatebs optimization as SDP⋄ Get dual of SDP: ({λn ≥ 0}N
n=1 are dual variables)⋄ Apply MFM and getλi iteratively by⋄ λ⋆
i = |g i |/√
pi ,g⋆i = λ(RRH + λ)−1f i
whereg i is ith row of [g1,g2, · · · ,gN ] (i.e.,needs inner iteration)R, f i are constants
⋄ Compute optimalbs by employing{λ⋆i }N
i=1
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 13 / 24
Transceiver design for Coordinated BS Systems Proposed Algorithms
Proposed Distributed Algorithm
⋄ For fixedB : Same as centralized
For fixedws, νs, τs
⋄ Formulatebs optimization as SDP⋄ Get dual of SDP: ({λn ≥ 0}N
n=1 are dual variables)⋄ Apply MFM and getλi iteratively by⋄ λ⋆
i = |g i |/√
pi ,g⋆i = λ(RRH + λ)−1f i
whereg i is ith row of [g1,g2, · · · ,gN ] (i.e.,needs inner iteration)R, f i are constants
⋄ Compute optimalbs 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 itePro(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
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 18 200
20
40
60
80
100
120
140
Number of iterations
Obj
ectiv
e fu
nctio
n
Small number of inner iteration is requiredIndeed distributed needs less computation than centralized
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 14 / 24
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 20 254
5
6
7
8
9
10
11
12
Number of iterations
Wei
ghte
d su
m r
ate
(bps
/Hz)
SNR=10dB
Proposed centralized algorithmProposed distributed algorithmExisting algorithm [1]
0 5 10 15 204
6
8
10
12
14
16
18
20
22
SNR (dB)
Wei
ghte
d su
m r
ate
(bps
/Hz)
Proposed centralized algorithmProposed distributed algorithmExisting algorithm [1]
Proposed algorithms have faster convergence than existingProposed algorithms have slightly higher WSR than existingDistributed algorithm achieves the same WSR as centralized
[1] Shi, S., Schubert, M., and Boche, H. ”Per-antenna power constrained rate optimizationfor multiuser MIMO systems”, Proc. WSA, Belrin, Germany, Feb., 2008.
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 15 / 24
Transceiver design for Coordinated BS Systems Drawbacks and Looking ahead
Drawbacks and Looking ahead
Drawbacks:
The proposed distributed algorithm is problem dependent (i.e., foreach problem we need to formulate its Lagrangian dual problem).
Looking ahead
The WSR max problem can be analyzed like in a conventionalmultiuser 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 andreceivers are interchanged.
Exploiting MSE duality for generalized power constraint shouldhelp to get problem independent distributed algorithm for manyclasses of transceiver design problems
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 16 / 24
Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements
System Model and Problem Statements
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
ObjectivesTo solve P1 and P2 by MSE duality approachTo show the benefits of the MSE duality solution approachTo show the extension of the duality for solving other transceiverdesign problems
P1 : minBk ,Wk
∑Kk=1
∑Ski=1 ηkiξ
DLki
s.t [∑K
k=1 Bk BHk ]n,n ≤ pn,
bHkibki ≤ pki , ∀n, k , i
P2 : min{Bk ,Wk}Kk=1
max ρkiξDLki
s.t [∑K
k=1 Bk BHk ]n,n ≤ pn,
bHkibki ≤ pki , ∀n, k , i
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 17 / 24
Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements
Existing MSE Uplink-downlink Duality (Revisited)
(a)
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
(b)
H1
H2
HK
n
TH
V1
V2
VK
d
d1
d2
dK
The duality can maintain ξDLki = ξUL
ki
The duality cannot ensure [∑K
k=1 BkBHk ]n,n ≤ pn and bH
kibki ≤ pki
P1 : min{Bk ,Wk}Kk=1
∑Kk=1
∑Ski=1 ηkiξ
DLki
s.t [∑K
k=1 Bk BHk ]n,n ≤ pn,
bHkibki ≤ pki
P2 : min{Bk ,Wk}Kk=1
max ρkiξDLki
s.t [∑K
k=1 Bk BHk ]n,n ≤ pn,
bHkibki ≤ pki
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 18 / 24
Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements
New MSE Downlink-Interference Duality
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
V1
V2
nI1S1
nIK 1
nIKSK
nI11
tHKSK
tHK 1
tH1S1
tH11
H111
H11S1
H21S1
H1KSK
H1K 1
HKK 1
HK 1S1
H2K 1
HKKSK
dK 1
d11
VK
HK 11
H2KSK
H211
d1
dKSK
d1S1d2
dK
P1 : min{Bk ,Wk}Kk=1
∑Kk=1
∑Ski=1 ηkiξ
DLki
s.t [∑K
k=1 Bk BHk ]n,n ≤ pn,
bHkibki ≤ pki
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements
New MSE Downlink-Interference Duality
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
V1
V2
nI1S1
nIK 1
nIKSK
nI11
tHKSK
tHK 1
tH1S1
tH11
H111
H11S1
H21S1
H1KSK
H1K 1
HKK 1
HK 1S1
H2K 1
HKKSK
dK 1
d11
VK
HK 11
H2KSK
H211
d1
dKSK
d1S1d2
dK
P1 : min{Bk ,Wk}Kk=1
∑Kk=1
∑Ski=1 ηkiξ
DLki
s.t [∑K
k=1 Bk BHk ]n,n ≤ pn,
bHkibki ≤ pki
⋄ Initialize Bk and updateWk by MMSE
Repeat
Transformation (DL to Interference )
⋄ Setd Iki ∼ (0, ηki ), nI
ki ∼ (0,Ψ + µki I),Ψ = diag(ψn)⋄ Getψn, µki iteratively
Key we show thatψn, µki > 0 always exist!
⋄ Setvki = wki and updatetki by MMSE
Transformation (Interference to DL )
⋄ Setbki = βtki , β2 =
∑Ki=1
∑Sij=1 ηij w
Hij Ri wij
∑Ki=1
∑Sij=1 tHij (Ψ+µij I)tij
⋄ UpdateWk by MMSE
Until convergence
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements
New MSE Downlink-Interference Duality
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
V1
V2
nI1S1
nIK 1
nIKSK
nI11
tHKSK
tHK 1
tH1S1
tH11
H111
H11S1
H21S1
H1KSK
H1K 1
HKK 1
HK 1S1
H2K 1
HKKSK
dK 1
d11
VK
HK 11
H2KSK
H211
d1
dKSK
d1S1d2
dK
P1 : min{Bk ,Wk}Kk=1
∑Kk=1
∑Ski=1 ηkiξ
DLki
s.t [∑K
k=1 Bk BHk ]n,n ≤ pn,
bHkibki ≤ pki
⋄ Initialize Bk and updateWk by MMSE
RepeatTransformation (DL to Interference )
⋄ Setd Iki ∼ (0, ηki ), nI
ki ∼ (0,Ψ + µki I),Ψ = diag(ψn)⋄ Getψn, µki iteratively
Key we show thatψn, µki > 0 always exist!
⋄ Setvki = wki and updatetki by MMSE
Transformation (Interference to DL )
⋄ Setbki = βtki ,wki =vkiβ, β2 =
∑Ki=1
∑Sij=1 ηij w
Hij Ri wij
∑Ki=1
∑Sij=1 tHij (Ψ+µij I)tij
⋄ DecomposeBk = Gk P1/2k , Wk = Gk P−1/2
k αk⋄ OptimizePk (increases convergence speed )
⋄ Again updateBk = Gk P1/2k andWk by MMSE
Until convergence
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements
New MSE Downlink-Interference Duality
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
V1
V2
nI1S1
nIK 1
nIKSK
nI11
tHKSK
tHK 1
tH1S1
tH11
H111
H11S1
H21S1
H1KSK
H1K 1
HKK 1
HK 1S1
H2K 1
HKKSK
dK 1
d11
VK
HK 11
H2KSK
H211
d1
dKSK
d1S1d2
dK
P2 : min{Bk ,Wk}Kk=1
max ρkiξDLki
s.t [∑K
k=1 Bk BHk ]n,n ≤ pn,
bHkibki ≤ pki
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements
New MSE Downlink-Interference Duality
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
V1
V2
nI1S1
nIK 1
nIKSK
nI11
tHKSK
tHK 1
tH1S1
tH11
H111
H11S1
H21S1
H1KSK
H1K 1
HKK 1
HK 1S1
H2K 1
HKKSK
dK 1
d11
VK
HK 11
H2KSK
H211
d1
dKSK
d1S1d2
dK
P2 : min{Bk ,Wk}Kk=1
max ρkiξDLki
s.t [∑K
k=1 Bk BHk ]n,n ≤ pn,
bHkibki ≤ pki
⋄ Initialize Bk and updateWk by MMSE
Repeat
Transformation (DL to Interference )
⋄ Setd Iki ∼ (0, 1), nI
ki ∼ (0,Ψ + µki I),Ψ = diag(ψn)⋄ Getψn, µki iteratively
⋄ Getβ2 , [β211, · · · β
2KSK
] asβ2 = Zx
x = [ψ1, · · · , ψN , µ11, · · · , µKSK], Z is constant
⋄ Setvki = βki wki and updatetki by MMSE
Transformation (Interference to DL )
⋄ Getβ2 , [β211, · · · β
2KSK
] asβ2 = Zx, Z is constant
Key we show thatψn, µki , β2ki , β
2ki > 0 always exist!
⋄ Setbki = βki tki and updateWk by MMSE
Until convergenceUnbalanced weighted MSE
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
Transceiver design for multiuser MIMO systems: Generalized duality System Model and Problem Statements
New MSE Downlink-Interference Duality
d2
d1 HH1
HH2
HHK
WH2
n1
nK
n2
WHK
WH1
d1
d2
dK
= d B
dK
V1
V2
nI1S1
nIK 1
nIKSK
nI11
tHKSK
tHK 1
tH1S1
tH11
H111
H11S1
H21S1
H1KSK
H1K 1
HKK 1
HK 1S1
H2K 1
HKKSK
dK 1
d11
VK
HK 11
H2KSK
H211
d1
dKSK
d1S1d2
dK
P2 : min{Bk ,Wk}Kk=1
max ρkiξDLki
s.t [∑K
k=1 Bk BHk ]n,n ≤ pn,
bHkibki ≤ pki
⋄ Initialize Bk and updateWk by MMSE
Repeat
Transformation (DL to Interference )
⋄ Setd Iki ∼ (0, 1), nI
ki ∼ (0,Ψ + µki I),Ψ = diag(ψn)⋄ Getψn, µki iteratively
⋄ Getβ2 , [β211, · · · β
2KSK
] asβ2 = Zx
x = [ψ1, · · · , ψN , µ11, · · · , µKSK], Z is constant
⋄ Setvki = βki wki and updatetki by MMSE
Transformation (Interference to DL )
⋄ Getβ2 , [β211, · · · β
2KSK
] asβ2 = Zx, Z is constant
Key we show thatψn, µki , β2ki , β
2ki > 0 always exist!
⋄ Setbki = βki tki ,wki = vki/βki and decompose
Bk = Gk P1/2k , Wk = Gk P−1/2
k αk⋄ OptimizePk , −Ensures balanced weighted MSE
−Increases convergence speed
⋄ Again updateBk = Gk P1/2k andWk by MMSE
Until convergence
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 19 / 24
Transceiver design for multiuser MIMO systems: Generalized duality Simulation Results
Simulation Results
10 15 20 25 30 350
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
Wei
ghte
d su
m M
SE
Proposed DualityAlgorithm in [1]
−25 −20 −15 −10 −5 07.5
8
8.5
9
9.5
10
σav2 (dB)
Tot
al B
S p
ower
Proposed DualityAlgorithm in [1]
10 15 20 25 30 350
0.05
0.1
0.15
0.2
0.25
SNR (dB)
Max
imum
sym
bol M
SE
Proposed DualityAlgorithm in [1]
−25 −20 −15 −10 −5 07
7.5
8
8.5
9
9.5
10
σav2 (dB)
Tot
al B
S p
ower
Proposed DualityAlgorithm 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 PowerConstraints for Network MIMO Systems”, Proc. IEEE ICC, Beijing, China, May, 2008.
P1P2
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 20 / 24
Transceiver design for multiuser MIMO systems: Generalized duality Simulation Results
Simulation Results
10 15 20 25 30 350
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
Wei
ghte
d su
m M
SE
Proposed DualityAlgorithm in [1]
−25 −20 −15 −10 −5 07.5
8
8.5
9
9.5
10
σav2 (dB)
Tot
al B
S p
ower
Proposed DualityAlgorithm in [1]
10 15 20 25 30 350
0.05
0.1
0.15
0.2
0.25
SNR (dB)
Max
imum
sym
bol M
SE
Proposed DualityAlgorithm in [1]
−25 −20 −15 −10 −5 07
7.5
8
8.5
9
9.5
10
σav2 (dB)
Tot
al B
S p
ower
Proposed DualityAlgorithm 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 PowerConstraints for Network MIMO Systems”, Proc. IEEE ICC, Beijing, China, May, 2008.
P1P2
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
Thesis Conclusions
Thesis Conclusions
In this PhD work, we accomplish the following main tasks:
We generalize the existing MSE duality to handle many practicallyrelevant transceiver design problems.
For stochastic robust design MSE-based problems, the duality canbe extended straightforwardly to imperfect CSI scenario.
For all of considered problems, the proposed duality algorithmsrequire less total BS power (and complexity) compared to theexisting solution approach which does not employ duality
The relationship between WSMSE and WSR problems have beenexploited. Consequently, the complicated nonlinear WSR problemcan be examined by its equivalent linear WSMSE problem
We also develop distributed transceiver design algorithms to solveweighted sum rate and MSE optimization problems
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 21 / 24
Future Research Directions
Future Research Directions
All of our algorithms are linear but suboptimal. So getting linearand optimal algorithm is still an open research topic (oneapproach could be to extend the well known Majorization theory toMultiuser MIMO setup).The proposed general duality is valid only for perfect CSI andimperfect CSI with stochastic robust design. The extension of theproposed duality to imperfect CSI with worst-case robust design isopen for future research.In all of our distributive algorithms, we assume that the globalchannel knowledge is available at the central controller (or at allBSs) prior to optimization. Thus, developing distributed algorithmwith local CSI knowledge is also an open research directionThe robust rate and SINR-based problems (i.e, in stochasticdesign approach) have not been examined. Hence, solving suchproblems is open research topic.
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 22 / 24
Future Research Directions
Selected List of Publications I
T. E. Bogale, B. K. Chalise, and L. Vandendorpe, Robusttransceiver 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 ofMIMO systems with arbitrary noise covariance matrices, 45thAnnual 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 optimizationfor downlink multiuser MIMO coordinated base station systems:Centralized and distributed algorithms, IEEE Trans. SignalProcess. 60 (2011), no. 4, 1876 – 1889.
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 23 / 24
Future Research Directions
Selected List of Publications II
, Weighted sum rate optimization for downlink multiuserMIMO systems with per antenna power constraint: Downlink-uplinkduality approach, IEEE International Conference On Acuostics,Speech and Signal Processing (ICASSP) (Kyoto, Japan), 25 – 30Mar. 2012, pp. 3245 – 3248.
, Linear transceiver design for downlink multiuser MIMOsystems: Downlink-interference duality approach, IEEE Trans. Sig.Process. 61 (2013), no. 19, 4686 – 4700.
Tadilo (PhD defense (UCL)) Transceiver design Dec. 2013 24 / 24