Transceiver design for single-cell and multi-cell downlink multiuser MIMO systems

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



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




(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


k αk UHk Γ

DLk Ukαk P−1/2

k

−2ℜ{P1/2k GH


ΓDLk = σ2

ek tr{Rbk GPGH}Rmk+

HHk GPGHHk + σ2IMk



k


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)




(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


k αk UHk Γ

DLk Ukαk P−1/2

k

−2ℜ{P1/2k GH


ΓDLk = σ2

ek tr{Rbk GPGH}Rmk+

HHk GPGHHk + σ2IMk



k


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)



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


k αk UHk Γ

DLk Ukαk P−1/2

k

−2ℜ{P1/2k GH


ΓDLk = σ2

ek tr{Rbk GPGH}Rmk+

HHk GPGHHk + σ2IMk



k


k Hk Uk Q1/2k }}

ΓULc =

∑Ki=1(σ

2ei tr{RmiUiQiUH

i }Rbi+

HiUiQiUHi HH

i ) + σ2IN




(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


k αk UHk Γ

DLk Ukαk P−1/2

k

−2ℜ{P1/2k GH


ΓDLk = σ2

ek tr{Rbk GPGH}Rmk+

HHk GPGHHk + σ2IMk



k


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




(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


k αk UHk Γ

DLk Ukαk P−1/2

k

−2ℜ{P1/2k GH


ΓDLk = σ2

ek tr{Rbk GPGH}Rmk+

HHk GPGHHk + σ2IMk



k


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


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




(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




(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


⋄ 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




(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





(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


⋄ 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


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


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


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



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



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




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



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




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



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.


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




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 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)




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 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 bs}+ 1)]

s.t [∑S

s=1 bsbHs ]n,n ≤ pn,

∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n



Proposed Centralized Algorithm

Reformulated WSR max problem

minτs,νs,bs,ws

∑Ss=1 κs[

νγssτs

+ τµss (wH

s (HHs BBHHs + σ2


s bs}+ 1)]

s.t [∑S


∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n





minτs,νs,bs,ws

∑Ss=1 κs[

νγssτs

+ τµss (wH

s (HHs BBHHs + σ2


s bs}+ 1)]

s.t [∑S


∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n

⋄ For fixedB : Optimizews, νs, τs (closed form solution)





minτs,νs,bs,ws

∑Ss=1 κs[

νγssτs

+ τµss (wH

s (HHs BBHHs + σ2


s bs}+ 1)]

s.t [∑S


∏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)





minτs,νs,bs,ws

∑Ss=1 κs[

νγssτs

+ τµss (wH

s (HHs BBHHs + σ2


s bs}+ 1)]

s.t [∑S


∏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





minτs,νs,bs,ws

∑Ss=1 κs[

νγssτs

+ τµss (wH

s (HHs BBHHs + σ2


s bs}+ 1)]

s.t [∑S


∏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!)



Proposed Distributed Algorithm


minτs,νs,bs,ws

∑Ss=1 κs[

νγssτs

+ τµss (wH

s (HHs BBHHs + σ2


s bs}+ 1)]

s.t [∑S


∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n





minτs,νs,bs,ws

∑Ss=1 κs[

νγssτs

+ τµss (wH

s (HHs BBHHs + σ2


s bs}+ 1)]

s.t [∑S


∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n

⋄ For fixedB : Same as centralized





minτs,νs,bs,ws

∑Ss=1 κs[

νγssτs

+ τµss (wH

s (HHs BBHHs + σ2


s bs}+ 1)]

s.t [∑S


∏Ss=1 νs = 1, νs > 0, τs > 0 ∀s,n


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





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!)


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


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.


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


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


bHkibki ≤ pki , ∀n, k , i



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


∑Kk=1

∑Ski=1 ηkiξ

DLki

s.t [∑K


bHkibki ≤ pki


max ρkiξDLki

s.t [∑K


bHkibki ≤ pki



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


∑Kk=1

∑Ski=1 ηkiξ

DLki

s.t [∑K


bHkibki ≤ pki




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


∑Kk=1

∑Ski=1 ηkiξ

DLki

s.t [∑K


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




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


∑Kk=1

∑Ski=1 ηkiξ

DLki

s.t [∑K


bHkibki ≤ pki


RepeatTransformation (DL to Interference )

⋄ Setd Iki ∼ (0, ηki ), nI


Key we show thatψn, µki > 0 always exist!

⋄ Setvki = wki and updatetki by MMSE


⋄ 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




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


max ρkiξDLki

s.t [∑K


bHkibki ≤ pki




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


max ρkiξDLki

s.t [∑K


bHkibki ≤ pki


Repeat


⋄ Setd Iki ∼ (0, 1), nI


⋄ Getβ2 , [β211, · · · β

2KSK

] asβ2 = Zx

x = [ψ1, · · · , ψN , µ11, · · · , µKSK], Z is constant

⋄ Setvki = βki wki and updatetki by MMSE


⋄ 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




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


max ρkiξDLki

s.t [∑K


bHkibki ≤ pki


Repeat


⋄ Setd Iki ∼ (0, 1), nI


⋄ Getβ2 , [β211, · · · β

2KSK

] asβ2 = Zx

x = [ψ1, · · · , ψN , µ11, · · · , µKSK], Z is constant

⋄ Setvki = βki wki and updatetki by MMSE


⋄ 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


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


10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

SNR (dB)

Max

imum

sym

bol M

SE


−25 −20 −15 −10 −5 07

7.5

8

8.5

9

9.5

10

σav2 (dB)

Tot

al B

S p

ower


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


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


−25 −20 −15 −10 −5 07.5

8

8.5

9

9.5

10

σav2 (dB)

Tot

al B

S p

ower


10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

SNR (dB)

Max

imum

sym

bol M

SE


−25 −20 −15 −10 −5 07

7.5

8

8.5

9

9.5

10

σav2 (dB)

Tot

al B

S p

ower


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)


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


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.



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.



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.


Engineering

Transceiver design for single-cell and multi-cell downlink multiuser MIMO systems