MSE uplink-downlink duality of MIMO systems under imperfect CSI

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)

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 Tadilo Endeshaw, Université 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

kkd C

Motivation of MSE duality Consider the following downlink system model

)(W K21 W,,W,W blkdiag kk SM

kW CkSN

kB C

Motivation of MSE duality cont’d 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


k

^

d

kd

Direct treatment of the above problem has

Complicated mathematical structure.

Difficult to examine.

Now, let us also see the following uplink system model


Motivation of MSE duality cont’d

K21 T,,T,TT

)(V K21 V,,V,V blkdiag

K21 H,,H,HH



The above uplink problem has - Simple mathematical structure. - Global optimal solution.

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.



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.



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


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.


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


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.


Application example

Application example cont’d


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.


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




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



where


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



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.



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



- With the optimal , are

updated by MMSE receiver.

Uplink channel

- First we ensure

- With the optimal , Update by

MMSE receiver.


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 k’s # of antennas Alg I: The proposed alternating optimization


kM

kM

We model Rc as


Simulation result (For case II)

Comparison of robust and naive designs K=2, N=4 and = 2 kM

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.


Thank You!


Engineering

MSE uplink-downlink duality of MIMO systems under imperfect CSI