1
Matrix-Field Water-Filling Architecture for
MIMO Transceiver Designs with Mixed Power
Constraints
NAME：Xiaoyu Liu
Abstract
The paper discuss MIMO transceiver design
under a new type of power constraint named mixed power
constraints. In the paper,author try to solve the optimization
problem in an analytical way instead of relying on some
famous software packagese.g.,CVXorSeDuMi. The specfic
formula of the optimal signal covariance matrix has been
derived. Based on the structure, a low complexity non-iterative
solution is given in the paper. This solution has a much clear
engineering meaning and is suitable for practical
implementations even for massive MIMO systems.
Motivation
For transceiver designs, power constraints are the most
natural and fundamental constraints,and the widely used power
constraint is sum power constraint. In the motivation,the author
introduce the performance gap between the individual power
constraint and sum power constraint .While sum power
constraint is very small and almost negligible. After description
the advantage and disadvantage of the individual power
constraint and sum power constraint.
The paper point out a new power constraint model called
mixed power model.This power constraint model includes both
sum power constraint and per-antenna power constraint models
as its special cases.
The motivation to choose this new power constraint is that
mixed power constraints and the final solutions can be derived
in much simple closed forms without too many mathematical
symbols.The main contributions of this work is the transceiver
design with mixed power constraint can realize any tradeoff
between sum power constraint and per-antenna power
constraint. The formula of the optimal solution has been
derived.
System model
At the beginning of this section, the capacity
maximization problem for a point-to-point MIMO system
under mixed power constraint is formulated as :
（
1
）
Q is the covariance matrix of the transmitted signal and H is the
MIMO channel matrix.Rn is the noise covariance matrix . The
symbol ej denotes the vector with the jth element being 1 and
the other elements being zeros. ψ k is a subset of the antenna
index.
To avoid case-by-case discussions exploiting the fact that
the considered objective function is continuous an auxiliary
variable
α
is introduced first and then maximizing the
following objective function will have the same optimal
solution as that of the original optimization problem.
(2)
take the limit α → 0 on the derived optimal solution to
achieve the exact optimal solution.
(3)
Matrix-field water-filling solution
The Lagrangian dual function of (3) is given as :
(4)
where in the final equation D is a diagonal matrix defined as
(5)
For the simplicity of analysis,defining a new kind of
matrix given as
and
H
H H
based on the Lagrangian dual function, the KKT
conditions of (3) can be derived to be
(6)
Using the first KKT condition in (6), the covariance
matrix Q can be solved to be
1
max log/ I R /
s.t. {e Qe } , 0
H
n
H
j j
j k
HQH
pk Q
1
0
m ax lim log/ I (H R H a I) Q /
k
H
n
a
Q
1
max log (H R H a ) Q
. {e Qe } , Q 0
H
n
Q
H
j j
j k
I I
s t pk
1
L(Q,{d }, ) log (H R H aI) Q {d p }
{Tr(DQ)} Tr ( Q)
H
k n k k
K
k
I
,
,
k k
j j
D d j
1
(I HQH )
( {e e }Q) 0
0, (Q )=0, 0
{e Qe } , Q 0,
k
H H
H
k k j j
j
k
H
j j k
j
H H D
d p Tr
d Tr
p
1 1
1 1 1 1
1 1
2 2 2 2
( D ) ( H H )
( I D D ) (H H )
H
H
Q
D D
1 1/2
(H R H )
H
n
H aI
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