# Maximum Data Rate Power Allocation for MIMO Spatial Multiplexing Systems with Imperfect CSI

**ABSTRACT** In MIMO systems, spatial multiplexing is a powerful technique for increasing channel capacity by transmitting multiple data streams in the same channel simultaneously. Moreover, additional performance can be extracted in the presence of channel state information (CSI) at the transmitter. However, channel estimation error usually exists in practical systems and leads to imperfect CSI. As a result, the system performance is degraded. Fortunately power allocation can mitigate the problem effectively. In this paper, the power allocation problem is investigated in the case of imperfect CSI with accurate system model. A greedy power allocation (GPA) algorithm with adaptive modulation scheme is proposed to maximize the system data rate while satisfying each data stream's bit error rate requirement. Simulation results show that GPA can reduce the effects of imperfect CSI and obtain better performance than other traditional algorithms, e.g. waterfilling and equal power allocation algorithms.

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**ABSTRACT:**This paper presents an overview of progress in the area of multiple input multiple output (MIMO) space-time coded wireless systems. After some background on the research leading to the discovery of the enormous potential of MIMO wireless links, we highlight the different classes of techniques and algorithms proposed which attempt to realize the various benefits of MIMO including spatial multiplexing and space-time coding schemes. These algorithms are often derived and analyzed under ideal independent fading conditions. We present the state of the art in channel modeling and measurements, leading to a better understanding of actual MIMO gains. Finally, the paper addresses current questions regarding the integration of MIMO links in practical wireless systems and standards.IEEE Journal on Selected Areas in Communications 05/2003; · 3.12 Impact Factor - SourceAvailable from: cttc.es[Show abstract] [Hide abstract]

**ABSTRACT:**We present the optimum design of a linear transmitter for a multi-input multi-output communication system when the input data consists of two QPSK streams and the receiver performs maximum likelihood detection. The transmitter design is optimal in the sense that it minimizes the worst-case pairwise error probability. We prove that the resulting linear transmitter constructs, as a function of the two input QPSK streams, a new signal constellation. Moreover, we show that this new signal constellation is such that the minimum distance among the points of the received constellation is maximized and, indirectly, the number of neighbors at minimum distance is also maximized.Communications, 2006. ICC '06. IEEE International Conference on; 07/2006 - SourceAvailable from: Gilles Burel[Show abstract] [Hide abstract]

**ABSTRACT:**We propose a Minimum Bit Error Rate (MBER) diagonal Precoder for Multi-Input Multi-Output (MIMO) transmission systems. This work is based on previous results obtained by Sampath et al. (1) in which the global transmission system (precoder and equalizer) is optimized with the Minimum Mean Square Error (MMSE) crite- rion. This process leads to an interesting diagonality property which decouples the MIMO channel into parallel and independent data streams and allows to perform an easy ML detection. This system is then optimized using a newdiagonal precoder that minimizes the BER. Our w ork is motivated by the fact that, from a prac- tical point of view, people are likely to prefer a system that minimizes the BER rather than the Mean Square Error. The performance improvement is illustrated via Monte Carlo simulations using a Quadratic Amplitude Modulation (QAM).Signal Processing. 01/2002; 82:1477-1480.

Page 1

Maximum Data Rate Power Allocation for MIMO

Spatial Multiplexing Systems with Imperfect CSI

Xin Jin1,2, Haiping Jiang3, Jinlong Hu1,Yao Yuan1, 2,Cuicui Zhao1, 2, Jinglin Shi1

1 Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China

2 Graduate University of Chinese Academy of Sciences, Beijing, China

3 GSM - Institut Fresnel, Ecole Centrale Marseille, Marseille, France

jxstar@gmail.com

Abstract—In MIMO systems, spatial multiplexing is a powerful

technique for increasing channel capacity by transmitting

multiple data streams in the same channel simultaneously.

Moreover, additional performance can be extracted in the

presence of channel state information (CSI) at the transmitter.

However, channel estimation error usually exists in practical

systems and leads to imperfect CSI. As a result, the system

performance is degraded. Fortunately power allocation can

mitigate the problem effectively. In this paper, the power

allocation problem is investigated in the case of imperfect CSI

with accurate system model. A greedy power allocation (GPA)

algorithm with adaptive modulation scheme is proposed to

maximize the system data rate while satisfying each data

stream’s bit error rate requirement. Simulation results show that

GPA can reduce the effects of imperfect CSI and obtain better

performance than other traditional algorithms, e.g. waterfilling

and equal power allocation algorithms.

I.

INTRODUCTION

Multiple-Input Multiple-Output (MIMO) technology has

been recognized as a key candidate for future wireless

communication systems. MIMO exploits multipath scattering

effects and extremely enhances channel capability [1, 2]. In

spatial multiplexing schemes, multiple independent data

streams can be transmitted simultaneously from individual

antennas. Under conducive channel conditions, such as rich

scattering, the receiver can separate different streams, yielding

a linear increase in channel capacity. Signal reception is

dependent on the criterion that receiver adopts, such as zero-

forcing (ZF), minimum mean-squared error (MMSE) and

maximum likelihood (ML) [3].

To maximize the system gains, the resources should be

effectively allocated among the antennas. A lot of efforts, such

as literatures [4,5], have been made to design better transmitter

structures so as to optimize the power allocation for each signal

stream by precoding. M. Payaró [4] proposes an optimum

design of a linear transmitter for MIMO systems with two

QPSK input data streams when both sides have perfect CSI and

the receiver performs ML detection. However the extension to

multiple streams becomes much more complicated. A

minimum bit error rate (BER) diagonal precoder is proposed in

[5]. Nevertheless, precoding requires high quality of channel

state information (CSI). When CSI is not accurate, the

performance of the aforementioned algorithms will be affected

significantly. Besides, the channel matrix is usually

transformed according to specific precoding algorithms, such

as singular value decomposition (SVD) [6] and block

diagonalization (BD) [7],

computational complexity.

which will increase the

Basically, power allocation at the transmitter will achieve

better system performance with perfect CSI knowledge.

However, perfect CSI can be hardly obtained in practical

systems, especially for systems equipped with a large number

of antennas. This is because neither the transmitter nor the

receiver has access to the ideal CSI. It is very likely that the

receiver has imperfect channel knowledge due to channel

estimation error. Moreover, the imperfect channel knowledge

may be further corrupted by delay, noise and interference in the

feedback channel. Consequently, it is more practical that the

power allocation algorithm is designed based on imperfect CSI.

Recently, Wang et al. in [8] optimize the transmitted power of

signal streams by applying minimum bit-error rate criterion and

the proposed scheme can reduce both transmitter complexity

and feedback overhead significantly. Moreover, power

allocation for multicarrier MIMO systems with imperfect or

partial CSI is considered in [9, 10] and maximum SNR or

transmission rate are employed as the optimization criteria.

In this paper, power allocation algorithm is investigated for

MIMO spatial multiplexing systems with imperfect CSI by

adopting accurate system model. We aim at maximizing the

system data rate while satisfying the system bit error rates

(BER) requirements. Accordingly, a greedy power allocation

(GPA) algorithm is proposed which is composed by a greedy

search scheme and an adaptive modulation scheme. Then,

simulation results show that GPA can mitigate the effects of

imperfect CSI and strike a good balance between the system

data rate and the BER requirements than traditional algorithms,

e.g. the waterfilling and the equal power allocation algorithms.

The reminder of this paper is organized as follows. Section

II describes MIMO system model, and Section III formulates

the power allocation problem and then a practical power

allocation algorithm is presented in Section IV. Numerical

results are shown in Section V and conclusions are drawn in

Section VI.

In this paper, bold uppercase (lowercase) letters denote

matrix (vector). For a matrixM ,

H

M ,

1

−

M ,

†

M denote M ’s

978-1-4244-2517-4/09/$20.00 ©2009 IEEE

Page 2

Hermitian, inverse and pseudo-inverse matrix, respectively.

I

is the identity matrix of size N

diagonal matrix. ( )E ⋅ denotes expectation operation.

N N

×

N

×

and

()diag ⋅ is a

II.

SYSTEM MODEL

In this section, the MIMO system model is presented. We

consider a link between a transmitter and a receiver. The

transmitter is equipped with

equipped with

r

N antennas.

It is assumed that the transmitted signal vector is d ,

∈

d

?

and ()

tt

NN

E

×

ddI. When each signal vector is

allocated with power

modeled as =s Pd, wheres is a N × vector andP is the power

t N antennas while the receiver is

1

t

N ×

H

=

ip , the transmitted signal vector can be

1

t

( diag

=

P

allocation vector, denoted as

12

,,,)

t

N

ppp

…

.

Then the received signal can be modeled as:

+

r = Hs n, (1)

channel matrix with zero mean unit

where H is the

variance i.i.d complex Gaussian entries, andn is the AWGN

vector,

1

~(0,)

r

nN

CN

×

nI.

Based on the channel inversion scheme, the MIMO channel

can be divided into a set of independent parallel Single Input

Single Output (SISO) channels. Each of the channels is used

to transmit a single signal stream which is modulated

independently from the other streams. If the transmitter has

the knowledge of perfect CSI and channel inversion scheme is

used, then the received signal vector can be written as:

=+

sG(Hs n)

?

where

=

G = H(H H) H .

In practice, channel estimation error may exist when the

channel matrix H is estimated at the receiver. Therefore, most

of the time, only an imperfect estimate ˆH of the channel

matrix H will be available. It can be modeled as:

ˆ

H = H

where

∈

E

?

is the channel estimation error matrix that is

uncorrelated with H and each of its entries is i.i.d. Gaussian-

distributed with a variance

σ composed of the real and

imaginary parts, i.e.

()2E

=

EE

In the case of imperfect channel knowledge, the ZF

estimation of the transmitted data vector is given by:

ˆ

ˆ =+

sG(Hs n)

ˆ

ˆ

()

==+

GHHE .

rt

NN

×

2

σ

= +

s Gn

, (2)

†1HH

−

+

E, (3)

rt

NN

×

2

e

σ

2

e

NN

rr

H

t

N

×

I

.

ˆˆ

=+

GHs Gn, (4)

where

††

III. PROBLEM FORMULATION

In this section, the power allocation problem is investigated

in order to maximize the system data rate while satisfying the

BER requirements.

From (4), the data-estimation error can be expressed as:

1

1

D

E

1

N

ˆ

= −

(() ( )) () ( )

((

?????????????

) ( )) ()

((

?????? ?????? ?

) ()) ()

HH

HH

HH

−

−

−

− =

s s

+

H

+

H

+

H

+−

+++

++++

HEHEHEHs ns

EEE Es

HEHHEn

. (5)

When each element of estimation errors matrixE has a small

variance

e

σ

, the data-estimation error functions can be

approximated with a Taylor series at

truncated after the linear terms, i.e.,

⎡

∂

⎢

− ≈ −

⎢

∂ℜ

⎢⎣

2

=

E0 , which is

[ ]

E

{}

[ ]

E

{}

[ ]

E

{}

[ ]

E

{}

[ ]

E

{}

[ ]

E

{}

[ ]

E

{}

[ ]

E

{}

,

,

,

=0

,

,

,

=0

1

,

,

,

=0

,

,

,

=0

D

ˆ

s s

D

()

N

N

rt

rt

rt

rt

rt

rt

rt

rt

rt

rt

rt

rt

n n

n n

n n

n n

n n

n n

HH

n n

n n

n n

n n

n n

n n

−

ℜ

⎤

⎥

⎥

⎥⎦

∂

+ℑ⋅

∂ℑ

⎡

+⎣

∂

+ℜ

∂ℜ

⎤

⎥

⎥

⎥

∂

+ℑ

∂ℑ

⎦

∑

∑

∑

∑

E

E

E

E

s

H HH

⋅

n

. (6)

Based on [11], (6) can be further computed as:

ˆ

(

− ≈ −

+

−

= −

H Es

Given the equation (7), the post-processing SINR of the

signal stream is defined as:

1

1

11

1111

)

() ()

( ) (

+

)()

HH

HHH

HHHHH

−

−

−−

−−−−

+

+

H EH n

−

s sH HH Es

H HHEn

H H H E E H H HH n

H n

. (7)

th

k

( )

k

[]

ˆ

s s s s

ˆ

[()() ]

H

kk

H

kk

E

−

E

γ=

−

ss

, (8)

where ( )kk

signal vector s , the transmit power can be expressed as:

=

ssP

i

refers to the ( , )th

k k

element of the matrix. For the

()

2

12

[],,,

t

H

N

E diag p pp

=

…

. (9)

The covariance matrix of ˆ −

R

−

=

≈

+

=

where PTis the total transmitted power, tr() ⋅ is the trace of the

matrix. The derivation in detail is given in Appendix.

Substituting equation (9) and (10) into (8), we can rewrite the

SINR of the

k signal stream as

s s can be written as:

) ]

−

]

EP E H

HE H H E

ˆ

s s

12-12

n

1

2

n

-1-1-1

2

e

2

n

2

n

2

e

-11

ˆ

s s s s

ˆ

[()(

[()()

{ (

E

σ

+

)}()

(2P2tr(( ) ))()

H

HHH

HHH

HH

T

E

E

σ

σ

σσ σ

−−

−

−

+

+

HH H

H

H HH H

(10)

th

Page 3

2

e

2

n

2

n

2

e

-11

[]

ˆ

s s s s

ˆ

[()() ]

[22tr((

≥

) )][(

σ ? ，the term

) ]

H

kk

k

H

kk

k

HH

T kk

E

−

E

p

γ

σσσ σ

−

=

−

=

++

ss

PH HH H

. (11)

According to [12], when

σ

H H

therefore can be neglected. Consequently, the SINR becomes:

p

γ

σσ+

=

where

[2][(

kneT

σσσ=+

PH H

In [13], the BER of the MIMO system using MQAM

modulation is expressed as:

rt

NN and

2

e

1

2

e

-1

tr(() )

H

in (11) can be approximated as zero, which

2

n

2

e

1

[2][() ]

1,2,,

kk

k

H

kTkk

…

t

p

kN

σ

−

==

PH H

, (12)

221

) ]

H

kk

−

.

(

k

)

1.6

Rate

0.2exp

21

k

M

k

BER

γ

−

−

⎡

⎢

⎣

⎤

⎥

⎦

k

≈

, (13)

where

k

Rate denotes the rate of the

th

stream, and

M ≥

2

log

k

BER

k

RateM

≤

=

.

(

k

)M

BER is tight within 1 dB when 4

k

and

signal stream transmit power can be written as:

p

σ

=−

Based on the discussions above, the power allocation

problem in each allocation cycle can be defined as follows.

The channel matrix is assumed constant over each allocation

cycle and is uncorrelated in time. Each active transmit antenna

will be allocated with certain power

Our main target is to allocate the total power among different

antennas so as to maximize the system data rate while meeting

each stream’s BER demand. Accordingly, the problem can be

formulated as:

N

f xRate

=

≤

∑

(

k

)3

10

M

−

. From equation (12) and (13), the

th

k

(21)

k

Rate

kk

A

=−

, (14)

where

(

k

)

ln5

1.6

M

k

k

ABER.

kp ( 0

≥

) to transmit data.

1

(

k

)

1

( )maxa

.(15b)

t

k

k

M

th

N

k

k

st BERBER

p

=

= (15 )

∑

(15c)

0d

t

T

k

P

p

≤

The objective function

current power allocation cycle. Constraints (15b) are the BER

requirement of each signal stream to guarantee that the signals

can be correctly received at the receiver. Constraints (15c) and

(15d) are the total power constraint with each power

assignment larger than zero.

≥ (15 )

( )f x is the sum of date rates at the

IV. POWER ALLOCATION ALGORITHM

In this section, we describe in detail how to solve the above

power allocation problem with a greedy power allocation

(GPA) algorithm.

To maximize the system data rate, it is obvious that during

each allocation cycle, the transmitter should transmit with its

maximum power

TP . So the problem can be rewritten as:

∑

1

(

k

)

1

( )f x maxa

. (16b)

t

N

k

k

M

th

N

k

k

Rate

st BERBER

p

=

=

= (16 )

≤

(16c)

0d

t

T

k

P

p

=

To solve this problem, we combine the greedy search

scheme and the adaptive modulation scheme together. For

each signal stream, the power is initiated as zero, and their

modulation orders are also set to zero. Then if the modulation

order of each signal stream is added by one, the required

incremental power is calculated according (14) as:

p

p

Rate

∂

Only one signal stream which requires the minimum

incremental power will be really allocated. In addition, the

algorithm should also guarantee each stream’s target BER

requirement by (13) and (15b). This makes each stream match

the best with the channel and the greatest spectral efficiency

can be achieved. Take the MIMO transmission scenario with 4

streams for example, which is described as Table I and II.

Provided that the power increment of the first stream, which is

used to upgrade its modulation order by one, is the smallest.

Then the first stream is selected and assigned with

Besides its modulation order is actually increased by one.

≥ (16 )

∑

k

k

k

Rate

∂

Δ =Δ

. (17)

1p

Δ

.

TABLE I

GPA STATE 1

Stream

index

Rate increment Power increment

1 0?1

1p

Δ

(min)

2 0?1

2p

Δ

3 0?1

3p

Δ

4 0?1

4p

Δ

TABLE II

GPA STATE 2

Stream

index

Rate increment Power increment

1 1?2

'

1p

Δ

2 0?1

'

2p

Δ

3 0?1

'

3p

Δ

(min)

4 0?1

'

4p

Δ

Page 4

Sequentially, the process in Table II begins. At this time the

third signal stream which is with smallest power increment is

chosen to allocate power of

Δ

terminated when the total power limit

overall GPA algorithm can be concluded as follows:

Step 1: Initialize the allocated power and modulation order

to zero for each signal stream.

Step 2: Calculate the required power increment

if each signal stream increases its modulation order by one.

Step 3: Select the signal stream k with the smallest

incremental power and allocate the corresponding power

Step 4: If all the power

Otherwise return to step 2.

'

3p . The allocation procedure is

TP is reached. The

kp

Δ

by (17)

kp

Δ

.

TP has been allocated, then stop.

V.

PERFORMANCE ANALYSIS

Simulation results and analysis are presented in this section.

We consider a system without coding for the sake of

straightforward comparison. In addition, the specific

propagation conditions are not considered in our model, e.g.

angles of the signal arrival and departure and their spread.

Moreover, rich scattering environment is assumed, where

independent Rayleigh fading between different antenna pairs

occurs. In our simulations, adaptive modulation (M-QAM

with M = 0, 1, 2, 3, 4, 5) is applied and the corresponding

transmission rates are 0, 1, 2, 3, 4 and 5, respectively.

Specially, the transmission rate of 0 means channel quality is

too bad to transmit data. In this case, no power will be

allocated to this antenna..

We analyze three different algorithms, i.e. GPA,

waterfilling algorithm (WF) [14] and equal power (EQ)

allocation algorithms respectively. For better comparison, the

same total power is assumed in all scenarios.

In Fig. 1, the number of received data bits normalized to

symbol duration is plotted as a function of the SNR, for a Nt =

Nr = 4 system with 0.05

e

σ = . The target BER of each data

stream is 10-3. SNR is defined as the ratio of the total power to

the noise power at the receive antenna. We can see that GPA

can achieve the greatest system data rate, which is superior to

WF and EQ. This is because GPA adopts adaptive modulation

scheme for each stream and chooses the stream with minimum

incremental power to allocate resource. As for the other

scenarios, e.g. with different estimation error variance

different number of antennas in Fig. 2 and Fig. 3, there will be

some similar simulation results.

In Fig. 2, the relationship between

rate is investigated in the Nt = Nr = 4 system with

0.1 or 0.15. With the increase of

data rate decreases. However, the increase of

influence on GPA than that on WF and EQ. As a matter of fact,

when the modulation order is set to 5 for all data streams, the

maximum system data rate can be calculated as 20 bits per

2

e

σ or

e

σ and the system data

σ = 0, 0.05,

σ , the system achievable

e

e

e

σ has less

024681012

0

2

4

6

8

10

12

14

16

SNR (dB)

data bits/symbol time

GPA

WF

EQ

Fig. 1. Data rates vs. SNR for 4×4 multiplexing system with

0.05

eσ =

024681012

0

2

4

6

8

10

12

14

16

18

20

SNR (dB)

data bits/symbol time

GPA,deltae=0

GPA,deltae=0.05

GPA,deltae=0.1

GPA,deltae=0.15

WF,deltae=0

WF,deltae=0.05

WF,deltae=0.1

WF,deltae=0.15

EQ,deltae=0

EQ,deltae=0.05

EQ,deltae=0.1

EQ,deltae=0.15

Fig. 2. Data rates vs. SNR for 4×4 system with different

e

σ

symbol time. From Fig. 2, it can be seen that in the case of

0

e

σ =

, which means the transmitter have perfect CSI, the

achievable data rates using GPA, WF and EQ are close to the

maximum system data rate at 12 dB. When

0.05, the system data rate of GPA decreases by 4 bits per

symbol time while the performance of WF and EQ decreases

by 6 and 9 bits per symbol time, respectively. This indicates

that GPA is more robust in practice which can effectively

mitigate the effect of imperfect CSI.

Then how the antenna number affects the system

performance is investigated in the case of

the system data rates increase when the antenna number goes

up with high SNR. Again, GPA achieves the best system

performance for all scenarios. When antenna number increases

with the SNR and total power fixed, the power per antenna

decreases. As a result, lower modulation order is applied in

order to meet the BER constraints. However, the system data

rate is still increased due to the spatial multiplexing technique.

Furthermore, it is noticeable that when SNR is low, the

e

σ varies from 0 to

0.05

e

σ = . In Fig. 3,

Page 5

024681012

0

5

10

15

20

25

30

SNR (dB)

data bits/symbol time

GPA,Nt=8,Nr=8

GPA,Nt=4,Nr=4

GPA,Nt=2,Nr=2

GPA,Nt=1,Nr=1

WF,Nt=8,Nr=8

WF,Nt=4,Nr=4

WF,Nt=2,Nr=2

WF,Nt=1,Nr=1

EQ,Nt=8,Nr=8

EQ,Nt=4,Nr=4

EQ,Nt=2,Nr=2

EQ,Nt=1,Nr=1

Fig. 3. Data rates vs. SNR for 1×1, 2×2, 4×4, 8×8 multiplexing systems

system data rates of WF and EQ decrease rather than increase

as the antenna number goes up. The main reason is that the

power allocated to each data stream using WF and EQ

becomes so little that the power can not guarantee its target

BER requirement. Consequently, nothing is transmitted on the

antenna which leads to the decrease of the system data rate.

VI. CONCLUSION

In this paper, the power allocation algorithm is investigated

in MIMO spatial multiplexing systems with imperfect CSI.

When the channel estimation error occurs in practice, CSI

becomes inaccurate, which leads to the degradation of the

system performance. Therefore, in order to keep the system

stable in the imperfect CSI condition, a greedy power

allocation (GPA) algorithm is proposed, which is composed

by the greedy search scheme and the adaptive modulation

scheme. Simulation results indicate that GPA can effectively

mitigate the effect of imperfect CSI and perform better than

two traditional power allocation algorithms, which can not

only maximize the system data rate but also satisfy the system

bit error rate (BER) requirements.

APPENDIX

In this appendix, we derive the covariance matrix of ˆ −

the presence of channel estimation error. Setting out from (7),

the covariance matrix

ˆ

R−

ˆˆ

R[( )() ]

E

−

−

≈]+

−

+

H E H H E

In addition, we have the following relations:

222

[2tr(

e

E

σ]=

EP EP I

σ=

E H H E

s s in

s s reads:

ˆ

s s

12-12

n

}−

H

1

2

n

-1-12

n

-1 -1

2

n

-1-1 -1

[()()

{()(){( ) }

{()( ) }

H

HHH

HHHH

HHH

E

σ

σ

EE

E

σ

σ

−

=−−

s s s s

H EP EHH H

H H EHH E H H

(18)

2

e

NNNN

)2

rrrr

H

T

σ

××

=

P I

, (19)

-12

e

-1

NN

{ (

E

)}2tr(() )

rr

HHH

×

H HI

. (20)

Consequently, using (19) and (20), (18) can be simplified as:

221

ˆ

R2tr()()

2 tr(() )(

ne

σ σ

σσσ σ =++

P

2

n

1

22-11

2

e

2

n

2

n

2

e

-11

()

)

(22tr(() ))()(21)

HH

e

HH

HH

T

σσ

H H

−−

−

−

−

≈+

+

s s

P H HH H

H H

H HH H

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