Thesis Organization

The focus of this thesis is on the joint beamforming design of the transmitter and receiver for multi-user MIMO-OFDM SDMA under channel estimation error. In order to clearly and completely describe the whole system, we separate the system into three topics and then analyze these topics in following chapters step by step.

The thesis is organized as follows. In chapter 2, we analyze the joint Tx/Rx MMSE beamforming design for single user MIMO channel case when the CSI is perfect known at both terminals and extend it to MIMO-OFDM system according to principles of chapter 1.1. In chapter 3, we investigate the joint beamforming design problem in multi-user MIMO-OFDM SDMA system with perfect CSI. Chapter 4 considers the case that the channel estimation contains errors. We apply some robust methods to improve the system performance. Furthermore, we use the moving average approach to enhance the system performance under slow time-variant channel environment. In last chapter, we conclude the thesis with some respectives.

--- Chapter 2

Joint Tx/Rx MMSE Beamforming Design for Single User MIMO-OFDM Downlink System

with Perfect CSI

---

In chapter 1, we have introduced the basics of OFDM system, channel models and properties of MIMO systems, and the MIMO-OFDM based system. We now concentrate on the joint Tx/Rx beamforming design for single user MIMO downlink system under the flat-fading channel and perfect CSI conditions. According the concepts of MIMO-OFDM system described in section 1.1.3, we then extend the joint beamforming design problem to MIMO-OFDM based system which obtains spectral-efficiency and combats the frequency selective channel environment.

2.1 Introduction

For MIMO system, space-time coding and spatial multiplexing are promising techniques for achieving high data rates requirement. Neither of the two techniques require CSI at transmit side. However, in some situations, channel information could

be available at the transmitter because of the feedback information from the receiver.

If the CSI is known at both of transmit and receive terminals, the optimal solution is provided by the eigen-value decomposition (EVD) weighting scheme combined with a water-pouring strategy [10]. However, instead of using the optimal solution, a sub-optimal solutions using a fixed number of data streams and a fixed identical modulation/coding scheme have been proposed [11]. Other joint designs subject to different constraints and criteria also have been presented [6] [12-14]. The joint Tx/Rx design diagonalizes the MIMO channel into eigen subchannels; achieve the symbol by symbol detection and the system structure can be scalable with respect to the number of antennas, size of the coding block, and transmit power. In this chapter, we focus on the joint MMSE beamforming design which minimizes the accumulative mean square error subject to the total transmitted power as the constraint.

2.2 Joint Tx/Rx Beamforming MIMO System Models 2.2.1 Single Carrier Flat-fading Case

The model of joint Tx/Rx beamforming design for single user MIMO downlink system in flat-fading channel environment is illustrated in Figure 2-1. We consider a wireless communication system with

M transmit and N

receive antennas under the flat-fading environment. Thus, the flat-fading MIMO channel H can be represented by a channel matrix with dimension

N × M

. The input symbol streams are passed through the transmit beamforming F (pre-filter) which is optimized for a known channel and then the pre-filter output is transmitted into the flat-fading MIMO

channel. The received signal is processed by the receive beamforming

G

(post-filter). In this thesis, we will not consider coding and modulation design and only concentrate on transmit and receive beamforming design.

… BS

s (k) t (k)

M 1 2

r (k)

…

N 1 2 MS

s (k)

…

…

Downlink

H G F

For a MIMO channel without any delay-spread, the joint beamforming design system equation can be written as

Gn GHFs

s

ˆ= + (2.2.1-1)

where H is the MIMO channel matrix described as above, sˆ is the

B

×1 received vector, and

s

is the

B

×1 transmitted vector. Note that

( ) ( M N )

rank

B = H ≤ min ,

(2.2.1-2)

is the number of parallel transmitted data streams. F is the

M

× transmit

B

beamforming matrix,

G

is the

B × N

receive beamforming matrix and n is the

× 1

N

noise vector. The transmit beamforming adds a redundancy of

M

− across

B

space, because the number of input symbols is just B but produces M output symbols transmitted simultaneously through M transmit antennas. That results the

Figure 2-1 Joint Tx/Rx beamforming design for single user MIMO system

performance improvement due to the diversity gain. Besides, the receive beamforming removes the redundancy that be introduced by the transmit beamforming and results

B output data for detection.

To derive the beamforming, we assume the following properties:

{ }

^{H = ;}

^E { }

^{H = ;}

^E { }

^{H =0;}

E ss I nn R

_nn

sn

(2.2.1-3)

where the superscript H represents the conjugate transpose (Hermitian) operation.

For simplicity of analysis, we assume that the transmitted signals are uncorrelated and normalized to unit power.

B = rank ( ) H

, the full-loaded case, the elements of MIMO channel matrix are uncorrelated with full rank. If

B < rank ( ) H

, under-loaded case, we can also apply it to our system. However the over-loaded case

B > rank ( ) H

is not possible for any practical system that transmits independent data streams more than rank of the MIMO channel H in order to have an acceptable performance.

2.2.2 OFDM-based Case

The OFDM technique which has attracted a lot of interest in the recent years as it can combat delay spread, easily deal with frequency selective channels and achieve the spectral efficiency described in section 1.2.1. Now, we apply the above MIMO system to the frequency selective channel with OFDM transmission, where the flat-fading conditions prevail on each subcarrier. The MIMO-OFDM system model in each subcarrier is similar to Figure 2-1. The difference is that the pro- and post-filter are processed at each subcarrier of OFDM. The MIMO-OFDM system equation has the similar form as the flat-fading case (2.2.1-1) and is shown as below where

subscript notation

k

of denotes the subcarrier index

Freq L

…

Freq 2

Note that, in MIMO-OFDM based system, the transmit beamforming is

Figure 2-2 Joint Tx/Rx beamforming processing at transmitter of a single user MIMO-OFDM system

Figure 2-3 Joint Tx/Rx beamforming processing at receiver of a single user MIMO-OFDM system

performed in frequency domain, that is, before the OFDM modulation. And the receive beamforming should be also performed in frequency domain, that is, after OFDM demodulation. Figure 2-2 and Figure 2-3 illustrate the detail operations at transmitter (BS) and receiver (MS) respectively.

2.3 Joint Tx/Rx Beamforming Design for Single User Case 2.3.1 Problem Description

Now we will design transmit and receive beamforming matrices F and

G

to minimize the accumulative MSE in the flat-fading MIMO channel environment.

However, as explained before, the flat-fading characteristic is preserved at each subcarrier of OFDM. Thus, we modify the above MIMO system and directly design

F and

k

G for the MIMO-OFDM system. The MSE matrix at each subcarrier of

_k MIMO-OFDM system is defined as the covariance matrix of the error vector and showed as below:

(

k k

) (

k k

) { }

k ^Hk

k

F , G MSE F , G E e e

E

= = (2.3.1-1)

where

e is the symbol estimation errors and defined as

_k

e

_k = ˆ

( s

_k−

s

_k

)

. Replacing the system equation (2.2.2-1) into (2.3.1-1), we obtain the MSE matrix as below:

( ) { } { ( )( ) }

We now use the assumptions in equation (2.2.1-3) and assume the channel matrix H

is fixed and known at the transmitter and the receiver The MSE matrix in equation (2.3.1-2) can be simplified as:

( ) ( )( )

Thus, the minimize MSE problem can be stated as follows:

{ } ^{( ( ) )}

where

p

_T,k is the transmitted power constraint at k-th subcarrier. Note that the above equation (2.3.1-4) is based on the Frobenius norm:

{ } { ( )

H

} ( { ( )

k k^H

} )

k k

k

E trace trace E

E e

² =

e e

=

e e

(2.2.1-5)

After formulating the problem of the joint beamforming design for transmitter and receiver over flat-fading channel and frequency-selective channel (OFDM-based). In the next section, we will continue to derive two other methods which have the same solution as the above transmit and receive beamformer.

2.3.2 Optimum Transmit and Receive Beamformings

2.3.2.1 Lagrange Multiplier Method

First, we use the method of Lagrange duality and Karush-Kuhn-Tucker (KKT) conditions to solve the joint design problem in equation (2.3.1-8). We add the Lagrange multiplier

μ

_k to form the Lagrangian shown as below:

( ) ( { } ) [ (

H Tk

) ]

Replacing the equation (2.3.1-3) into above equation, we obtain

( ) [ ( )( )

The following KKT conditions are necessary and sufficient to solving the optimal transmit and receive beamforming

F and

_k

G .

_k

F and

_k

G are optimal solutions

_k

From the equation (2.3.2.1-3) and (2.3.2.1-2), we can obtain

,

= 0

and from the equation (2.3.2.1-4) and (2.3.2.1-2), we can obtain

=0 To obtain above two equations, we have to use the fact:

( )

( ∂ trace AXB ) ( ) / ∂ X = BA

(2.3.2.1-8)

( )

(

∂

trace AX

^H

B )

/

( )

∂

X

=

0

(2.3.2.1-9)

Now we are going to solve the two equations (2.3.2.1-6) and (2.3.2.1-7) to obtain the optimal transmit and receive beamformer. First of all, we define the SVD of following equation:

( ) (

k k

)

^H

and

S

is a diagonal matrix with B nonzero singular values with decreasing order.

S ~

k

is a diagonal matrix with zero singular values;

U ~

_k

and

V ~

_k

are orthogonal matrices with dimensions

M × ( M − B )

which form a basis of the null space of

k k H

k

R H

H

_nn⁻¹_, . Note that we have assume that the rank of

H is B for simplicity.

_k

Applying the similar approaches used in [7], we can obtain the transmit and receive beamforming matrices with structures as follows:

k k

k

V Φ

F

F

= (2.3.2.1-11)

1 ,

=

^H_{k k}^H − _k

k

Φ

G

U H R

nn

G

k (2.3.2.1-12)

Where

Fk

Φ and Φ

_Gk are diagonal matrices with nonnegative values and with

dimension

B

× . Thus, the transmit and receive beamforming matrices diagonalize

B

the MIMO channel matrix into a set of eigen subchannels. We will explain the results in section 2.3.3. The diagonal matrices

Fk

Φ and

Gk

Φ in above two equations are

given by:

(

^{12 12 1}

)

+¹²

− −

− −

= _{k k k}

k

S S

Φ

_F

μ

(2.3.2.1-13)

(

^{12 12 1}

)

+^{12 −12}

− −

−

= _{k k k k}

k

S S S

Φ

_G

μ

(2.3.2.1-14)

The subscript notation

+

denoted that the negative elements of the diagonal matrices are replaced by zero and

μ

_k in the above two equations is chosen to satisfy the transmit power constraint and given by:

( )

¹

,

12 12

−

−

+

⎟⎠

⎜ ⎞

⎝⎛

=

k k

T

k

k

P trace

trace S

μ S

(2.3.2.1-15)

Up to now, we have showed the Lagrange multiplier approach to derive transmit and receive beamforming matrices. In next section, we show another method to derive these beamforming matrices. These approaches are different, but the results are identity since both of them are looking for the optimal solutions.

2.3.2.2 Two Step Method

Recall that the minimized accumulative MSE problem is stated in (2.3.1-4) and the accumulative MSE matrix is given in equation (2.3.1-3). We now use the two-step derivation approach to design the system. In first step, we derive the optimal receive beamforming matrix

G by assuming that the transmit beamforming matrix is fixed

_k and then leave the difficult part which is to derive the transmit beamforming matrix

F to next step.

k

The optimal receive beamforming solution

G

_{k ,opt} that minimizes the MSE matrix is the same as the Wiener solution which is known to minimize the

( )

( MSE

_{k k}

)

trace F , G

and is given by the following equation

( )

( G ) 0

G =

∇

trace MSE

_k

k (2.3.2.2-1) And then the optimal solution

G

_{k ,opt} can be obtained as below:

(

,

)

¹

,

+

−

=

_{k k k k k}^H _k^H _k

opt

k

F H H F F H R

nn

G

(2.3.2.2-2)

The optimal receive beamforming is exactly the Wiener filter solution. Replacing the optimal receive matrix

G

_{k ,opt} into the MSE matrix, we obtain the following concentrated error matrix:

( ) ( )

Thus, the joint beamforming design problem is simplified to the design of the transmit beamforming with the receive beamforming matrix given by Wiener solution (2.3.2.2-2). Note that, without any constraint, the minimization of (2.3.2.2-3) will lead to the trivial solution of increasing to infinity of the norm of

F . Thus, the solution of

_k the optimization problem with transmit power as a constraint:

( )

actually equivalent to the elements of equation (2.3.2.1-13) which is the matrix form.

If we replace the transmit beamforming into the optimal receive beamforming matrix derived in first step, we can obtain the same optimal receive solution given in section 2.3.2.1.

2.3.3 Equivalent Decomposition of MIMO-OFDM system

The major characteristic of the joint beamforming design is to convert the

mutually cross-coupled MIMO transmission system into a set of parallel eigen subchannels (also termed channel eigenmodes) system. From the results of previous section, the matrix model of MIMO-OFDM system at k-th subcarrier similar to flat-fading case can be illustrated in Figure 2-4.

k f,

Φ V

_k

H

_k ¹

R

nn

H

U

_k^{H H}_k ⁻ _,k

Φ

g

+

Downlink

F

k

n

^k

G

_k

sˆ

k

s

k

Mathematically, the product of optimal beamforming matrices

F

_{k ,opt} and

G

_{k ,opt} in equation (2.3.2.1-11) and (2.3.2.1-12) and the channel matrix

H becomes:

_k

k k

k k H k H k opt

k k opt

k

H F Φ

G

U H R

nn

H V Φ

F

G

k

1 , ,

,

=

− (2.3.3-1)

Using the SVD in equation (2.3.2.1-10), it can be rewritten as:

k

k k

k H k k k H k opt

k k opt

k

H F Φ

G

U U S V V Φ

F

Φ

G

S Φ

F

G

_{, ,}

=

_k

=

_k (2.3.3-2)

Since

Gk

Φ

,

S and

_k

Fk

Φ are all diagonal matrices, the MIMO channel are

decoupled into parallel eigen subcarriers. Figure 2-5 illustrates the equivalent MIMO-OFDM system at k-th subcarrier.

Figure 2-4 The matrix model of a single user MIMO-OFDM system at k-th subcarrier

M

k

Φ

f,_1,

k

Φ

f,_2,

k

Φ

f,N_, k

f,

Φ V

_k

H

_k

+ U

_k^H

H

^H_k

R

_nn⁻¹_,k

Φ

_g

F

k

n

^k

G

_k

sˆ

k

s

k

k ,

s

1

k ,

s

2

k

s

N ,

k

Φ

g,_1,

k

Φ

g,_2,

k

Φ

g,N_,

k ,

ˆ

1

s

k ,

ˆ

2

s

k

ˆ

N ,

s

k ,

λ

1

k ,

λ

2

k

λ

N ,

The major interest of the diagonalized structure is that all the matrix equations can be substituted with scalar ones with the consequent great simplification and only the symbol by symbol detection need to be performed.

2.4 Conclusions

In this chapter, we have shown the system model for flat-fading channel environment and extend to OFDM-based system which can combat the frequency selective channel and achieve the spectral efficiency. Each subcarrier of MIMO-OFDM based system can be treated as flat-fading case and solved by the same ways. We have also developed two forms of joint MMSE transmitter-receiver beamforming: the Lagrangian and the two-step methods. Both approaches result in the same close-form solution.

The joint design problem shows that the optimal beamforming matrices

F

_{k ,opt}

Figure 2-5 Equivalent decomposition of MIMO-OFDM system at k-th subcarrier

and

G

_{k ,opt} in equation (2.3.2.1-11) and (2.3.2.1-12) cascaded with the channel matrix

H in between will result in a diagonal matrix. That is, the original mutually

k

cross-coupled MIMO transmission system is decoupled into a set of parallel eigen subchannels system. It means that the matrix equations can be simplified to scalar ones so that we can perform symbol by symbol detection similar to a set of parallel SISO systems. This joint design approach makes the number of antennas, the size of the coding block, and the transmit power become scalable.

--- Chapter 3

Joint Tx/Rx MMSE Beamforming Design for Multi-user MIMO-OFDM SDMA Downlink

System with Perfect CSI

---

In this chapter, we extend the single user joint Tx/Rx beamforming design to the multi-user case so that we can achieve the multiple access via the space domain the so called spatial-division multiple access (SDMA) [15] [18]. Compare to the conventional frequency-division multiple access (FDMA) and time-division multiple access (TDMA), the SDMA can conduct multi-user transmission at the same time and frequency. That is, we can reuse the frequency bandwidth. However, the multi-user interference (MUI) can potentially cause performance degradation. The technique of null-spacing [18] is proposed to handle the MUI problem.

3.1 Introduction

Different from most of the SDMA systems which assume a single antenna equipped at mobiles, the MIMO system we consider is the system with multiple

antennas used at both transmitter and receiver. We will further extend the MIMO SDMA system to MIMO-OFDM SDMA system by the concepts mentioned in chapter 2. In the multiple access system, the MUI is a major problem which causes significant performance loss. A way of solving this problem is to use the null-space constraint to decouple the multi-user MIMO SDMA joint design problem into several single user problems which have been described in previous section, where each problem only depends on each single user MIMO channel. In other words, the product of the MIMO channel and the null-space matrix at transmit side results in a block-diagonal matrix, which means the MUI between each user is completely removed. Thus, each user terminal only has to deal with its own inter-stream interference.

In this chapter, we first model the single carrier flat-fading MIMO SDMA system which combines the joint Tx/Rx beamforming design with the null-space technique.

Then we extend such system to MIMO-OFDM SDMA case which preserves the flat-fading property at each subcarrier. Thus, the beamforming and null-space matrices have to be designed based at each subcarrier. That is, we have to perform the pre-filter with the null-space constraint before OFDM modulation at the transmitter and at the receiver. The post-filter is also performed after OFDM demodulation. The null-space matrix design technique will be introduced in section 3.3.1. Thereafter, we will introduce the combination of the joint Tx/Rx beamforming and the null-space constraint to deal with the multi-user MIMO-OFDM SDMA downlink system.

3.2 Joint Tx/Rx Beamforming MIMO SDMA System Models 3.2.1 MIMO SDMA under Single Carrier Flat-fading Channel

Figure 3-1 illustrates a multi-user MIMO SDMA downlink system under single carrier flat-fading channel. We consider the transmit side (BS) equipped with M antennas simultaneously communicates with

U

user terminals (mobile station or MS). Each user terminal has

N receiver antennas. The BS transmits several data

^u symbol streams towards the

U

user terminals simultaneously.

C data streams are

¹ transmitted towards user terminal 1,

C data streams are transmitted towards user

² terminal 2, and so on.

H

… BS … ^C

¹

C

^U

… …

1

M 2

s¹

(k)

s^U

(k)

t(k)

… MS-1

MS-U … … ¹

N

¹

1 N

^U r¹

(k)

r^U

(k)

… …

C

¹

C

^U s¹

(k)

s^U

(k)

Downlink

If we do nothing at BS, each user terminal will receive the mixture of all data streams and needs to recover its own streams. Note that the receiver antennas

N of

^u each user terminal is greater or equal to the number of data streams

C in order to

^u make sure an acceptable performance.

The joint Tx/Rx beamforming design combined with null-space matrix can be

Figure 3-1 Joint Tx/Rx beamforming design for a multi-user MIMO SDMA downlink system

depicted in Figure 3-2.

3.2.2 MIMO-OFDM SDMA

By using the OFDM technology, the transmit beamforming and null-space matrix is designed based on each subcarrier and performed before OFDM modulation at the transmit side. At each user terminal, the receive beamforming is also designed based on each subcarrier and performed after OFDM demodulation. And at each subcarrier, the flat-fading conditions prevail and can be treated as above single carrier flat-fading MIMO system shown in Figure 3-2.

3.3 Joint Tx/Rx Beamforming Design for Multi-user Case 3.3.1 Null-space Constraint Design

Now we introduce the design of the null-space matrix which block-diagonalizes

Figure 3-2 Joint Tx/Rx beamforming design for a multi-user MIMO downlink system with null-space matrix

the MIMO channel. The following design is based on MIMO-OFDM system where the subscript notation

k

denotes the subcarrier index. In order to remove the MUI between each user, a null-space matrix denoted by

W is designed that the product

_k of the MIMO channel matrix and the null-space matrix

H

_k

W

_k at k-th subcarrier

results a block-diagonal matrix with u-th block in the diagonal which is u-th user’s data streams. That is the MUI is completely eliminated and leaves only each user’s inter-stream interference which can be deal with by each user’s processing.

)

( R T

H

∑

^u×

…

) ( ²

2 N M

H ×

) ( ¹

1 N M

H ×

) ( N M H

^{U U}

×

)

( N M

H ∑

^u

×

First of all, the multi-user MIMO channel matrix

H at k-th subcarrier can be

_k

viewed as a vertical concatenation of

U

MIMO subchannels matrix

H which

^u_k means the BS to u-th user’s MIMO subchannel at k-th subcarrier and with dimension

M

N

^u× . We illustrate the whole multi-user MIMO channel by Figure 3-3.

In order to block-diagonalize the whole MIMO channel matrix

H , we have to

_k design the null-space matrix

W with horizontal concatenation of

_k

U

sub-matrices

Figure 3-3 The vertical concatenation representation of a multi-user MIMO channel matrix

u

W depicted in Figure 3-4. The MIMO channel matrix and null-space matrix at k-th

k

subcarrier can be represented in mathematic form as follows (where the superscript

T denotes the transport operation):

[

^Uk^T

]

^T

T k T k

k

H H H

H

= ^{1 2} L (3.3.1-1)

[

k¹ k² k^U1

]

k

W W W

W

= L (3.3.1-2)

W

1

W

²

L W

^U

We can see that the block-diagonal condition is fulfilled if each column of

W

_k^u lies in the null-space of

H

^u_{k ,C} where

H

^u_{k ,C} is obtained by removing

N rows from

^u

H and has dimension

k ^U

N M

u i i

u

×

∑

≠

= ,1

. We can represent it by the following mathematic

equation:

{ } H H W 0

W

_k^u

∈ null

^u_k,C

⇔

^u_k,C

⋅

_k^u

=

(3.3.1-3)

For example,

W , the first columns of

_k¹

W , is a set of orthogonal basis of the null

_k space of

H

¹_k,C;

W is built by the orthogonal basis of the null space of

_k²

H

²_k,C, and

so on. It is easy to see that each

W has

_k^u

D columns where

_k^u

D is given by the

_k^u dimension theorem and is given by:

∑

≠

=

−

=

^{U u}

u

k

M N

D

(3.3.1-4)

⎟ =

⎠

⎜ ⎞

⎝

⎛ ×

∑

=

1 U

u

D

u

M W

Figure 3-4 The horizontal concatenation representation of a multi-user null-space matrix

The null-space matrix

W which block-diagonalizes the channel matrix

_k^u

H

_k

can be illustrated by Figure 3-5. The block-diagonalized matrix means that the MUI is completely eliminated and each user terminal receives only its own data streams.

…

⁾

( ¹

1 N ×M

H

) (

N^U M

U

×

H

W

1

W

^U

1

⎟ ⎠

⎜ ⎞

⎝

⎛ × ∑

= U u

D u

M

)

W

( ∑ N

^u

× M H

× =

… …

…

… … …

O

0

0 0

0

0 0

1

1

D

N ×

U

U

D

N ×

L

3.3.2 Joint Tx/Rx Design with Null-space Constraint

We have introduced how to design the null-space matrix in previous section.

Now we combine the joint Tx/Rx beamforming design with null-space constraint to deal with multi-user MIMO-OFDM SDMA system. The multi-user MIMO-OFDM SDMA system on k-th subcarrier can be modeled as shown in Figure 3-6 which is similar to the flat-fading case as shown in Figure 3-2. The difference between the two systems is that we process these operations in time or frequency domain. For MIMO-OFDM based system, these operations are performed in frequency domain while for MIMO system they are processed in time domain. Thanks to the null-space constraint that the transmit beamforming matrix can be calculated independently for each user and each user terminal only needs to know its part of the multi-user MIMO channel to calculate the receive beamforming matrix. It is reasonable for practical systems. Figure 3-7 and Figure 3-8 illustrate the details of joint Tx/Rx beamforming design for multi-user MIMO-OFDM SDMA downlink system at transmitter and

Figure 3-5 The product of the MIMO channel and the null-space matrix

receiver respectively.

User 1 data

User 2 data

User U data Demux

Figure 3-6 Joint Tx/Rx beamforming for multi-user MIMO-OFDM downlink system with null-space matrix

Figure 3-7 Joint Tx/Rx beamforming for multi-user MIMO-OFDM downlink system at transmitter

CP^-1

User u-th data

… … … …

3.3.3 Derivation of The Joint Tx/Rx Design with Null-space Constraint

We now describe the idea by using mathematic equations and explain the joint Tx/Rx MMSE beamforming design with the null-space constraint for multi-user MIMO-OFDM SDMA downlink system as follows. The superscript notation

u

denotes user index, and for each user the joint beamforming design is performed over the equivalent channel

H

^u_k

W

_k^u which means we have taken the block-diagonalization

constraint into account and results MUI free.

The system equation for user

u

and subcarrier

k

is And the problem is formulated as

{ } ( )

T^uk

Figure 3-8 Joint Tx/Rx beamforming for multi-user MIMO-OFDM downlink system at u-th user receiver

where

e is error vector and equal to

^u_k

s

^u_k− and

s

ˆ^u_k

p

_T^u_,k denotes the transmit power

constraint of user

u

at subcarrier

k

. Combining the system equation (3.3.3-1) and

constraint of user

u

at subcarrier

k

. Combining the system equation (3.3.3-1) and

在文檔中 多重輸入多重輸出正交分頻多工空間分割存取下鏈系統在通道估計錯誤下之共同波束設計 (頁 27-0)