Conclusions

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 above equations, we have

{ }

In the same way, we use the Frobenius norm and then the minimization problem can be rewritten as minimizing accumulative MSE matrix

( )

subject to ,

,

And we have the same assumptions similar to (2.2.1-3), that is

{ } s

^u_k

s

^u_k^H

= I ^; E { } n

^u_k

n

^u_k^H

= R

_nn^u ,_k

^; E { } s

^u_k

n

^u_k^H

= 0 ^;

Furthermore, we can solve the optimization problem by Lagrangian or two-step approaches. Following show the Lagrange duality method which transforms the

constrained optimization problem into an unconstrained one.

where

μ

_k^u is the Lagrange multiplier and the KKT conditions are

( ^{, ,} ) ^{= 0}

We obtain the similar results in chapter 2.

ku

The meanings of above parameters are equivalent to the description in chapter 2.

Particularly,

μ

_k^u is chosen to satisfy the transmit power constraint and given in following equation:

( )

¹

,

12 12

−

−

+

⎟⎠

⎜ ⎞

⎝⎛

= u

k u

k T

u u k

k

P trace

trace S

μ S

(3.3.3-17)

We can expect that the same results can be obtained from the two-step approach where we use the equivalent MIMO channel

H

^u_k

W

_k^u instead of

H .

^u_k

3.4 Simulation Results and Comments

We will show the performance of above joint Tx/Rx beamforming design for multi-user MIMO-OFDM SDMA downlink system by computer simulations. In this set-up, we first assume that the channel estimations are perfect known at both transmit and receive terminals; the elements of the MIMO channel are independent-identically-distribution (i.i.d) complex Gaussian distribution with zero mean and variance 1 and the channel length is

L .

_C

At BS which is equipped with M transmit antennas, each user’s data are QPSK modulated and de-multiplexed into B parallel paths and each user is equipped with

N

receive antennas. Each path is processed by OFDM modulation. Before passing to OFDM, we have to perform the transmit beamforming for each subcarrier of each user and then process the null-space matrix for each subcarrier of all user. We can see procedures in Figure 3-7. In OFDM, we assume the length of FFT is L and the length of CP is

L which is larger or equal than the channel length

_CP

L in order to

_C

keep the orthogonality between each subcarrier. And then launch the OFDM packets via M transmit antennas. At each receive terminal, thanks to the null-space matrix, each user only receives it own data. After performing OFDM demodulation, the

output signal is processing to the receive beamforming for each subcarrier and then passes to an appropriate interleaver to obtain correct data streams.

Figure 3-9 shows the bit error rate (BER) curves of typical multi-user MIMO-OFDM SDMA systems where the BS equipped with 6 to 8 transmit antennas communicates simultaneously with 3 users. Each user is equipped with 2 receive antennas. The FFT length L is 64. Each OFDM packet contains 640 data symbols and 100 MIMO channel realizations described above are simulated and generated independently for each packet. The total transmit power per symbol period across all antennas is normalized to 1. The SNR is defined as the total transmitted power normalized with the noise variance at each subcarrier.

0 5 10 15 20 25 30

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

Total Tx Power / Received Noise Power

BER

Joint Tx−Rx MMSE Beamforming Design for MIMO−OFDM System Under Perfect Channel Estimation

Multi−user case;M=6;N=2;U=3 Single user case;M=6;N=6;U=1 Multi−user case;M=7;N=2;U=3 Single user case;M=7;N=6;U=1 Multi−user case;M=8;N=2;U=3 Single user case;M=8;N=6;U=1

Figure 3-9 Joint Tx/Rx MMSE beamforming design for MIMO-OFDM system under perfect CSI

We call the MIMO system “fully loaded” when the number of parallel streams is equal to the number of BS antennas. When the number of BS antennas is greater than the number of parallel streams, the system is called ”under-loaded”. In this case, the diversity gain can be expected. However, it is impossible to simultaneously transmit more parallel streams (over-loaded case) without inducing irreducible MUI.

In Figure3-9, we also show the single user case which has 6 receive antennas.

From the simulation results, we can summarize the following observations.

z Since the single user case with the same number of receive antennas has more degrees of freedom for spatial processing at the receiver, it has a better performance than the multi-user case.

z When the number of BS antennas increases, we can have of course a better performance because of the diversity gain obtained from the transmit beamforming.

z In multi-user case, adding one antenna at the BS provides a diversity gain of 1 to all users.

z The performance difference between the single user and the multi-user case becomes negligible when the number of BS antennas increases.

3.5 Conclusions

In this chapter, we consider the joint design Tx/Rx beamforming for multi-user MIMO-OFDM downlink communications using SDMA under perfect channel estimation. In multi-user system, the MUI is the major problem which needs to be

handled. We use a null-space technique to decouple the multi-user MIMO SDMA joint design problem into several single user problems, In other words, the product of the MIMO channel and the null-space matrix results a block-diagonal matrix which means the MUI between each user is completely removed. Thus, each user terminal only has to handle its own inter-stream interference. Furthermore, we extend this flat-fading channel case to frequency selective channel environment by using the MIMO-OFDM based system. Using the null-space constraint, the several decoupled single user designs also have the properties that the system structure can be scalable with respect to the number of antennas, size of the coding block, and transmit power.

--- Chapter 4

Robust Design of Joint Tx/Rx MMSE Beamforming with Excellent Channel Estimation Error immunity

for Multi-user MIMO-OFDM SDMA Downlink System

---

We have discussed the joint Tx/Rx MMSE beamforming design for the multi-user MIMO-OFDM SDMA system in chapter 3. The channel information is assumed to be perfectly known at both transmit and receive terminals. However, the channel estimation always contains errors in real communication systems and only imperfect CSI can be obtained. In a practical wireless environment, channel information has to be estimated periodically because of the time-varying characteristic of channel, especially in the mobile environment. The imperfect CSI has a significant impact on the performance. In this chapter, we will focus the robust beamforming design to enhance the system performance.

4.1 Introduction

If the channel estimation is perfect, we can design the optimal solutions of

transmit and receive beamforming matrices in single user MIMO-OFDM case.

Furthermore, for a multi-user MIMO-OFDM SDMA system, a perfect null-space matrix that removes the MUI between each user also can be obtained. However, if the channel estimation contains errors, the null-space and Tx/Rx beamforming matrices will be designed imperfectly in multi-user joint design system (see Figure 4-1).

H

… BS … ^C

¹

C

^U

… …

1

M 2

s

¹

(k)

s

^U

(k) t(k)

…

MS-1

MS-U … … ¹

N

¹

1 N

^U

r

¹

(k)

r

^U

(k)

… …

C

¹

C

^U

s

¹

(k)

s

^U

(k) ( )

( ) ^H

F H W

ˆ ˆ

( ) ^ˆ

1

H G

( ) ^H

^ˆ

G

^U

H ˆ Hˆ

Downlink

The imperfect null-space design will induce MUI between each user and the imperfect transmit and receive beamforming designs will cause the inter-stream interference at each user terminal. That is why the imperfect CSI will cause significant performance degradation. This is an unavoidable problem in realistic wireless communication systems that the channel estimation always contains errors and only the imperfect CSI can be obtained.

The CSI at receiver can be obtained via the training sequence or pilot symbols that allows to estimate the channel. The CSI at transmitter can be obtained by using feedback channels from the receiver to the transmitter. In many cases, a sufficiently

Figure 4-1 Joint Tx/Rx beamforming design for multi-user MIMO downlink system under imperfect CSI

accurate channel information at receiver can be assumed, however, the CSI at transmitter is always far from sufficient accuracy. Hence one can assume that the receiver has perfect CSI to design receive beamforming and the transmitter has imperfect CSI to design transmit beamforming and null-space matrices. But in this chapter, we will consider more general case that both CSI at transmit and receive sides are imperfect to design null-space and Tx/Rx beamforming matrices in multi-user joint design MIMO-OFDM SDMA system.

We are going to apply two robust approaches to our multi-user joint design problem to against the performance degradation caused by MUI and inter-stream interference [6] [18]. Both of these robust methods have the similar performance over fast time-variant and slow time-variant environments. Therefore, we apply the moving average approach that has better performance over slow time-variant environment.

Note that all the robust methods need the statistic properties of the estimation errors.

In the end, we will show the simulation results and give some comments.

4.2 Robust Design of Joint Tx/Rx MMSE Beamforming 4.2.1 Problem Description

We now consider interferences induced by the imperfect null-space matrix and Tx/Rx beamforming due to the channel information error. Recall that for each subcarrier of OFDM system the product of MIMO channel and null-space matrix results a block-diagonalized matrix which means MUI is perfect removed. That is

u

u

i

k i

k

W

=

0

; if ≠

H

(4.2.1-1)

where the subscript notation denotes subcarrier index and the superscript notation represents the user terminal index. And each user’s Tx/Rx beamforming is also designed due to the exact channel information

H . Figure 4-2 illustrates again the

ⁱ_k major block-diagonalized operation at each subcarrier.

1

Moreover, we explain the operation by another point of view in order to derive the robust null-space matrix against MUI. Assume that t is the total transmitted _k signal at subcarrier

k

and it can be formulated as beamforming design MIMO-OFDM SDMA system becomes

u

By the equation (4.2.1-1), the above equation can be rewritten as

Figure 4-2 Joint Tx/Rx beamforming for multi-user MIMO-OFDM downlink system with null-space matrix

u

In (4.2.1-4), it is easy to see that the mutual interferences between co-channel users are removed and leaves only each user’s joint design problem to cope with its own inter-stream interference.

Now we assume that the MIMO channel information contains errors and can be represented as

k k

k

H H

H ˆ = + Δ

(4.2.1-5) where Δ

H

_k is channel information error and its elements are independent random variables with zero mean and variance

σ

_Δ²_Hk . Since the BS has the imperfect CSI

Hˆ

k, not the exact CSI

H , the null-space matrix is obtained under the condition

_k

u

And we also obtain the imperfect transmit beamforming

Fˆ

_k^u that is designed from the

equivalent channel

H W ^ˆ

ⁱ_k

^ˆ

_k^u. Thus, the total transmitted signal at each subcarrier becomes

And the received signal of user

u

becomes

u be affected by MUI. Furthermore, at user

u

’s terminal, since the first term of

(4.2.1-8), both of transmit and receive beamforming will be designed imperfectly which leads to the inter-stream interference. In the following section, we will first consider the MUI caused by imperfect null-space matrix (the second term effect), and then consider the inter-stream interference caused by imperfectly designed beamformings (the first term effect).

4.2.2 Robust Designs

4.2.2.1 Robust Design Against MUI

In this section, we derive the robust null-space matrix which minimizes the expected power of MUI by using the statistics of the channel estimation error. Before deriving it, we first consider the statistical dependence among the exact channel

H ,

_k the estimated channel

Hˆ

_k and the estimated channel error Δ

H

_k. In general, Δ

H

_k is independent of

H but dependent on

_k

Hˆ

_k. However, if

Δ H

_k is much smaller

than

H

_k , it can be assumed that Δ

H

_k is approximately independent of

Hˆ

_k. Thus, in order to derive the robust null-space matrix, we will assume that

k k

k

k

H H H

H ˆ ⊥ ; ˆ ⊥ Δ

(4.2.2.1-1)

We now investigate the effect that when user

u

’s signal is transmitted, how it will induce interference to affect other users. The received signal at subcarrier

k

and all user terminals when user

u

’s signal is transmitted is

u k k u k u k u k k all

k

H W F s H t

y = ˆ ˆ =

(4.2.2.1-2) Obviously, if the null-space is perfect designed, that is, if we know the exact CSI,

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