Summary

First of all, we stress the advantages of STBC and give the details of OSTBC and Alamouti STBC scheme in this Chapter. Because the space-time block coded MIMO communication system has proven capable of achieving high spectral efficiency and high link quality, it is believed to play an important rule in next generation wireless communication systems. Therefore, we here put our emphasis on a point-to-point space-time block coded MIMO system model. In the point-to-point MIMO communication case, the optimal ML detector provides a simple but linear receiver which maximizes the output SNR performance. Extending the above result, a system model for the multi-user space-time block coded MIMO is given in Section 2.2.3.

Nevertheless, for multi-user case, ML detector in Section 2.2.2 becomes highly non-optimal since the co-channel interference dominates the receiver performance instead of the channel noise term. And it has a much more complicated structure and higher complexity for multi-user case. Therefore, in the following chapters, we will search for a simple receiver scheme for the multi-user space-time block coded MIMO systems.

Chapter 3

Linear Receivers for

Space-time Block Coded Multi-user MIMO System with Perfect Channel Estimation

In Chapter 3, we consider the general case of space-time block coded multi-user MIMO wireless communication systems. The scheme can be presented as that both the receiver and the multiple transmitters are equipped with multiple antennas. In addition, OSTBC is used to send the data simultaneously from each transmitter to the receiver.

Here an optimal receiver scheme is provided for this system. It is able to suppress multi-access interference (MAI) and noise while decoding the data which is sent from the transmitter-of-interest by leverage of the spatial resource at the same time. The spatial resource comes from the array gain provided by multiple antennas at the transmitter and the receiver [18]. In other words, the proposed receiver is designed to minimize the filtered interference average power subject to the constraints that all received symbol gains of the transmitter-of-interest are maximal. Given that the exact channel estimation is available at the receiver, the equalizer scheme is regard as an associated reduced complexity implementation in comparison with joint ML detector.

We also applied the successive interference cancellation (SIC) mechanism to utilize the in-built structure of the space-time signature matrix [13]-[14]. Finally, the simulation results of the proposed receiver are provided at the end of this chapter.

3.1 Problem Formulation and System Model

In the multi-user MIMO case, the receiver performance is dominated by the signal-to-interference-plus-noise ratio (SINR) instead of the signal-to-noise ratio (SNR). Therefore, in order to design the receiver for a space-time block coded multi-user MIMO system, we have to define the interference term in the received signal first. Based on the system model proposed in Section 2.2.3, the received signal can be represented as

+

Y = HX V , (3.1)

where H = ⎣⎡H H₁ ...₂ H_q...H_Q⎤⎦ denotes the whole channel matrix of the system and

H denotes the channel matrix between the q th transmitter and the receiver. Q is q

the total number of transmitters. X= ⎢⎡⎣X X_{1 2}...X_q...X_Q⎤⎥⎦ is the total transmission matrix of all users, V is the channel noise vector. The following assumptions are made in the sequel:

1. The transmitted symbol vector s_q = ⎡⎢⎣s s_{q1 q2}...s_qK⎤⎥⎦, 1≤ ≤q Q , of q th user is zero mean and uncorrelated to each other. In the other words, it means the expectation of any two transmitted symbol vectors are

{ }

^{q p* ( ),}

E s s =δ q−p (3.2)

where E y denotes the expectation of the random variable y , and { } δ^{( )}. is

the Kronecker delta function.

2. For a Rayleigh fading channel, the elements of q th channel matrix , 1 ,

q ≤ ≤q Q

H are modeled as i.i.d zero-mean complex Guassian random variables with its variance equal to 0.5.

3. Each element of the noise vector V is i.i.d zero-mean complex Guassian random variables with variance σ in order to model the AWGN channel noise. _V²

For the advantages over joint design of interference suppression and decoding, we rewrite (3.1) into the space-time signature form which is similar to (2.23) to combine the effect of channel and space-time coding upon the transmitted symbol vector

= +

Y As V , (3.3)

where s = ⎢⎡⎣s s_{1 2}...s_Q⎤⎥⎦ is a presented vector of all users’ transmitted symbol vectors

and A= ⎢⎡⎣A H

( ) ( )

1 ^...A H2 ...A H

( ) ( )

_q A H_Q ⎤⎥⎦ with 1≤ ≤q Q is denoted as the space-time signature matrix of all transmitters. Then we can use this representation form to characterize the MAI term in the system. From (3.3), we can see the received signal is the combination of each transmitter's signal through its own space-time signature as

( )

^{1 1}

( )

^{2 2 ...}

( )

^{Q Q}

= + + + +

Y A H s A H s A H s V . (3.4) Now assuming without any loss of generality that the first transmitter is the

transmitter-of-interest, we can observe that the first term on the right hand side of (3.4) comes from the desired signal. And the other terms in (3.4) denote the interference comes from non-desired signal including MAI and channel noise. The separated structure of signal, multi-user interference and noise suggests us to obtain the corresponding description of (3.4)

= + _{I I}

Y Ds H s + V, (3.5)

where D:=A H

( )

₁ denotes the space-time signature matrix mapping to the desired

symbol vector s:=s₁ in the receiver, and H^I := ⎢⎡⎣A H A H

( ) ( )

2 3 ...A H

( )

_Q ⎤⎥⎦ denotes the space-time signature matrix mapping to the other users’ symbol vectors

2 3

:= ⎢⎡⎣ ..._Q⎤⎥⎦

sI s s s in the system. In the following procedure of receiver design, we assume that the perfect channel estimation is available at receiver which means that we know the exact information of D and H . _I

3.2 GSC-Based Interference Suppression

Based on the system model proposed in the previous section, there are several existing linear equalization methods such as zero-forcing (ZF) equalizer or minimum mean square error (MMSE) equalizer...etc[19]-[20]. These linear equalizers are capable of proving simple recovery solution to the desired signal but having limited performance. On the contrary, other nonlinear equalizers can provide the additional performance gain at the expense of higher computation complexity, for example, joint maximum likelihood (ML) equalizer. In this section, we first provide constrained optimization as a typical nonlinear approach of signal recovering. Then we transform the constrained optimization problem into an unconstrained one by generalized sidelobe canceller (GSC) technique. As a result, we proposed a low complexity but optimal solution to the multi-user system in this section.

3.2.1 Constrained Optimization

The diversity and array gain provided by STBC in the multi-user MIMO system

not only can resist the channel fading effect but also can reject interference comes from the other users. To fully use the extra degrees-of-freedom in system model, one typical solution for signal recovering is the constrained optimization [21]-[23]. The first step in developing the constrained algorithm is to define the optimum weighting matrix W . Here we follow the same assumption in Section 3.1 that the first transmitter is the user-of-interest without any generality loss. Using the system model (3.5), we can express the output vector of this equalizer as

( )

interpreted as the receiver weight vector for the k th symbol of desired signal.

The second step is to determine the constraints in the receiver. We have two main goals to be achieved by the constraints at the same time: one is able to maintain an undistorted response to the transmitter-of- interest’s signal. In other words, we try to seek a weighting matrix W to linearly combine the desired signal in the maximum ratio sense. As results, the linear weighting matrix must satisfy

H ≅ H 1

W Y D Ds ; (3.7)

the other goal is to suppress the MAI and the channel noise under the condition that the first goal is set up. It can be implemented via minimizing the total filtered output power of interference-plus-noise as much as possible while keeping the desired signal maximal. The goal can be expressed as the followed mathematic form:

( )

{

²

}

minE || ^H _{I I} + ||

W W H s V . (3.8)

Commonly, the closed-form optimal solution to satisfy the above two constraints are

solved by Lagrange multiplier method [24]. Although it does represent the solutions to the constrained optimization problems, it is computationally complex in the sense that a correlation matrix of the received signal must be estimated regularly and then inverted in order to arrive at the solutions.

3.2.2 GSC-based Equalizer

Here we use GSC as an efficient tool to solve the above constrained optimization problem instead of Lagrange multiplier method. The advantage over GSC principle is its ability to transform a constrained problem into an unconstrained one. The main idea of GSC is depicted in Figure 3.1.

Figure 3.1. Structure of GSC equalizer under the perfect channel estimation.

We can see the linear weighting matrix W is decomposed into three parts:

= −

W D BU , (3.9)

where D∈C^{MT K×} forms a conventional match filter matrix along the upper path to satisfies the first constraint (3.7). The superscript H is the complex conjugate (Hermitian) transpose. Therefore it is designed to equal the transmitter-of-interest’s

D^H Matched Filter

B^H Blocking Matrix

U^H

Adaptive Weighting Matrix

Estimated signal s

Received signal Y

space-time signature. As the upper path’s output is reduced by the lower path where the amount subtracted is the least-squares estimation of the noise and interference. To guard from subtracting the desired signal, the lower path is allocated with a blocking matrix B that nulls desired signal components. The blocking matrix

(MT K MT)

C ^{− ×}

∈

B has the property that B D^H =0, so that any component of the desired signal arriving at the lower path will be blocked. The adaptive weighting matrix

MT K MT

C ^{− ×}

∈

U is designed to use the remaining degrees of freedom to suppress the noise and other user’s interference power.

Following the procedure illustrated in Figure 3.1, the output power of GSC equalizer becomes

where the output power of upper path can be represented as

( )

: ^H ₁

d = + I I +

y D Ds H s V , (3.11)

and the output power of lower path can be represented as

( )

: ^H ₁

b = + I I +

y B Ds H s V . (3.12)

We observe that contaminated term in the upper path output is := ^H _{I I} + ^H

i D H s D V . (3.13)

Therefore if we want to minimize the filtered noise power and interference power, the adaptive weighting matrix U must satisfy the following cost function:

{ }

: ²

minJ =E || − ^H _b ||

U i U y . (3.14)

Simply following the standard procedures of GSC technique[25], the optimal weighting matrix U_opt has to satisfy the linear equation

H H

: ^{H 2 MT MT}

and we can get the optimal GSC weighting matrix

opt = − opt

W D BU (3.18)

3.3 GSC/SIC-Based Interference Suppression

In the above section we provide a GSC-based receiver to reject multi-user interference and noise at the same time. However by observing the algebraic structure of system model (3.3), we discover that there exists a in-built group partition at the transmit symbol vector s . The partition is according to the transmitter and each group transmits its symbol vector through its own space-time signature matrix. This basic structure gives rise to the multilayered space–time architecture. Here we use the SIC mechanism [13]-[14] to do the multistage detection and cancellation in the extension of proposed GSC-based receiver in Section 3.2. The basic idea is supposed that the first user’s symbol vector s is recovering successfully by the GSC-based receiver. After ₁ decoding s , we subtract the contribution of this signal from the received signals at all ₁ receive antennas. In other words, the communication system now are with less transmit antennas and the same number of receive antennas in comparison with the original one.

We next use GSC-based receiver to recover the second user’s signal then subtract its contribution from the received signals at all receive antennas. Proceeding in this manner, we observe that by subtracting the contribution of previously recovered user’s signal from the received signals at receive antennas the space–time code affords an

extra diversity gain [13]. This scheme can lead a straightforward power allocation application. In fact, powers at the different layers could be allocated based on the diversity gains. For example, the allocated powers can be decreased geometrically in terms of the diversity gains. In other words, it can also solve the near-far problem commonly happened in the multi-user case. The detailed procedures of the proposed receiver are described in the following algorithm:

GSC/SIC-Based Interference Suppression Receiver Algorithm Initialization:

1 =

( )

1

D A H ,H_I = ⎢⎡⎣A H

( )

2 ...A H

( )

_p ⎤⎥⎦,Y₁ =Y Recursive: For 1≤ ≤i Q

Step 1) B_i is the blocking matrix of D _i

( )

( ( ))

Step 2) R_{I i,}:= H H_{I i,} ^H_{I i,} +σ_{v MT K i}²I _{− −1}

( )

Step 3) U_{q i,} = B R B^H_{i I i i,} ⁻¹B R D _i^H _{I i i,} Step 4) W_{q i,} =D_i −B U _{i q i,}

( )

^{( )}

Step 5) s_i =ϑ W y_i^T _i ,ϑ . is the decision slice

( )

Step 6) y_i+1 =y_i − D_{i i}sˆ

( ) ( ) ( )

Step 7) D_i+1 =A H_i+1 ,H_{I i,+1} = ⎢⎡⎣A H_2+i ...A H_Q ⎤⎥⎦

3.4 Low Computational Complexity Scheme:

Alamouti Code

The computational complexity of the above proposed receiver mainly comes from the two places: one is in solving for the blocking matrix via B D^H =0; the other one is in multiplying of two large matrix H in order to get _I R_in =H H_I ^H_I +σ_{v MT}²I . In this section, we provide a low complexity scheme with Alamouti code employed in the system. Because of the peculiar structure of Alamouti code block, we can obtain the blocking matrix B and the multiplication R through much simpler calculations. _I

The typical approach for obtaining B∈C^{QK K×} is through the singular value decomposition (SVD) of the match filter matrix D∈C^{QK K×} . Since B is constructed as a basis of the left null space of D , the computational complexity cannot be reduced due to the large user number Q . In the Alamoti code case, the matrix D is composed of Q Alamouti code blocks as

where K = . In consequence of this distinctive structure of D , we discover that the 2 blocking matrix B has the similar structure which is also composed of Q code blocks as

[ ]

where K = in Alamouti case. We observe an important fact that the two columns 2 of B are composed of the same elements in the different order. It means that as long as we know any one column’s information of B , we can find the other one and complete the matrix B . It inspires us that we only need to find a column b∈C^QK×1 which is satisfied

H =0

b D , (3.21)

then we can find the matrix B to satisfied B D^H = 0 by copying and reordering the column b . Therefore we use the Gram–Schmidt process to find the desired column. It is a method for orthogonalizing a set of vectors in an inner product space, commonly the Euclidean space Rⁿ . The Gram–Schmidt process takes a finite, linearly independent set S =

{

v₁,...,v_n

}

and generates an orthogonal set

{

₁

}

' ,..., _n

S = u u that spans the same subspace as S . To do the process, we firstly define the projection operator by

proj , columns of D are orthogonal to each other in nature, we directly assign that

1 = 1= 1

u v d , u₂ =v₂ =d . Then ₂ v₃ ∈C^KQ×1 is randomly assigned any vector which is orthogonal to v . The Gram–Schmidt process works as follows: ₁

1 = 1 = 1 greatly reduced the flop counts from

( )

The scheme of Alamouti code also gives some additional computational advantage besides the calculation of the blocking matrix in GSC-based equalizer: the multiplying of the large matrix H_I ∈C^{QK QK×} is presented as a symmetric form as

where m n, , and p are scalars. As a result, the total number of flops for computing

* ^H

I I

H H is determined as KQ³−KQ² comparing to original K Q^{3 3} −K Q^{3 2} flops. Here the total approximate flop counts which are required to compute the regular OSTBC code scheme and the Alamouti code scheme are respectively summarized as below: times of computational load at the highest order.

3.5 Computer Simulations

Throughout the simulations in this section, Q = transmitters are assumed. 2 Each transmitter is with N = transmit antennas and the full-rate Alamouti’s 2 OSTBC (T =2, K = ) is used. In addition, we assume a single receiver of 2 M = 2 receive antennas. For simplicity, the interfering transmitter uses the same OSTBC as the transmitter-of-interest. Here channel model is assumed to be independent Rayleigh fading channel and the perfect channel estimation is available in receiver end. QPSK modulation is used. All plots are averaged over at least 1000 independent simulation runs.

In the first example, two transmitters are under the assumption of equal power.

Figure 3.2 shows the bit error rates (BERs) of the GSC-based receiver and the

GSC/SIC-based receiver versus SNR. We can observe that the GSC-based receiver with the SIC mechanism offers about 1~2 dB gain. This would benefit from the increased received diversity obtained by SIC mechanism. In Figure 3.3, the BERs of all the receivers tested are displayed versus SNR. The simulations compare the proposed GSC/SIC-based with several existing methods: the Stamoulis’s method [5], the Naguib’s approaches [6], and the minimum variance (MV) receiver [7]. As we can see, the proposed receiver provides better performance over the whole tested SNR range as compared to the other receivers. As expected, the performance of MV is limited by the finite sample effect. And although the Stamoulis’s decoupled based detector is free from error-propagation problem but its diversity gain is fixed.

Moreover both the Stamoulis’s method and the Naguib’s approaches have the limitation of the Alamouti code usage and two transmit antennas. In contrast to these restrictions, the proposed GSC/SIC-based receiver is free for any OSTBC and any number of transmit antennas.

In the second example, two transmitters are assumed with unequal power to model the near-far problem in a multi-user’s system. In this case, we assume that the power of transmitter is known at the receiver end. The decoding order is based on the amount of power. Figure 3.4 shows the BERs of the GSC-based receiver with and without SIC mechanism versus SNR and Figure 3.5 shows the BERs of all the receivers tested and are displayed versus SNR. Compared to Figure 3.2 and Figure 3.3, the improvement of performance confirm the advantage of SIC mechanism in the unequal power case.

0 5 10 15 20 25 10^-5

10^-4 10^-3 10^-2 10^-1 10⁰

GSC GSC/SIC

Figure 3.2. BER performances of the GSC-based receiver with and without the SIC mechanism (equal-power case)

0 5 10 15 20 25

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Stamoulis method Minimum variance Naguib method GSC

GSC/SIC

Figure 3.3.BER performances of the proposed receiver and other existing methods (equal-power case)

0 5 10 15 20 25 10^-5

10^-4 10^-3 10^-2 10^-1 10⁰

GSC GSC/SIC

Figure 3.4. BER performances of the GSC-based receiver with and without the SIC mechanism (unequal-power case)

0 5 10 15 20 25

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Stamoulis method Minimum variance Naguib method GSC

GSC/SIC

Figure 3.5. BER performances of the proposed receiver and other existing methods (unequal-power case)

3.6 Summary

In Section 3.1, we define the interference term and the system model of a space time block coded multi-user MIMO system. Under the assumption that perfect channel estimation is known at the receiver end, we use the system model to derive the optimal constrained equalizer to reject MAI and noise in Section 3.2. A GSC-based equalizer is also provided to transform the constrained problem into an unconstrained one in the same section. In addition, due to the multi-group structure in the system model, we apply the SIC mechanism to implement the multi-stage detection and interference cancellation in Section 3.3. Since Alamouti code is famous for its full rate and full diversity and commonly used in OSTBC, in Section 3.4 we also derive a low computational complexity scheme for Alamouti case. Finally computer simulation results are available in Section 3.5. It shows the SIC mechanism do improve the performance of GSC-based receiver and this proposed GSC/SIC-based receiver do have the comparable performance with other existing methods for the space time block coded multi-user MIMO system.

Chapter 4

Robust Linear Receivers for

Space-time Block Coded Multi-user MIMO System with Imperfect

Channel Estimation

The proposed receiver in Chapter 3 and other existing receivers for space-time block coded multi-user MIMO system have a major shortcoming: they use the assumption that the channel estimate is perfect at the receiver. When channel estimation error occurs, the performance of these receivers will degrade since their design does not take the channel estimation error into account. In practice, channel estimation error does happen. It comes from limited or outdating training symbols and leads to serious performance problems as do the effects of MAI and channel noise. In this chapter, we propose a robust receiver which is able to combat the imperfect channel estimation. By modeling the channel estimation error as a random variable and

The proposed receiver in Chapter 3 and other existing receivers for space-time block coded multi-user MIMO system have a major shortcoming: they use the assumption that the channel estimate is perfect at the receiver. When channel estimation error occurs, the performance of these receivers will degrade since their design does not take the channel estimation error into account. In practice, channel estimation error does happen. It comes from limited or outdating training symbols and leads to serious performance problems as do the effects of MAI and channel noise. In this chapter, we propose a robust receiver which is able to combat the imperfect channel estimation. By modeling the channel estimation error as a random variable and

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