Summary

In wireless communication systems, spatial multiplexing techniques are widely used to exploit the characteristics of multipath channels and to improve the capacity of transmission without allocating extra bandwidth. On the aspect of diversity techniques, which is implemented to reduce the effects of multipath fading and to enhance the reliability of transmission without increasing the transmit power or sacrificing the bandwidth, it is popular to make use of space diversity which can be classified into two categories, transmit diversity and receive diversity. In this chapter, we focus on two high-rate wireless communication systems, simple spatial multiplexing and double space-time transmit diversity. Spatial multiplexing systems can provide a high date rate but no diversity gain can be obtained. Thus, a compromised transmission scheme, double space-time transmit diversity, is introduced. Compared to conventional SISO systems, DSTTD can provide a double rate and double diversity at the same time.

The QR decompositions of spatial multiplexing and DSTTD systems are characterized in Section 2.3. Based on the results of QR decomposition, the structures of the upper triangular matrix are shown in Equation (2.23) and (2.29). It is useful for us to detect the received signals and provide better performance. Furthermore, in DSTTD systems, we derive the diagonal entries of the upper triangular matrix and express them in terms of the determinants of the channel matrix and its partitioned matrices. Then, some performance comparisons are illustrated in Section 2.5. Since QR detector is an unordered SIC detector, its performance is slightly poorer than the performance of MMSE detector. In the next two chapters, we will exploit the precoding matrix to allocate the transmit power and reduce the average bit error rate (BER). That is, we focus on the power allocation scheme by using the statistical properties of with QR-based SIC detection in spatial multiplexing and DSTTD systems.

Chapter 3

Power Allocation for Minimum BER in Spatial Multiplexing Systems

Equation Section 3

In this chapter, we will discuss the power allocation scheme with QR-based successive detection for spatial multiplexing MIMO systems over flat fading channels.

Our consideration is confined to uncoded quadrature phase-shift keying (QPSK) signals and the channels are independent and identically distributed Rayleigh fading. Given that the channel state information (CSI) is perfectly available at the receiver and only signal-to-noise ratio (SNR) is known at the transmitter, we design a precoding matrix to allocate transmit power at the transmitter under average channel realizations.

Furthermore, at the receiver, the received signals are detected with a QR-based successive symbol detecting scheme which is described in the previous chapter. For simplicity, the precoding matrix is restricted to be a diagonal power loading matrix so as to reduce the computational complexity. The optimization criterion is determined based on minimizing the overall average bit error rate (BER) of this transmission scheme. From the theory in [15], the design of the precoding matrix is based on the minimization of the lower bound for average BER. It can be proved that minimizing the lower bound for average BER will lead to minimizing the upper bound for the block error rate simultaneously. Following that, we exploit the power loading factors derived

from the closed-form solutions which we obtain by averaging the lower bound for BER over the channel realizations. Finally, some performance comparisons and discussions will be illustrated in the end of this chapter.

3.1 Bound for BER of QR-Based Successive Detection

Power loading schemes allocate the transmit power across symbols under the constraint of constant power per block. At the transmitter, for four symbols

transmitted at the same time, we denote the transmit power allocated to the th symbol as and define the power loading matrix as

, 1, 2, 3,

where is the power loading factor ,and the block transmit power constraint must be normalized as

i 0

Assuming that the channels are flat fading, we insert the power loading matrix into the system model in Equation (2.24). If the receiver replies the channel state information (CSI) to the transmitter, the transmitter can determine the power loading factors by the CSI. The block diagram with the transmit power allocation scheme is shown in Figure 3.1.

x ˆx H

Figure 3.1: Block diagram of a spatial multiplexing MIMO system with the QR receiver with the transmit power allocation scheme

4 4 ×

The received signals are multiplied from the left by the unitary matrix from QR decomposition, and they can be written as

Q

H detected as follows:

= H

Assume that there is no error in the previous symbol detection, and then we can obtain . It is obvious that the i th modified received signal is determined by the th transmitted symbol and the transformed channel noise. As long as the symbol in each stage is correctly detected and, hence, there is no layer-wise error

ˆ_{i ii i i}

y

=

R p x

+

n

i

The power loading factor represents the transmit power allocated to the i th sub-channel and is the i th sub-channel gain. The average energy of the symbols transmitted from each antenna is normalized to be one. Therefore, the total symbol energy is equal to , so that the average power of received signal at each receive antenna is also . The real part and imaginary part of noise have the same variance

n

T

SNR

. The average signal-to-noise ratio (SNR) is defined by

0

E N

s

ρ

, where the total symbol energy and noise variance are defined as

and , respectively. At some

ρ

, the instantaneous BER under the th sub-channel is given by

{ } 4 4 instantaneous BER of the symbol block given a fixed channel realization is expressed as

It is noted that the above discussion is under the error propagation free case; that is, error propagation is not taken into account. In this case, Equation (3.6) is merely a lower bound for average BER, and we rewrite it as

4 4

where the subscript indicates the lower bound for BER. This is a lower bound with QR-based successive symbol detection without considering the error propagation.

L

SNR regime because the error propagation is small enough to be neglected when SNR is high. If the error propagation occurs, the detection error of previous symbol decisions will affect the detection of the present symbol. This effect causes that the average BER with the error propagation is slightly higher than the average BER in the error propagation free case.

After the lower bound for average BER is presented, we will introduce the upper bound for average block error rate, which is of our major concern. First, let us define the detection error in the th symbol when there may be errors in the detection of previous symbols. We denote the received signal vector which would have been detected before by , and the transmitted signal vector which will be detected before by case so that Equation (3.8) can be written as

ˆ

In the above equation, the last inequality is due to the fact that and for high SNR. Furthermore,

(3.10)

, the last equality is obtained which can be easily derived by some manipulations. Combining (3.9) and (3.10), the result is shown as

1 ˆ 1

eLi eLj eLj eUi

j i j i

where the subscript indicates the upper bound for BER. Equation (3.11) represents the upper bound for the BER based on the consideration that there may be detection errors in the previous detected symbols. The upper bound for the average BER of four symbols is given by

U

The last equality is achieved because the detection order follows the upper triangular structure of the matrix. In view of the block error rate, let

i =

0 in (3.11), we have

It is obvious that the block error rate is upper bounded by four times the lower bound for the average BER . This is an important result for us to determine the power allocation factors. If a power allocation matrix is designed to minimize the lower bound for the average BER, it simultaneously minimizes the upper bound for the block error rate as well. From the above derivations, the minimization of lower bound for the average BER is reasonable because we can minimize the upper bound for the block error rate at the same time. It implies that the decision performance can be

{ ˆ P x ≠ x

P

eL

potentially improved even in the presence of cross-layer error propagation. In the next section, we will propose a power allocation scheme to minimize the overall average BER over average channel realizations with this lower bound for BER.

3.2 Optimal Power Allocation for Minimum Upper Bound of Overall Average BER

In the previous section, it has been shown that, given a fixed channel realization, the design criterion for power allocation to minimize the average BER can be modified to minimize the lower bound for average BER

4

instead of minimization of the upper bound for the block error rate. Now we would like to consider the channel probability density function (pdf) in deriving the error probability. We consider the general case assuming ; i.e., there are parallel transmission links in the spatial multiplexing MIMO systems, and the lower bound in (3.14) is expressed as

M

=

n

T =

n M

First of all, we have to determine the distribution of

R

_ii . From Lemma 2.2, we have

known that is a chi-square random variable with

2(

degrees of freedom; that is,

2

R

_ii2

M − + i 1)

Let

r

_i

= R

_ii for convenience, and with some variable transformations, it is shown

Hence, the lower bound for the overall detection error rate of

x

_i is given by

2( ) 1 2

The lower bound for the overall average BER for the data block is given by

2( ) 1 2

It is very difficult to calculate (3.19) if we use the definition of Q-function directly, and there will be no closed-form solution for it. Thus, we substitute Q-function with its Chernoff bound

and obtain an upper bound of (3.19) given by

2 2 2

After some rearrangements, we can rewrite this bound as

2 1 2

In order to calculate the above integrand, we use a known integration formula shown as follows:

In the above equation, let

x

=

r

_i,

p = + 1 ρ p

_i²

2

, and , the upper bound in Equation (3.22) can be expressed as

n = M − i

where is the number of transmission links. Based on the above result, we can find a set of to minimize this upper bound, and, hence, minimize

M

p

i

P .

_eL

To find the set of to minimize the upper bound with the power constraint on the elements , we can formulate an optimization problem as follows:

p

i

which can be solved by the Lagrange multipliers technique. First, we define

2 ( 1)

By Lagrange multipliers, the optimization problem is transformed into

(3.27) 0.

∇ =

F

Assuming

a

_i =

p

_i², we can get

2 obtain the constraint with an unknown variable

λ

as follows:

2

From the above derivations, a method of determining the power loading factors by minimizing the average BER upper bound in (3.24) subject to the power normalization constraint is proposed. We propose to determine the power loading factors by minimizing the upper bound of the error probability averaged with respect to the channel distribution. As long as is determined, each power loading factor will be determined as well. However, it is very difficult to calculate an explicit solution for

, though it is possible to find a suitable solution for a set of . Since we have already derived a closed-form expression for the upper bound of

λ p

_i

the optimization problem can be solved via numerical search instead (e.g. by using

fmincon function in Optimization Toolbox in MATLAB). Some average BER

performances are shown in the next section followed by a brief summary.

3.3 Computer Simulations

We use fmincon function in MATLAB [10] which employs sequential quadratic programming to find the optimal power loading factors that minimize the upper bound for the overall average BER. We, for example, consider a spatial multiplexing MIMO wireless communication systems with QPSK modulation scheme. First, we compare the evaluations of the upper bound for the lower bound for overall average BER without and with the transmit power allocation scheme proposed. The evaluation results are shown in

4 4 ×

Figure 3.2. It can be seen that in theory, the lower bound for the overall average BER has a 5-dB improvement at the high SNR regime when the power loading scheme is applied. However, this is just the theoretical result, and the computer simulations are shown in Figure 3.3. In Figure 3.3, it can be seen that at medium-high SNR the BER performance with power loading is improved by about 2 dB compared with that without power loading. The performance of QR receiver with power loading closely approaches that of ZF-VBLAST receiver, whose detecting procedure is more complex. In Figure 3.4, the performances with different modulation orders for QR receiver with and without power loading are shown. Since in the above derivation, for different modulation orders, the design criterion can be modified with different BER performance bounds [21], it can be seen that the performances are improved as long as the power loading factors for different modulation orders are determined. However, for

for example, the improvement with 16QAM becomes around 1 dB. This is because that the number of nearest neighbors in the modulation constellation increases when the constellation size gets larger.

Figure 3.2: Evaluations of upper bounds for the lower bound of overall average BER in spatial multiplexing systems with the QR receiver with QPSK modulation

4 4 ×

Figure 3.3: Average BER performances of spatial multiplexing systems with different receivers with QPSK modulation

4 4 ×

Figure 3.4: Average BER performances of spatial multiplexing systems with the QR receivers with different modulation orders

4 4 ×

3.4 Summary

In Section 3.1, a BER performance analysis based on spatial multiplexing MIMO systems with QR-based successive symbol detection is introduced. For simplicity, we consider the error propagation free case and derive the lower bound for the average BER. Following that, it is proved that minimization of this lower minimizes the upper bound for the block error rate, and thus essentially minimizes the average BER.

Nevertheless, in the above discussion, we focus on the design criterion under a fixed channel realization. With the probability distribution based on QR decomposition over channel realizations shown in Chapter 2, we can obtain the lower bound for the overall average BER. An upper bound for the expression of this lower bound is obtained by using Chernoff bound of Q-function, and then this upper bound can be expressed in closed form. In the proposed power allocation scheme, the transmit power is allocated based on the power loading factors which minimize this closed-form expression. Due to that the power loading factors are derived from the performance bound over random channel realizations, and there is no need to calculate them for each given channel realization, the computational complexity is quite low. Furthermore, from Equation (3.30), the bound is only affected by the number of transmission links and SNR . This simplicity makes it possible to construct a precoding table based on different SNR for a given spatial multiplexing MIMO system. In simulations, the results show that this power allocation scheme can enhance the performances of spatial multiplexing MIMO systems by 1-2 dB, and it can adapted according to different modulation orders.

M ρ

Chapter 4

Power Allocation for Minimum BER in Double Space-Time Transmit

Diversity Systems

Equation Section 4

In the previous chapter, the performance analysis of spatial multiplexing MIMO systems with the QR-based successive detection is given and a transmit power allocation scheme based on it is proposed. Now, we focus on the double space-time transmit diversity (DSTTD) systems which combines the advantages of higher data rates and transmit diversity. Since DSTTD systems can be considered as involving two parallel links in wireless transmission, with each link being space-time encoded by an Alamouti encoder, a DSTTD system can be analyzed as a spatial multiplexing MIMO system with transmit diversity provided at the transmitter. In Chapter 2, the R matrix of QR decomposition in DSTTD systems is shown to have a special structure;

we will examine the distribution of each diagonal entry of the R matrix in this chapter.

Although the exact probability density functions (pdf) of the diagonal entries of the R matrix may not be easily determined, they can still be estimated and approximated reasonably to be chi-square distributed. Therefore, this result makes it possible for us to derive the design criterion of the power allocation scheme for DSTTD systems in a way similar to the derivations in Chapter 2.

2 2 ×

4.1 Distribution of Diagonal Elements of R Matrix in QR Decomposition

The QR decomposition of channel matrix in DSTTD systems is shown below:

11 12 13 14 11 13 14

determined. First, we examine the value of . From Appendix A, it can be shown that

Because

H

₁ and

H

₃ are Alamouti block matrices, their determinants are given by

2 2

and

In the model of DSTTD systems, we assume that is a complex zero-mean Gaussian random variable and the variances of the real part and the imaginary part of it are 0.5. It can be readily proved that

h

ij

2

h

_ij 2 has a chi-square distribution with two

degrees of freedom. As a result, we can write Equation (4.4) as

2 2 2 2

According to the fact that the sum of chi-square random variables is still a chi-square random variable, and its degrees of freedom is the sum of the degrees of freedom of the summands, 2 R₁₁² is also a chi-square random variable with eight degrees of freedom.

Compared with the result of spatial multiplexing systems, the pdf of 2 R₁₁² is expressed as

which is identical to the pdf of 2 R₁₁² in spatial multiplexing systems. Therefore, let

1 1

Afterwards, we examine the distribution of

R

₃₃ . From Appendix A, it is given that

For the convenience for the derivations below, we denote , and thus we rewrite Equation (4.8) as

det( ) theorem of the determinant of partitioned matrices:

t( ) H d

₁

d

₂

d

₃

d

₄

Theorem 4.1: Let M

be a square matrix partitioned as

By taking determinants for both sides, we have

1

Applying Theorem 4.1 in (4.9), we can obtain

1

In the above equation, after some calculations, it can be shown that

1 3 2

Consequently, Equation (4.14) can be expressed as

1 4 3 2

To keep the consistency, we want to determine the distribution of 2 R₃₃², which is written as follows:

2 1 4 2 3

Although, based on the derivations for

R

₁₁ , it can be shown that

are chi-square-distributed with four degrees of freedom, there is no particular distribution suitable for this random variable. However, it can be observed that

2 ,

d i =

_i 1, 2, 3, 4

2

2 R33

consists of four i.i.d. chi-square random variables and a zero-mean variable with an unknown distribution. Based on this observation, we use the gamma distribution to approximate 2 R₃₃² because the chi-square distribution can be considered as a special case of the gamma distribution. The pdf of gamma distribution is given by

where and are the parameters which affect the shape and the scale of gamma distribution respectively. Here, we use dfittool in MATLAB to adjust the parameters of the gamma pdf and find the fit pdf which approaches the distribution of

k θ

2

2 R33

closely. In Figure 4.1, when and in (4.18), we can see that the curve of the gamma pdf is quite close to the sampled data of

2

k = θ = 2

2

2 R33 .

Figure 4.1: Comparison between the sampled data of 2 R₃₃² and the gamma pdf with and

2

k = θ = 2

Therefore, this approximation makes it reasonable and accurate to assume that 2 R₃₃² is a random variable following the gamma distribution with and , and its pdf is given by

It is obvious that (4.19) is the pdf of a chi-square distribution with four degrees of freedom as well. With some derivations similar to Chapter 3, denoting

r

₃

= R

₃₃ , the estimated pdf of

R

₃₃ can be expressed as

2 average BER can be calculated.

4.2 Optimal Power Allocation for Minimum Upper Bound of Overall Average BER

Consider a DSTTD system with QR successive detection at the receiver.

Consider a DSTTD system with QR successive detection at the receiver.

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