*2.1.2. Open-loop single user MIMO (SU-MIMO) transmission*

*NT* antennas *MR* antennas

The key to the performance gain in MIMO systems lies in the additional degree-of-freedom provided by the spatial domain and associated with multiple antennas. These additional degree-of-freedom can be exploited and utilized in the same way as the frequency and time resources have been used in the classical Single Input Single Output (SISO) systems. The initial promise of an increase in capacity and spectral efficiency of MIMO systems ignited by the work of Telatar [1] and Foschini [2] has now been validated where by adding more antennas to the transmitter and receiver, the capacity of the system has been shown to increase linearly with the *NT* or *M***R**,which is minimum, i.e the min(*NT* , *M***R**) [12]. This capacity can be extracted by making use of three transmission techniques, namely: spatial multiplexing, spatial diversity,

From classical communication and information theory, channel characteristics play a crucial rule in the system design, in that both transmitter and receiver design are highly dependent on it [13, 14]. In MIMO system, the knowledge of the Channel State Information (CSI) is an important factor in system design. CSIT and CSIR refer to the CSI at the transmitter and receiver respectively. Basically, in the state-of-art communication system design, there is a common assumption that the receiver has perfect CSI. With this fair assumption, all MIMO performance gains are exploited. Further improvement in the performance is dependent on the availability and quality of CSI at the transmitter [8, 15]. Accordingly, the accessibility and utilization of CSI at the transmitter is one of the most important criteria of MIMO research classification in the last decade. Next sub-sections gives a brief overview of the most critical processing techniques and types of gains that we can extract from the single user point-to-point MIMO link in both open-loop systems (CSI is available at the receiver) and closed-loop systems (CSI

Receiver

Transmitter

*2.1.1. MIMO channel gain*

and beam-forming.

**Figure 1.** Block Diagram of single User MIMO System

is available at both transmitter and receiver).

4 Selected Topics in WiMAX

When there is no CSI at the transmitter, this is called open-loop MIMO configuration. There are two types of performance gains that can be extracted - multiplexing gain and diversity gain [16]. Multiplexing gain is the increase in the transmission rate at no cost of power consumption. This type of gain is achieved through the use of multiple antennas at both transmitter and receiver. In a single user MIMO system with spatial multiplexing gain configuration, different data streams can be transmitted from the different transmit antennas simultaneously. At the receiver, both linear and nonlinear decoders are used to decode the transmitted data vector. Spatial multiplexing gain is very sensitive to long-deep channel fades. Thus, in such commu‐ nication environment, the designer can solve this problem by resolving to system design that can extract MIMO diversity gain with the help of time or frequency domain.

Diversity gain is defined as the redundancy in the received signal [17]. It affects the probability distribution of received signal power favorably. In single user MIMO system, diversity gain can be extracted when replicas of information signals are received through independent fading channels. It increases the probability of successful transmission which, in turn increases the communication link reliability. In the single user MIMO system, there are two types of diversity methods that are popular, namely: transmit diversity and receive diversity.

Receive diversity is applied on a sub-category of MIMO system where there is only one transmit antenna and *M***R** receive antennas, also called Single Input Multiple Output (SIMO). In this case the MIMO channel **H** is reduced to the vector of the form:

$$\mathbf{H} = \mathbf{h} = [h\_1 h\_2 \cdots h\_{M\_R}] \tag{4}$$

with **s** denoting the transmitted signal with unit variance, the received signal **y**∈**C***M***R**×1 can be expressed as:

$$\mathbf{y} = \mathbf{h}\mathbf{s} + \mathbf{n} \tag{5}$$

The received signal vector from all receiving antennas is combined using one of the many combining techniques like Selection Combining (SC), Maximal Ratio Combining (MRC) or Equal Gain Combining (EGC) to enhance the received Signal to Noise Ratio (SNR) [18]. The most notable drawback of these diversity techniques is that most of the computational burden is on the receiver which may lead to high power consumption on the receiver unit.

On the other hand, MIMO transmit-diversity gain can be extracted by using what is called Space Time Codes (STC) or Space Frequency Code (SFC) [12, 19, 20]. Unlike receive diversity, transmit diversity requires simple linear receive processing to decode the received signal. STC and SFC are almost similar in many aspects except that one of them uses the time domain while the other uses frequency domain. Space-time codes are further classified into Space-Time Block Codes (STBC) and Space-Time Trellis Codes (STTC) families. In general, STTC families achieve

#### 6 Selected Topics in WiMAX

better performance than STBC families at the cost of extra computational load. A well known example and starting point for understanding the STBC transmit diversity techniques is the basic method of Alamouti code [4] which has diversity gain of the order of 2*M***R**. However, the main limitation of the basic Alamouti method is that it works only for two transmit antennas. However, latest advances in MIMO diversity techniques extends this method to the case of MIMO channel with more than two transmit antennas through what is known today as Orthogonal Space Time Block Codes (OSTBC) [21].

## *2.1.2.1. Channel capacity of open-loop single user MIMO system*

Without the CSI at the transmitter, the MIMO channel capacity is defined and obtained in [1, 22, 23]. Specifically, for the time-invariant communication channel, the capacity is defined as the maximum mutual information between the MIMO channel input and the channel output and is given by:

$$\mathbf{C}^\* = \alpha \log\_2 \left\| \mathbf{I} + \frac{1}{\sigma\_n^2} \mathbf{H} \mathbf{R}\_s \mathbf{H}^\* \right\| \quad \text{bits/s} \tag{6}$$

Andconsequently, for the case of MISO configuration (*NT* transmit antenna and one receive

<sup>1</sup> log (1 ) bit/s *<sup>T</sup>*

*N*

*n T*

Conversely, for the time varying communication channel, the capacity in equation (8) becomes

<sup>1</sup> { log } bit/s *<sup>T</sup> n T*

s

Unlike the capacity gains defined in equations (8-10) which can be extracted by spatial multiplexing or diversity, the system capacity in equation (11) is unidentified and it has no significant practical meaning. Thus, in such cases, the system designer can use some kind of system outage metric for the performance evaluation. The quantity called the outage capacity can be defined by the probability that the channel mutual information is less than some

*N*

<sup>1</sup> { : log } *n T*

s

When the CSI is available at both transmitter and receiver, all kinds of MIMO gains (diversity, spatial multiplexing and beam forming) can be extracted and optimized. In practice, CSI can be acquired at the transmitter either through feedback channels in Frequency Division Duplex (FDD) systems or just taking the dual transpose of the received channel in the case of timeinvariant Time Division Duplex (TDD) systems[24]. To extract the maximum spatial multi‐ plexing gain, transmission optimization should be done by what is called channel precoder and decoder [25, 26]. For single user MIMO channel, firstly the precoder is designed, multiplied with the user's data, and launched through *NT* transmit antennas at the transmitter site. At the receiver, the received signal from the *M***<sup>R</sup>** receive antennas is processed by the optimized

*N*

2 2

s

*SIMO F*

*<sup>p</sup> <sup>C</sup>*

ergodic 2 2

w

*<sup>p</sup> C E*

out 2 2

w

linear decoder. The general form of the precoded received signal is written as:

*<sup>p</sup> pro prob <sup>C</sup>*

w

2

\*

\*

= +< **H I HH** (12)

**y HFs n** = + (13)

= + **h** (10)

On MU-MIMO Precoding Techniques for WiMAX

http://dx.doi.org/10.5772/56034

7

= +**I HH** (11)

antennas) the channel capacity reduces to:

random or ergodic [7] and is defined by:

*2.1.3. Closed-loop single user MIMO transmission*

constant *C*:

where *ω* is bandwidth in Hz and **Rs** is the covariance matrix of the transmitted signal and *PT* =*tr*(**Rs**) is the total power-constraint. So, for the single user MIMO channel with a Gaussian random matrix with i.i.d elements, the channel capacity will be maximized by distributing the total transmit power over all transmit antennas equally. Thus, in this uniform power allocation scenario, the input covariance matrix **Rs** must be selected such that:

$$\mathbf{R}\_s = \frac{P\_T}{N\_T} \mathbf{I}\_{N\_T} \tag{7}$$

With power constraint inequality of the form *tr*(**Rs**)≤*PT*, where:*PT* is the total transmitting power, the substitution of the power constraint in the average capacity formula of equation (6) yields:

$$\mathbf{C}^\* = \alpha \log\_2 \left\| \mathbf{I} + \frac{1}{\sigma\_n^2} \frac{p\_T}{N\_T} \mathbf{H} \mathbf{H}^\* \right\| \text{ bit/s} \tag{8}$$

[18], and in the case of SIMO configuration (one transmit antenna and *M***R** receive antennas) the channel capacity reduces to:

$$\mathcal{C}\_{\text{SDMO}} = o \log\_2 \left( 1 + \frac{P\_T}{\sigma\_n^2} \left\| \mathbf{h} \right\|\_F^2 \right) \text{ bit/s} \tag{9}$$

Andconsequently, for the case of MISO configuration (*NT* transmit antenna and one receive antennas) the channel capacity reduces to:

$$C\_{SIMO} = o \log\_2(1 + \frac{1}{\sigma\_n^2} \frac{p\_T}{N\_T} \left\| \mathbf{h} \right\|\_F^2) \text{ bit/s} \tag{10}$$

Conversely, for the time varying communication channel, the capacity in equation (8) becomes random or ergodic [7] and is defined by:

$$\mathcal{C}\_{\text{ergodic}} = E\{o\log\_2\left\|\mathbf{I} + \frac{1}{\sigma\_n^2} \frac{p\_T}{N\_T} \mathbf{H} \mathbf{H}^\*\right\|\} \text{ bit/s} \tag{11}$$

Unlike the capacity gains defined in equations (8-10) which can be extracted by spatial multiplexing or diversity, the system capacity in equation (11) is unidentified and it has no significant practical meaning. Thus, in such cases, the system designer can use some kind of system outage metric for the performance evaluation. The quantity called the outage capacity can be defined by the probability that the channel mutual information is less than some constant *C*:

$$pro\_{\text{out}} = prob \{ \mathbf{H} : o \log\_2 \left\| \mathbf{I} + \frac{1}{\sigma\_n^2} \frac{p}{N\_T} \mathbf{H} \mathbf{H}^\* \right\| < \mathbf{C} \} \tag{12}$$

## *2.1.3. Closed-loop single user MIMO transmission*

better performance than STBC families at the cost of extra computational load. A well known example and starting point for understanding the STBC transmit diversity techniques is the basic method of Alamouti code [4] which has diversity gain of the order of 2*M***R**. However, the main limitation of the basic Alamouti method is that it works only for two transmit antennas. However, latest advances in MIMO diversity techniques extends this method to the case of MIMO channel with more than two transmit antennas through what is known today as

Without the CSI at the transmitter, the MIMO channel capacity is defined and obtained in [1, 22, 23]. Specifically, for the time-invariant communication channel, the capacity is defined as the maximum mutual information between the MIMO channel input and the channel output

> <sup>1</sup> log bits/s *<sup>s</sup> n*

where *ω* is bandwidth in Hz and **Rs** is the covariance matrix of the transmitted signal and *PT* =*tr*(**Rs**) is the total power-constraint. So, for the single user MIMO channel with a Gaussian random matrix with i.i.d elements, the channel capacity will be maximized by distributing the total transmit power over all transmit antennas equally. Thus, in this uniform power allocation

> *T T s N T P N*

> > \*

2

*F*

With power constraint inequality of the form *tr*(**Rs**)≤*PT*, where:*PT* is the total transmitting power, the substitution of the power constraint in the average capacity formula of equation

> <sup>1</sup> log bit/s *<sup>T</sup> n T*

*N*

[18], and in the case of SIMO configuration (one transmit antenna and *M***R** receive antennas)

*n*

s

SIMO 2 <sup>2</sup> log (1 ) bit/s *<sup>T</sup>*

2 2

s

*<sup>p</sup> <sup>C</sup>*

*<sup>P</sup> <sup>C</sup>* w

w

2 2

s

\*

= +**I HR H** (6)

**R I** = (7)

= +**I HH** (8)

= + **h** (9)

Orthogonal Space Time Block Codes (OSTBC) [21].

*C*

w

scenario, the input covariance matrix **Rs** must be selected such that:

and is given by:

6 Selected Topics in WiMAX

(6) yields:

the channel capacity reduces to:

*2.1.2.1. Channel capacity of open-loop single user MIMO system*

When the CSI is available at both transmitter and receiver, all kinds of MIMO gains (diversity, spatial multiplexing and beam forming) can be extracted and optimized. In practice, CSI can be acquired at the transmitter either through feedback channels in Frequency Division Duplex (FDD) systems or just taking the dual transpose of the received channel in the case of timeinvariant Time Division Duplex (TDD) systems[24]. To extract the maximum spatial multi‐ plexing gain, transmission optimization should be done by what is called channel precoder and decoder [25, 26]. For single user MIMO channel, firstly the precoder is designed, multiplied with the user's data, and launched through *NT* transmit antennas at the transmitter site. At the receiver, the received signal from the *M***<sup>R</sup>** receive antennas is processed by the optimized linear decoder. The general form of the precoded received signal is written as:

$$\mathbf{y} = \mathbf{H} \mathbf{F} \mathbf{s} + \mathbf{n} \tag{13}$$

where **F** is the transmit precoding matrix. Different constraints and conditions are used to design the single user MIMO precoding matrix. Generalized method of joint optimum precoder and decoder for single user MIMO system based on Minimum Mean Square Error (MMSE) approach is proposed in [15]. In this method, minimum mean square error perform‐ ance criteria is used. As the name suggests, the framework is general and leads to flexible solution for performance criterias such as minimum BER and maximum information rate. The main drawbacks of this method are its high computational complexity and the restrictions on the number of antennas. In addition, there are many other simple and linear methods of precoding such as zero forcing, Singular Value Decomposition (SVD) [6, 8] or code book based techniques [27, 28]. Although these methods are simple, they have quite acceptable perform‐ ance. On the other hand, the spatial diversity gain can also be optimized by precoding when some kind of CSI is available at the transmitter. The precoding across the space-time block code in [19] or transmit antenna selection method in [29] are two other notable closed-loop spatial diversity gain optimization techniques.
