**2. Preliminaries**

## **2.1. Single user MIMO (SU-MIMO) channel model**

In wireless communications, channel modeling and link parameter design are the core problems in designing communication system. To understand the MIMO system, MIMO channel modeling and the related assumptions behind the practical system realization should therefore be first discussed and summarized. Basically, it all begins by the designer identifying the channel type from among the three types of wireless communication channel namely: direct path, frequency selective and frequency-flat channels. Direct path, also called Line-of-Sight (LOS) channel is the simplest model where the channel gain consistes of only the free space path loss plus some complex Additive White Gaussian Noise (AWGN). This simplified model is typically used to design the various microwave communication links such as terrestrial, near space satellite, and deep space communication links. The other two types of wireless channel models are the frequency flat and frequency selective channels which both describes the channel gains due to the complex propagation environment where both the free space path loss, shadowing effects and multipath interference are obvious and the objective of the link model is to account for multipath and Doppler shift effects. In frequency selective channel model, the link gain between each transmit and receive antennas is represented by multiple and different impulse response sequences across the frequency band of operation. This is in contrast to the frequency flat fading which has single constant scalar channel gain across the band. Frequency selectivity has crucial Inter-Symbol Interference (ISI) effect on the high speed wireless communication system transmission. Technically there are three ways to mitigate the negative effect of ISI. Two of which are transmission techniques namely: spread spectrum transmission and Multicarrier modulation transmission while the third is the equalization techniques as a receiver side mitigation method. In a MIMO system, researchers generally make common assumption that the channel-frequency-response is flat between each pair of transmit and receive antennas. Thus, from the system design point of view, the system designer can alleviate the frequency selectivity effect in a wideband system by subdividing the wideband channel into a set of narrow sub-bands as in [7, 8] using Orthogonal Frequency Division Multiplexing (OFDM). Figure 1 shows point-to-point single user MIMO system with *NT* transmit antennas and *M***R** receive antennas. The channel from the multiple transmit antenna to the multiple receiving antenna is described by the gain matrix **H**. With the basic assumption of frequency-flat fading narrow band link between the transmitter and the receiver, **H** will be given by:

$$\mathbf{H} = \begin{vmatrix} h\_{11} & \cdots & h\_{1,N\_T} \\ \vdots & \ddots & \vdots \\ h\_{M\_R,1} & \cdots & h\_{M\_R,N\_T} \end{vmatrix} \in \mathbf{C}^{M\_R \times N\_T} \tag{1}$$

is equal to *ω*. Thus, the received signal vector **y** *n* ∈**C***M***R**×1

*<sup>T</sup>* ∈**C***NT* ×1

Gaussian elements with zero mean and variance *σ<sup>n</sup>*

given by:

*n* ∈**C***M***R**×1

described by:

where **s** *n* = **s**1⋯**s***NT*

power spectral density.

at the time instants *n* can be

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

3

On MU-MIMO Precoding Techniques for WiMAX

<sup>2</sup> <sup>=</sup>*ωNo* and *No* is the noise

**y Hs n** [] [] [] *n nn* = + (2)

denotes the complex transmitted signal vector, and **n**

where *σ<sup>n</sup>*

denotes the AWGN vector which is assumed to have independent complex

2 **I***M***<sup>R</sup>**

The channel matrix **H** is assumed to have independent complex Gaussian random variables with zero mean and unit variance. This statistical distribution is very useful and reasonable assumption to model the effect of richly scattering environment where the angular bins are fully populated paths with sparsely spaced antennas. Justification of these MIMO link assumptions is a very important point for system design. On choosing system operation bandwidth *ω*, the system designers and researchers always assume that the channel frequency response over the bandwidth of MIMO transmission is flat, as practically, it is very hard to implement equalizers for mitigating ISI across all the multiple antennas in a MIMO system. It is to be noted that the twin requirements of broadband transmission to support the high rate applications and narrow band transmission to facilitate the use of simple equalizers to mitigate ISI can be met by utilizing OFDM based physical layer transmission. In physically rich scattering environment (e.g. typical urban area signal propagation) with proper antenna array spacing, the common assumption is that the elements of the Channel matrix are independent identically distributed (i.i.d.). Though, beyond the scope of this study, in practice, if there is any kind of spatial correlation, this will reduce the degrees of freedom of the MIMO channel and consequently this would then result in a decrease in the MIMO channel capacity gain [3, 9]. The assumption of i.i.d channel is partially realizable by correctly separating the multiple element antennas. In some deployment scenarios, where there are not enough scatterers in the propagation environment, the i.i.d assumption is not practical to statistically model the MIMO fading correlation channel. A part of the last decade research was focused on the study of this kind of channel correlation effects [10]. In general, fading correlation between the elements of MIMO channel matrix **H** can be separated into two independent components, namely: transmit correlation and receive correlation [11]. Accordingly, the MIMO channel model **H** can be

1/2 1/2

where **H***<sup>w</sup>* is the channel matrix whose elements are i.i.d, and **R***<sup>r</sup>*

transmit correlation matrix respectively.

*r wt* **H R HR** = (3)

1/2 and **R***<sup>t</sup>*

1/2 are the receive and

where **h***i*, *<sup>j</sup>* denotes the complex channel gain between the *jth* transmit antenna to the *ith* receive antenna, while 1≤ *j* ≤ *NT* , 1≤*i* ≤*M***R**. We further assume that the channel bandwidth is equal to *ω*. Thus, the received signal vector **y** *n* ∈**C***M***R**×1 at the time instants *n* can be given by:

**2. Preliminaries**

2 Selected Topics in WiMAX

receiver, **H** will be given by:

11 1,

é ù ê ú <sup>=</sup> <sup>Î</sup> L MO M L

*h h*

*h h*

**H**

,1 ,

*M M N*

*R R T*

*T*

*N*

where **h***i*, *<sup>j</sup>* denotes the complex channel gain between the *jth* transmit antenna to the *ith* receive antenna, while 1≤ *j* ≤ *NT* , 1≤*i* ≤*M***R**. We further assume that the channel bandwidth

*R T*

(1)

*M N*

**C**

´

**2.1. Single user MIMO (SU-MIMO) channel model**

In wireless communications, channel modeling and link parameter design are the core problems in designing communication system. To understand the MIMO system, MIMO channel modeling and the related assumptions behind the practical system realization should therefore be first discussed and summarized. Basically, it all begins by the designer identifying the channel type from among the three types of wireless communication channel namely: direct path, frequency selective and frequency-flat channels. Direct path, also called Line-of-Sight (LOS) channel is the simplest model where the channel gain consistes of only the free space path loss plus some complex Additive White Gaussian Noise (AWGN). This simplified model is typically used to design the various microwave communication links such as terrestrial, near space satellite, and deep space communication links. The other two types of wireless channel models are the frequency flat and frequency selective channels which both describes the channel gains due to the complex propagation environment where both the free space path loss, shadowing effects and multipath interference are obvious and the objective of the link model is to account for multipath and Doppler shift effects. In frequency selective channel model, the link gain between each transmit and receive antennas is represented by multiple and different impulse response sequences across the frequency band of operation. This is in contrast to the frequency flat fading which has single constant scalar channel gain across the band. Frequency selectivity has crucial Inter-Symbol Interference (ISI) effect on the high speed wireless communication system transmission. Technically there are three ways to mitigate the negative effect of ISI. Two of which are transmission techniques namely: spread spectrum transmission and Multicarrier modulation transmission while the third is the equalization techniques as a receiver side mitigation method. In a MIMO system, researchers generally make common assumption that the channel-frequency-response is flat between each pair of transmit and receive antennas. Thus, from the system design point of view, the system designer can alleviate the frequency selectivity effect in a wideband system by subdividing the wideband channel into a set of narrow sub-bands as in [7, 8] using Orthogonal Frequency Division Multiplexing (OFDM). Figure 1 shows point-to-point single user MIMO system with *NT* transmit antennas and *M***R** receive antennas. The channel from the multiple transmit antenna to the multiple receiving antenna is described by the gain matrix **H**. With the basic assumption of frequency-flat fading narrow band link between the transmitter and the

$$\mathbf{y}[n] = \mathbf{H}\mathbf{s}[n] + \mathbf{n}[n] \tag{2}$$

where **s** *n* = **s**1⋯**s***NT <sup>T</sup>* ∈**C***NT* ×1 denotes the complex transmitted signal vector, and **n** *n* ∈**C***M***R**×1 denotes the AWGN vector which is assumed to have independent complex Gaussian elements with zero mean and variance *σ<sup>n</sup>* 2 **I***M***<sup>R</sup>** where *σ<sup>n</sup>* <sup>2</sup> <sup>=</sup>*ωNo* and *No* is the noise power spectral density.

The channel matrix **H** is assumed to have independent complex Gaussian random variables with zero mean and unit variance. This statistical distribution is very useful and reasonable assumption to model the effect of richly scattering environment where the angular bins are fully populated paths with sparsely spaced antennas. Justification of these MIMO link assumptions is a very important point for system design. On choosing system operation bandwidth *ω*, the system designers and researchers always assume that the channel frequency response over the bandwidth of MIMO transmission is flat, as practically, it is very hard to implement equalizers for mitigating ISI across all the multiple antennas in a MIMO system. It is to be noted that the twin requirements of broadband transmission to support the high rate applications and narrow band transmission to facilitate the use of simple equalizers to mitigate ISI can be met by utilizing OFDM based physical layer transmission. In physically rich scattering environment (e.g. typical urban area signal propagation) with proper antenna array spacing, the common assumption is that the elements of the Channel matrix are independent identically distributed (i.i.d.). Though, beyond the scope of this study, in practice, if there is any kind of spatial correlation, this will reduce the degrees of freedom of the MIMO channel and consequently this would then result in a decrease in the MIMO channel capacity gain [3, 9]. The assumption of i.i.d channel is partially realizable by correctly separating the multiple element antennas. In some deployment scenarios, where there are not enough scatterers in the propagation environment, the i.i.d assumption is not practical to statistically model the MIMO fading correlation channel. A part of the last decade research was focused on the study of this kind of channel correlation effects [10]. In general, fading correlation between the elements of MIMO channel matrix **H** can be separated into two independent components, namely: transmit correlation and receive correlation [11]. Accordingly, the MIMO channel model **H** can be described by:

$$\mathbf{H} = \mathbf{R}\_r^{1/2} \mathbf{H}\_w \mathbf{R}\_t^{1/2} \tag{3}$$

where **H***<sup>w</sup>* is the channel matrix whose elements are i.i.d, and **R***<sup>r</sup>* 1/2 and **R***<sup>t</sup>* 1/2 are the receive and transmit correlation matrix respectively.

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

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

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

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).

1 2 [ ] *MR*

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

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

is on the receiver which may lead to high power consumption on the receiver unit.

with **s** denoting the transmitted signal with unit variance, the received signal **y**∈**C***M***R**×1

**H h** = = *hh h* L (4)

On MU-MIMO Precoding Techniques for WiMAX

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

5

**yh n** = +*s* (5)

can be

diversity methods that are popular, namely: transmit diversity and receive diversity.

can extract MIMO diversity gain with the help of time or frequency domain.

In this case the MIMO channel **H** is reduced to the vector of the form:

expressed as:

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