**2.2. Multiuser MIMO channel**

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

\*

= +**I HR H** (14)

) is (*r*), so *i* =1⋯*r*. The solution of the optimization problem given

(15)

is the

) and **P***<sup>i</sup>*

spatial diversity gain optimization techniques.

following optimization:

8 Selected Topics in WiMAX

of the covariance matrix (**HH**\*

filling in space solution which is given by:

*2.1.3.1. Channel capacity of the closed-loop single user MIMO*

*C*

Consider the general capacity formula for MIMO system given in [2]:

subject to:

*i*

å

*i T*

*P P*

£

2 2

s

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

This capacity depends on the channel realization **H** and the input covariance matrix **Rs**. Taking into account the availability of the CSI at the transmitter, there exists for any practical channel realization, an optimum choice of the input covariance matrix **Rs**such that the channel capacity is maximized subject to the transmit power constraint [7]. This capacity is calculated from the

<sup>2</sup> <sup>2</sup> max log (1 )

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

w

= +

å

where *λ<sup>i</sup>* is the *ith* eigenvalue of the single user MIMO covariance matrix (**HH**\*

*i n*

transmit power on the *ith* channel. For the purpose of generalization, we assume that the rank

in equation (15) [7] shows that the maximum capacity is achieved by what is called the water-

*i i*

l

s

Unlike the simple SU-MIMO channel, Multi-user MIMO (MU-MIMO) channel is a union of a setof SU-MIMOchannels.InMU-MIMOsystemconfiguration,there are twomaincommunica‐ tion links - the downlink channel (One-to-many transmission link) which is also known as MU-MIMOBroadcastChannel(MU-MIMO-BC)andtheuplinkchannel(Many-to-OneTransmission link)whichisalsoknownasMU-MIMOMultipleAccess(MU-MIMO-MAC)channel.Inaddition to the conventional MIMO channel gains, in MU-MIMO we can make use of the multi-user diversity gain to send simultaneously to a group of users or receive data from multiple users at the same time and frequency. As depicted in figure 2, MU-MIMO system configuration can be described as follows: Central node/base station equipped with *NT* transmit antennas transmit‐ ting simultaneously to *B* number of users in the downlink MU-MIMO-BC channel, where *kth* user is equipped with *Mk* receive antennas, *k* =1, ⋯, *B*. In the reverse uplink MU-MIMO-MAC the base station receiving data from the multiple users simultaneously.

Regardless of its implementation complexity, it is generally known that the minimum meansquare-error with successive interference cancelation (MMSE-SIC) multi-user detector is the best optimum receiver structure for the MU-MIMO-MAC channel [30]. To simultaneously transmit to multiple users in the downlink MU-MIMO-BC, Costa's Dirty-Paper Coding (DPC) or precoding is needed [31] to mitigate the Multi-User Interference (MUI). Both linear and nonlinear precoding transmission techniques have been heavily researched in the last decade with much preference given to the linear precoding methods owing to their simplicity [32-37]. In the downlink MU-MIMO-BC channel at some *kth* user, the received signal is given by:

$$\mathbf{y}\_k = \mathbf{H}\_k \mathbf{F}\_k \mathbf{s}\_k + \mathbf{H}\_k \sum\_{\substack{i=1 \\ i \neq k}}^B \mathbf{F}\_i \mathbf{s}\_i + \mathbf{n}\_k \tag{17}$$

where **H***<sup>k</sup>* <sup>∈</sup>**C***Mk* <sup>×</sup>*NT* is the channel from the base station to the *kth* user. **F***<sup>k</sup>* <sup>∈</sup>**C***NT* <sup>×</sup>*Mk* is the *kth* user precoding matrix, while **s***<sup>k</sup>* <sup>∈</sup>**C***Mk* ×1 is the *kth* user transmitted data vector and **n** *k* is the received additive white noise vector at the *kth* user antenna front end.

*2.3.1. Zero-forcing MIMO receiver*

receiving methods.

of the transmitted symbol vector can be written as

where the decoder is calculated from **G**=(**H**∗**H**)

*2.3.2. Minimum mean-square-error MIMO receiver*

*2.3.3. Maximum likelihood MIMO receiver*

vector **s** that minimizes the ML criteria of the form:

factor **G** is designed to maximize the expectation criteria of the form:

Zero-Forcing (ZF) decoder is a simple linear transformation of the received signal to remove the inter-channel interference by multiplying the received signal vector by the inverse of the channel matrix [38]. In fact, if perfect CSI is available at the receiver, the zero-forcing estimate

−1

inverse of the MIMO channel matrix. In ZF, the complexity reduction comes at the expense of noise enhancement which results in some performance losses compared to other MIMO

Unlike ZF receiver which completely force the interference to zero, the MIMO MMSE receiver tries to balance between interference mitigation and noise enhancement [39]. Thus, at low SNR values the MMSE outperforms the ZF receiver. In the MMSE MIMO receiver the decoding

By analytically solving this MMSE criterion for MIMO channel, the factor **G** is found to be:

With successive Interference Cancelation (SIC), additional nonlinear steps are added to the original ZF and MMSE equalizers. The resulting versions are ZF-SIC and MMSE-SIC decoding methods. In short, in SIC, the data layer symbols are decoded and subtracted successively from the next received data symbol starting with the highest SINR received signal at each decoding stage. The main drawback of this kind of receive structure is however, the error propagation.

The Maximum Likelihood (ML) decoder is an optimum receiver that achieves the best BER performance among all other decoding techniques. In ML, the decoder searches for the input

2

2 1 ( ) *<sup>n</sup>* s

**y G Hs n s Gn** = + =+ ( ) (18)

*E*{[ ][ ] }\* **Gy s Gy s** - - (19)

\* -\* **G HH I H** = + (20)

*<sup>F</sup>* **y Hs** - (21)

**H**∗, which is also known as the pseudo

On MU-MIMO Precoding Techniques for WiMAX

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

11

**Figure 2.** Block Diagram of MU-MIMO System Configuration

Before returning to the general question like how to design the precoder, what is the best Precoder, and what we can gain by incorporating multi-user mode in the WiMAX standards, we summarize the different MIMO system configurations in figure 3. Basically, there are two main modes, SU-MIMO and MU-MIMO. Essentially MU-MIMO is a closed-loop transmission system which means that the channel-state information is required at the transmitter for any transmission. There are two modes of operation for SU-MIMO configurations – closed-loop where the CSI is required at the transmitter and open-loop where the CSI is not required to be used at the transmitter. Also the diagram indicates at each end the type of MIMO gain that can be extracted by each mode of operation and specific configuration.
