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

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

$$\mathbf{C} = \{ \log\_2 \left\| \mathbf{I} + \frac{1}{\sigma\_n^2} \mathbf{H} \mathbf{R}\_s \mathbf{H}^\* \right\| \} \text{ bit/s} \tag{14}$$

2 o

*<sup>P</sup> is*

 g

MAC the base station receiving data from the multiple users simultaneously.

, some cut-off SNR

In short this meants that, technically we have to allocate more power to the strong eigen modes and less power to the weak ones. It is also clear that this capacity is proportional to the

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-

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:

1

**y H F s H Fs n** =+ + å (17)

is the *kth* user transmitted data vector and **n**

*B*

*k kkk k ii k i i k* = ¹

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*

received additive white noise vector at the *kth* user antenna front end.

user precoding matrix, while **s***<sup>k</sup>* <sup>∈</sup>**C***Mk* ×1

(16)

9

On MU-MIMO Precoding Techniques for WiMAX

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

is the *kth*

*k* is the

*i o*

g g

g g

1 1

*i i o o i*

<sup>ï</sup> - ³ <sup>=</sup> <sup>í</sup> <sup>ï</sup> <sup>&</sup>lt; <sup>î</sup>

0 given that :

g g

ì

*i i*

l

=

s

*n*

*T*

g

min(*NT* , *M***R**).

**2.2. Multiuser MIMO channel**

*P P*

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 following optimization:

$$\begin{aligned} \text{C = max} & \sum\_{i} \alpha \log\_2(1 + \frac{\lambda\_i P\_i}{\sigma\_n^2})\\ \text{subject to:} & \\ & \sum\_{i} P\_i \le P\_T \end{aligned} \tag{15}$$

where *λ<sup>i</sup>* is the *ith* eigenvalue of the single user MIMO covariance matrix (**HH**\* ) and **P***<sup>i</sup>* is the transmit power on the *ith* channel. For the purpose of generalization, we assume that the rank of the covariance matrix (**HH**\* ) is (*r*), so *i* =1⋯*r*. The solution of the optimization problem given in equation (15) [7] shows that the maximum capacity is achieved by what is called the waterfilling in space solution which is given by:

$$\begin{aligned} \frac{P\_i}{P\_T} &= \left| \frac{1}{\chi\_o} - \frac{1}{\chi\_i} \right| & \gamma\_i &\ge \gamma\_o \\ \frac{P\_i}{0} & & \gamma\_i &< \gamma\_o \\ \text{given that :} \\ \gamma\_i &= \frac{\lambda\_i P}{\sigma\_n^2}, \quad \gamma\_o \text{ is some cut-off SNR} \end{aligned} \tag{16}$$

In short this meants that, technically we have to allocate more power to the strong eigen modes and less power to the weak ones. It is also clear that this capacity is proportional to the min(*NT* , *M***R**).
