**8.1. Zero Forcing and Block Diagonalization methods**

Popular low-complexity techniques include both Zero Forcing (ZF) and Block Diagonalization (BD)[27][28] methods. Algorithms for the ZF as well as BD methods are presented in [29]. The aim of these solutions is to improve the sum rate capacity of the communication system under a given power constraint. These performances could be achieved by canceling inter-user interference. Zero Forcing Dirty Paper Coding (DPC) [30] represents a famous technique for data precoding where the channel is subject to interference which is assumed to be known at the transmitter. The precoding matrix is equal to the conjugate transpose of the upper triangular matrix obtained via the QR decomposition of the channel matrix.

#### *8.1.1. MU-MIMO with Block Diagonalization precoding*

We consider a communication system model with a broadcast MIMO channel where the transmitter is a base station equipped with *N* antennas and the receiver consists of *K* users *Uk*; *k* = 1... *K* (See figure 6(b)). The received signal at user *Uk*; *k* = 1... *K* with dimension (*Mk* × <sup>1</sup>) is expressed as :

$$Y\_k = H\_k \cdot V\_{\rm BD}{}^{(k)} \cdot X\_k + B\_k \qquad ; \quad k = 1, \ldots, K \tag{23}$$

10.5772/57133

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http://dx.doi.org/10.5772/57133

*8.1.2. MU-MIMO with Zero Forcing precoding*

**8.2. Beamforming for linear precoding**

but not orthogonal to the *<sup>l</sup>* − *th* column of *Hk* [26].

. . . .

.

*xK*

**Figure 12.** MU-MIMO with beamforming

where:

station.

*Yk* <sup>=</sup> *Hk* · *Vt*(*k*)

• *Bk*(*Mk* × <sup>1</sup>) is an additive noise signal vector.

✲ *<sup>x</sup>*<sup>1</sup>

Case of Zero Forcing strategy, each transmitted symbol to the *l* − *th* antenna (among *M* antennas of user *Uk*) is precoded by a vector which is orthogonal to the columns of *Hj*, *<sup>j</sup>* �= *<sup>k</sup>*

Beamforming paradigms represent another class of linear precoding for MU-MIMO systems. For the communication model with beamforming (Figure 12), we consider a MU-MIMO system with *K* multiple antenna users *U*1,..., *UK* at the receive side which are respectively equipped with *M*1,..., *MK* antennas. At the transmit side, a multiple antenna base station

*Tx*<sup>1</sup> *Rx*<sup>1</sup>

*TxN RxM*<sup>1</sup>

*HK*

. .

✲

❯

*VtBF Vr*(1)

Transmit side Receive side

✲ ✲

*Rx*<sup>1</sup>

. .

. .

✲

. .

*Vr*(*K*) *BF*

*BF* · *xj* <sup>+</sup> *Bk* ; *<sup>k</sup>* <sup>=</sup> 1, . . . , *<sup>K</sup>* (26)

*YK zK*

*BF*

✲

✲

. .

*H*<sup>1</sup> *Y*<sup>1</sup> *z*<sup>1</sup>

Multi User MIMO Communication: Basic Aspects, Benefits and Challenges

*RxMK*

. . .

with *N* antennas transmits data signals *x*1,..., *xK* to users *U*1,..., *UK*.

The received signal vector at user *Uk*; *k* = 1, . . . , *K* is expressed as :

*K* ∑ *j*=1, *k*�=*j*

*Hk* · *Vt*(*j*)

• *Hk*(*Mk* × *<sup>N</sup>*) is the complex channel matrix between receiver *Uk* and the transmit base

*BF* · *xk* <sup>+</sup>


We assume in the following that users *U*1,... *UK* have the same number of antennas which will be denoted by *M*. Block Diagonalization strategy defines a set of precoding matrices *VBD*(*k*)(*<sup>N</sup>* × *<sup>M</sup>*) associated to users *<sup>U</sup>*1,..., *UK*. These matrices form an orthonormal basis such that:

$$\left[\left[V\_{BD}\right]^{(k)}\right]^\* \cdot V\_{BD}\left^{(k)} = I\_M \quad \text{; } k = 1 \dots K \tag{24}$$

and the Block Diagonalization algorithm achieves :

$$H\_k \cdot V\_{BD}{}^{(j)} = 0 \quad ; \; \forall \; \; j \neq K \tag{25}$$

The aim of these conditions is to eliminate multi-user interference so that to maximize the achievable throughput.

The performance of downlink communication scenarios with precoding techniques depends on the SNR level. In fact, it has been shown in [27] that SU-MIMO achieves better performances than MU-MIMO at low SNRs. However, the BD MU-MIMO achieves better performances at high SNRs. As such, switching between SU-MIMO and MU-MIMO is optimal for obtaining better total rates over users.
