**7.1. Capacity region of UL-MU-MIMO with single antenna users**

We consider the UL-MU-MIMO with *N* multiple antenna BS and *K* single antenna users. The performed communication scheme is depicted in Figure 9. The equivalent MIMO channel for the presented communication model is given by:

$$H = [H\_1, \dots, H\_K] \tag{14}$$

*Hk*(*<sup>N</sup>* <sup>×</sup>1); *<sup>k</sup>* <sup>=</sup> 1, . . . , *<sup>K</sup>* represents the Single Input Multiple Output (SIMO) channel between user *Uk*; *k* = 1, . . . , *K* and the BS. Case of two users (i.e. *K*=2), the capacity region is defined as the set of rates (*R*1, *R*2) associated to users *U*<sup>1</sup> and *U*2. We consider the notations:


12 Recent Trends in Multiuser MIMO Communications

**Figure 8.** MU-MIMO coordinated network in cellular network [20]

Division Multiple Access (CDMA) techniques.

**7. Capacity region of multi-user MIMO system**

3. DL-MU-MIMO with multiple antenna users and single antenna BS

Subscriber Line (DSL)[22][13].

communication scenario.

We address the following scenarios :

1. UL-MU-MIMO with single antenna users 2. UL-MU-MIMO with multiple antenna users

**Digital Subscriber Line (DSL):** MU-MIMO systems are not only performed by multi-cell systems but also find their applications in other systems such as the downlink of a Digital

The performance of MU-MIMO could be improved via the use of Orthogonal Frequency Division Multiplexing (OFDM) or Orthogonal Frequency Division Multiplexing Access (OFDMA) for multiple access scenarios in frequency selective channels. MU-MIMO systems could also improve multi-user diversity by performing High Data Rate (HDR) or Code

There is no closed form for the channel capacity of multi-user MIMO systems. For this purpose, the performances of such systems will be analyzed in terms of the capacity region. This metric [23] could be defined in the usual Shannon sense as the highest rates that can be achieved with arbitrarily small error probability. Firstly, the capacity [24] needs to be evaluated for each user. Then, the capacity region is determined as the entire region for which maximum achievable rates are reached. The evaluation of the capacity region is strongly related to some constraints and should be determined according to the performed

**Figure 9.** UL-MU-MIMO with *N* multiple antenna BS and *K* single antenna users

The capacity region [25] is evaluated by determining the individual rate constraint for each user. Assuming that user *U*<sup>1</sup> has the entire channel, an upper bound of the maximum achievable rate is given by :

$$R\_1 \le \log\_2\left(1 + \frac{||H\_1||^2 \cdot P\_1}{N\_0}\right) \tag{15}$$

� · � indicates the Frobenius norm.

Similarly, an upper bound for the maximum achievable rate for user *U*<sup>2</sup> is:

$$R\_2 \le \log\_2\left(1 + \frac{\|H\_2\|^2 \cdot P\_2}{N\_0}\right) \tag{16}$$

Finally, the sum rate constraint which is obtained when both users are acting as two transmit antennas of a single user has an upper bound expressed as :

$$R\_1 + R\_2 \le \log\_2\left[\det\left(I\_N + \frac{H.diag(P\_1, P\_2).H^\*}{N\_0}\right)\right] \tag{17}$$

10.5772/57133

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

(20)

17

(21)

(22)

The sum rate constraint of UL-MU-MIMO with multiple antennas users is expressed as :

*K* ∑ *k*=1

*Hk*.*Dk*.*H*<sup>∗</sup> *k*

Multi User MIMO Communication: Basic Aspects, Benefits and Challenges

 

*N*<sup>0</sup>

�

�

*RxM*<sup>1</sup>

. . .

*Rx*<sup>1</sup>

*U*<sup>1</sup>

❃

*U*<sup>2</sup>

⑦

*RxM*<sup>2</sup>

. . .

*Rx*<sup>1</sup>

 *IN* +

In the case of downlink scenario, the upper bounds of the users rates are analogously determined as the uplink scenario. Nevertheless, the effect of interference could not be neglected. In fact, for the scenario with two multiple antenna users *U*<sup>1</sup> and *U*<sup>2</sup> and one antenna base station, the upper bounds of the rates achievable by users *U*<sup>1</sup> and *U*<sup>2</sup> become:

> <sup>1</sup> <sup>+</sup> �*H*1�<sup>2</sup> · *<sup>P</sup>*<sup>1</sup> *<sup>N</sup>*<sup>0</sup> + �*H*1�<sup>2</sup> · *<sup>P</sup>*<sup>2</sup>

> > <sup>1</sup> <sup>+</sup> �*H*2�<sup>2</sup> · *<sup>P</sup>*<sup>2</sup> *N*<sup>0</sup>

 det

**7.3. DL-MU-MIMO with multiple antenna users and single antenna BS**

�

�

*<sup>R</sup>*<sup>1</sup> <sup>+</sup> ... <sup>+</sup> *RK* <sup>≤</sup> log2

*<sup>R</sup>*<sup>1</sup> <sup>≤</sup> log2

*<sup>R</sup>*<sup>2</sup> <sup>≤</sup> log2

Here, the signal of user *U*<sup>2</sup> is considered as interference for user *U*1.

*Tx*<sup>1</sup>

The DL-MU-MIMO system uses precoding techniques which are usually linear.

BS

**Figure 11.** DL-MU-MIMO with multiple antenna users and single antenna BS

**8. Precoding techniques**

and

The capacity region for the UL-MU-MIMO is presented in Figure 10 where two users with single antennas are considered.

**Figure 10.** Capacity region of UL-MU-MIMO for two single antenna users

Case of *K* users, the capacity region is determined as a function of several constraints and *K*! corner points are determined for evaluating the boundary of the capacity region. For rates *R*1,..., *RK* respectively associated to users *U*1,..., *UK*, the sum rate is determined for an optimal receiver [25] as:

$$\sum\_{k \in \mathcal{S}} R\_k \le \log\_2 \left[ \det \left( I\_N + \frac{1}{N\_0} \sum\_{k \in \mathcal{S}} P\_k \cdot H\_k \cdot H\_k^\* \right) \right]; \quad \mathcal{S} \subset \{1, \dots, K\} \tag{18}$$

#### **7.2. Capacity region of UL-MU-MIMO with multiple antenna users**

The capacity region could be obtained for the generalized case where the base station has *N* antennas and user *Uk*; *k* = 1, . . . , *K* is equipped with multiple antennas of number *Mk* > 1. An upper bound of the maximum achievable rate for user *Uk* is given by :

$$R\_k \le \log\_2\left[\det\left(I\_N + \frac{H\_k \cdot D\_k \cdot H\_k^\*}{N\_0}\right)\right] \quad ; \; k = 1, \ldots, K \tag{19}$$


The sum rate constraint of UL-MU-MIMO with multiple antennas users is expressed as :

$$R\_1 + \dots + R\_K \le \log\_2 \left[ \det \left( I\_N + \frac{\sum\_{k=1}^K H\_k D\_k H\_k^\*}{N\_0} \right) \right] \tag{20}$$

#### **7.3. DL-MU-MIMO with multiple antenna users and single antenna BS**

In the case of downlink scenario, the upper bounds of the users rates are analogously determined as the uplink scenario. Nevertheless, the effect of interference could not be neglected. In fact, for the scenario with two multiple antenna users *U*<sup>1</sup> and *U*<sup>2</sup> and one antenna base station, the upper bounds of the rates achievable by users *U*<sup>1</sup> and *U*<sup>2</sup> become:

$$R\_1 \le \log\_2\left(1 + \frac{||H\_1||^2 \cdot P\_1}{N\_0 + ||H\_1||^2 \cdot P\_2}\right) \tag{21}$$

and

14 Recent Trends in Multiuser MIMO Communications

single antennas are considered.

log2 

an optimal receiver [25] as:

user *Uk*.

∑ *k*∈*S* *Rk* <sup>≤</sup> log2

*Rk* <sup>≤</sup> log2

 det *IN* +

antennas of a single user has an upper bound expressed as :

*<sup>R</sup>*<sup>1</sup> <sup>+</sup> *<sup>R</sup>*<sup>2</sup> <sup>≤</sup> log2

<sup>1</sup> <sup>+</sup> �*H*2�2·*P*<sup>2</sup> *N*<sup>0</sup> 

**Figure 10.** Capacity region of UL-MU-MIMO for two single antenna users

 det *IN* +

 det *IN* +

✻

*R*2

Finally, the sum rate constraint which is obtained when both users are acting as two transmit

The capacity region for the UL-MU-MIMO is presented in Figure 10 where two users with

log2 

Case of *K* users, the capacity region is determined as a function of several constraints and *K*! corner points are determined for evaluating the boundary of the capacity region. For rates *R*1,..., *RK* respectively associated to users *U*1,..., *UK*, the sum rate is determined for

The capacity region could be obtained for the generalized case where the base station has *N* antennas and user *Uk*; *k* = 1, . . . , *K* is equipped with multiple antennas of number *Mk* > 1.

*Hk* · *Dk* · *<sup>H</sup>*<sup>∗</sup>

*N*0

• *Dk*(*Mk* × *Mk*) is a diagonal matrix formed by the power allocated at transmit antennas at

• *Hk*(*<sup>N</sup>* × *Mk*) links the *<sup>N</sup>* antenna base station to the *Mk* antenna user; *<sup>k</sup>* = 1, . . . , *<sup>K</sup>*.

*k*

1 *<sup>N</sup>*<sup>0</sup> <sup>∑</sup> *k*∈*S*

**7.2. Capacity region of UL-MU-MIMO with multiple antenna users**

An upper bound of the maximum achievable rate for user *Uk* is given by :

<sup>1</sup> <sup>+</sup> �*H*1�2·*P*<sup>1</sup> *N*<sup>0</sup> 

*Pk* · *Hk* · *<sup>H</sup>*<sup>∗</sup>

*k* 

*<sup>H</sup>*.*diag*(*P*1, *<sup>P</sup>*2).*H*<sup>∗</sup> *N*0

✲

*R*<sup>1</sup> + *R*<sup>2</sup> = log2 det 

*R*1

*IN* <sup>+</sup> *<sup>H</sup>*.*diag*(*P*1,*P*2).*H*<sup>H</sup> *N*<sup>0</sup>

; *S* ⊂ {1, . . . , *K*} (18)

; *k* = 1, . . . , *K* (19)

(17)

$$R\_2 \le \log\_2\left(1 + \frac{||H\_2||^2 \cdot P\_2}{N\_0}\right) \tag{22}$$

Here, the signal of user *U*<sup>2</sup> is considered as interference for user *U*1.

**Figure 11.** DL-MU-MIMO with multiple antenna users and single antenna BS

#### **8. Precoding techniques**

The DL-MU-MIMO system uses precoding techniques which are usually linear.
