**2. MIMO-UWB interference mitigation by interference alignment**

IA is a promising technique which achieves the maximum degrees of freedom (DoF) for *K* users in interference channels [4]. This can be achieved by a combination of linear precoding

©2013 Blume et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2013 Blume et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 134 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Interference Alignment for UWB-MIMO Communication Systems <sup>3</sup>

at the transmitters and interference suppression at the receivers. IA permits to force interfering signals at each receiver in one subspace and the desired signal in another orthogonal subspace [5].

Consider a *K*-user UWB Multi Band Orthogonal Frequency Division Multiplexing (MB-OFDM) interference channel with *Mj* transmit antennas at transmitter *j* and *Ni* receive antennas at receiver *i*. All users transmit *ds* streams using *N* sub-carriers. Every transmitter communicates with his desired receiver and causes interference to other pairs of transmitter and receiver. The discrete-time complex received signal over the *n*th subcarrier at the *i*th receiver over a flat channel is represented as[21],[28]:

$$\mathbf{y}\_{i}^{n} = \sum\_{j=1}^{K} \mathbf{H}\_{ij}^{n} \mathbf{V}\_{j}^{n} \mathbf{x}\_{j}^{n} + \mathbf{z}\_{i}^{n} = \mathbf{H}\_{ii}^{n} \mathbf{V}\_{i}^{n} \mathbf{x}\_{i}^{n} + \sum\_{j=1, j \neq i}^{K} \mathbf{H}\_{ij}^{n} \mathbf{V}\_{j}^{n} \mathbf{x}\_{j}^{n} + \mathbf{z}\_{i}^{n} \tag{1}$$

and the interference suppression matrix for receiver *i* is given by:

**2.2. Artificial channel diversity algorithm**

and robustness.

detail.

**3.1. Antenna selection criterion**

**Figure 1.** Transmitter block diagram of a MIMO UWB MB-OFDM communication system.

For successful applying the IA principle a sufficient orthogonality between all channels is required. In real-world indoor environments and MIMO UWB systems such orthogonality is not guaranteed; instead, the small distances between neighboring antennas and a possible low scattering could lead to high correlation among the several channels of the communication system. Therefore, the orthogonal component of the desired signal to the plane of the aligned undesired signals would be less pronounced, leading to worse overall system performance

To overcome this problem, an artificial channel diversity technique is applied utilizing an antenna selection algorithm. The goal of this algorithm is to maximize the *orthogonality* of the desired signal on the plane of the aligned undesired ones. Here each transmitting node has *Q*i antennas and only the best *M*i antennas will be selected for maximizing the orthogonality of the desired component on the undesired signals plane. A brut force iterative process is carried out for all available combinations to choose the best selection that realizes this maximum orthogonality [8]. In the next section, the antenna selection criteria are illustrated in more

As mentioned before we propose an antenna selection algorithm in order to increase the required orthogonality directly leading to a more robust communication system in terms of minimum Bit Error Rate (BER). Note that antenna selection is a widely known approach in order to capture diversity and to improve the SNR of the communication systems [19],[14].

In the following we consider a *K*-user MIMO system with perfect IA. The selection algorithm consists of choosing the best *M* out of the *L* available transmit antennas. Denote by *Sk* the selected subset indices of the transmit antennas of *k* users [6]. The goal of the selection is to find *Sk* for all users {*k* = 1 : *K*} which maximizes the average SNR*<sup>S</sup>* for the multi-user system by increasing the projected desired signal into the interference-free space. This can be achieved by minimizing the principal angles between the desired signal subspace and

**3. Low complexity signal processing antenna selection algorithm**

**U***<sup>i</sup>* = null([**H***ij***V***j*]) ∀*j* �= *i* (9)

Interference Alignment for UWB-MIMO Communication Systems 135

where **y***<sup>n</sup> <sup>i</sup>* is the *Ni* <sup>×</sup> 1 received vector at receiver *<sup>i</sup>* , **<sup>H</sup>***<sup>n</sup> ij* is the *Ni* × *Mj* flat frequency domain channel matrix over *n*th subcarrier between *j*th transmitter and *i*th receiver, **V***<sup>n</sup> <sup>j</sup>* is the *Mj* × *ds* unitary precoding matrix which is applied for the transmitted *Mj* <sup>×</sup> 1 vector **<sup>x</sup>***<sup>n</sup> <sup>j</sup>* from the *j*th transmitter, and **z***<sup>n</sup> <sup>i</sup>* is the *Ni* × 1 zero mean unit variance circularly symmetric additive white Gaussian noise vector at receiver *i*. The Channel State Information (CSI) is assumed to be perfectly known at each node. To reconstruct the transmitted *ds* signal at the *i*th receiver, the received signal is decoded using a unitary linear suppression interference matrix **U***<sup>n</sup> <sup>i</sup>* . The reconstructed data **<sup>y</sup>** at receiver *<sup>i</sup>* is defined as:

$$\hat{\mathbf{y}}\_{i}^{n} = \mathbf{U}\_{i}^{nH} \mathbf{H}\_{ii}^{n} \mathbf{V}\_{i}^{n} \mathbf{x}\_{i}^{n} + \sum\_{j=1, j \neq i}^{K} \mathbf{U}\_{i}^{nH} \mathbf{H}\_{ij}^{n} \mathbf{V}\_{j}^{n} \mathbf{x}\_{j}^{n} + \mathbf{U}\_{i}^{nH} \mathbf{z}\_{i}^{n} \tag{2}$$

For perfect interference alignment, the following conditions need to be fulfilled [1]:

$$\text{rank}\left(\mathbf{U}\_{i}^{n^{H}}\mathbf{H}\_{il}^{n}\mathbf{V}\_{i}^{n}\right) = d\_{\text{s}} \qquad \forall i \tag{3}$$

and

$$\mathbf{U}\_{i}^{nH}\mathbf{H}\_{i\mathbf{j}}^{n}\mathbf{V}\_{\mathbf{j}}^{n} = \mathbf{0} \qquad \forall \mathbf{j} \neq \mathbf{i} \tag{4}$$

According to (3) and (4), the received signal after processed by the linear suppression interference matrix is:

$$\widehat{\mathbf{y}}\_{i}^{n} = \mathbf{U}\_{i}^{nH} \mathbf{H}\_{ii}^{n} \mathbf{V}\_{i}^{n} \mathbf{x}\_{i}^{n} + \mathbf{U}\_{i}^{nH} \mathbf{z}\_{i}^{n} \tag{5}$$

### **2.1. Closed-form interference alignment**

In order to achieve a closed-form IA solution, 3 users interference channel (*K*=3) has been considered, where each node has *M* = 2*d* antennas, and each user wishes to achieve *d* degrees of freedom by applying the IA principles. The conditions of IA given in (3) and (4) are obtained by setting the precoding matrices as [4]:

$$\mathbf{V}\_1 = \text{sign}(\mathbf{H}\_{31}^{-1}\mathbf{H}\_{32}\mathbf{H}\_{12}^{-1}\mathbf{H}\_{13}\mathbf{H}\_{23}^{-1}\mathbf{H}\_{21})\tag{6}$$

$$\mathbf{V\_2} = (\mathbf{H\_{32}^{-1}H\_{31}V\_1})\tag{7}$$

$$\mathbf{V\_3} = (\mathbf{H\_{23}^{-1}H\_{21}V\_1})\tag{8}$$

and the interference suppression matrix for receiver *i* is given by:

$$\mathbf{U}\_{i} = \text{null}([\mathbf{H}\_{ij}\mathbf{V}\_{j}]) \qquad \forall j \neq i \tag{9}$$

**Figure 1.** Transmitter block diagram of a MIMO UWB MB-OFDM communication system.
