**2.2. Artificial channel diversity algorithm**

2 Will-be-set-by-IN-TECH

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

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

> *K* ∑ *j*=1,*j*� *i*

*<sup>i</sup>* is the *Ni* × 1 zero mean unit variance circularly symmetric additive white

**H***<sup>n</sup> ij***V***<sup>n</sup> j* **x***n <sup>j</sup>* <sup>+</sup> **<sup>z</sup>***<sup>n</sup>*

*<sup>i</sup>* (1)

*<sup>j</sup>* is the *Mj* × *ds*

*<sup>i</sup>* (2)

*<sup>j</sup>* from the *j*th

*<sup>i</sup>* . The

*ij* is the *Ni* × *Mj* flat frequency domain

*<sup>i</sup>* ) = *ds* ∀*i* (3)

*<sup>i</sup>* (5)

<sup>23</sup> **H**21) (6)

<sup>32</sup> **H**31**V**1) (7)

<sup>23</sup> **H**21**V**1) (8)

*<sup>j</sup>* = 0 ∀*j* � *i* (4)

*<sup>i</sup>* <sup>=</sup> **<sup>H</sup>***<sup>n</sup> ii***V***<sup>n</sup> i* **x***n <sup>i</sup>* +

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>*

> *K* ∑ *j*=1,*j*� *i*

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

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

<sup>31</sup> **<sup>H</sup>**32**H**−<sup>1</sup>

**V**<sup>2</sup> = (**H**−<sup>1</sup>

**V**<sup>3</sup> = (**H**−<sup>1</sup>

<sup>12</sup> **<sup>H</sup>**13**H**−<sup>1</sup>

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

rank (**U***<sup>n</sup> i <sup>H</sup>***H***<sup>n</sup> ii***V***<sup>n</sup>*

> **U***n i <sup>H</sup>***H***<sup>n</sup> ij***V***<sup>n</sup>*

**y***n <sup>i</sup>* <sup>=</sup> **<sup>U</sup>***<sup>n</sup> i <sup>H</sup>***H***<sup>n</sup> ii***V***<sup>n</sup> i* **x***n <sup>i</sup>* <sup>+</sup> **<sup>U</sup>***<sup>n</sup> i <sup>H</sup>***z***<sup>n</sup>*

**V**<sup>1</sup> = eign(**H**−<sup>1</sup>

**U***n i <sup>H</sup>***H***<sup>n</sup> ij***V***<sup>n</sup> j xn <sup>j</sup>* <sup>+</sup> **<sup>U</sup>***<sup>n</sup> i <sup>H</sup>***z***<sup>n</sup>*

channel matrix over *n*th subcarrier between *j*th transmitter and *i*th receiver, **V***<sup>n</sup>*

unitary precoding matrix which is applied for the transmitted *Mj* <sup>×</sup> 1 vector **<sup>x</sup>***<sup>n</sup>*

receiver over a flat channel is represented as[21],[28]:

*<sup>i</sup>* is the *Ni* <sup>×</sup> 1 received vector at receiver *<sup>i</sup>* , **<sup>H</sup>***<sup>n</sup>*

**y***n <sup>i</sup>* = *K* ∑ *j*=1 **H***<sup>n</sup> ij***V***<sup>n</sup> j* **x***n <sup>j</sup>* <sup>+</sup> **<sup>z</sup>***<sup>n</sup>*

reconstructed data **<sup>y</sup>** at receiver *<sup>i</sup>* is defined as:

**y***n <sup>i</sup>* <sup>=</sup> **<sup>U</sup>***<sup>n</sup> i <sup>H</sup>***H***<sup>n</sup> ii***V***<sup>n</sup> i* **x***n <sup>i</sup>* +

**2.1. Closed-form interference alignment**

by setting the precoding matrices as [4]:

[5].

where **y***<sup>n</sup>*

and

transmitter, and **z***<sup>n</sup>*

interference matrix is:

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 and robustness.

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 detail.
