**3.1. Antenna selection criterion**

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

#### 4 Will-be-set-by-IN-TECH 136 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Interference Alignment for UWB-MIMO Communication Systems <sup>5</sup>

the interference-free subspace. Minimizing the principal angles is equivalent to maximizing so-called *canonical correlations*.

Let *ζ*<sup>1</sup> and *ζ*<sup>2</sup> be subspaces in the complex plane **C**. Considering the dimension of *ζ*<sup>1</sup> is smaller than or equal to the dimension of *ζ*<sup>2</sup> (dim *ζ*<sup>1</sup> = *d*<sup>1</sup> � dim *ζ*<sup>2</sup> = *d*2). The canonical correlations are defined as the cosines of the principal angles between any two linear subspaces, which can uniquely defined as [3]

$$\cos \theta\_{\bar{l}} = \max\_{\mathbf{a}\_{\bar{l}} \in \mathbb{Z}\_{\bar{l}}} \max\_{\mathbf{b}\_{\bar{l}} \in \mathbb{Z}\_{\bar{z}}} \mathbf{a}\_{\bar{l}}^H \mathbf{b}\_{\bar{i}\_{\bar{l}}} \qquad \bar{i} = 1, \dots, d\_{1'} \tag{10}$$

where **a***<sup>i</sup>* and **b***<sup>i</sup>* are principal vectors of *ζ*<sup>1</sup> and *ζ*<sup>2</sup> respectively, subject to **a***<sup>H</sup> <sup>i</sup>* · **<sup>a</sup>***<sup>i</sup>* <sup>=</sup> **<sup>b</sup>***<sup>H</sup> <sup>i</sup>* · **b***<sup>i</sup>* = 1 and **a***<sup>H</sup> <sup>i</sup>* · **<sup>a</sup>***<sup>j</sup>* <sup>=</sup> **<sup>b</sup>***<sup>H</sup> <sup>i</sup>* · **b***<sup>j</sup>* = 0, *i* �= *j*.

If **Q**<sup>1</sup> and **Q**<sup>2</sup> are orthonormal bases of the two subspaces *ζ*<sup>1</sup> and *ζ*<sup>2</sup> respectively, the canonical correlations are obtained as singular values of **Q***<sup>H</sup>* <sup>1</sup> **<sup>Q</sup>**<sup>2</sup> <sup>∈</sup> **<sup>C</sup>***d*1×*d*<sup>2</sup> as follows [3]

$$\mathbf{Q}\_1^H \mathbf{Q}\_2 = \mathbf{P}\_1 \boldsymbol{\Lambda} \mathbf{P}\_2^H \quad , \tag{11}$$

transfer function is given by **H***<sup>k</sup>* = **U***<sup>H</sup>*

and by applying a thin QR decomposition

*C* =

Since **U***<sup>k</sup>* is a unitary matrix, meaning

*K* ∑ *k*=1 log � � � � 1 *σ*2 � **Q***Uk* **R***Uk* �*<sup>H</sup>* � **Q***Vk* **R***Vk*

*C* =

*C* =

Thereafter, (22) can be re-formulated as

*K* ∑ *k*=1

*K* ∑ *k*=1

*C* =

not result in the maximum *C* because the term

*K* ∑ *k*=1 log ⎛ ⎝ � 1 *σ*2

participating in the linear combinations yielding the columns of **H***kk*.

log �� <sup>1</sup>

log �� <sup>1</sup>

*C* =

Note that at high SNR, (17) can be approximated as

*K* ∑ *k*=1 log � � � � **I***<sup>N</sup>* + 1 *<sup>σ</sup>*<sup>2</sup> **<sup>U</sup>***<sup>H</sup>*

*C* �

*K* ∑ *k*=1 log � � � � 1 *<sup>σ</sup>*<sup>2</sup> **<sup>U</sup>***<sup>H</sup>*

(16) reduces to

where

and **Q***Uk*

follows

can be written as

such that

diag(*α*1, .., *αds* ).

*<sup>k</sup>* **H***kk***V***k*, for *k* = 1, .., *K*. Then the sumrate equation in

Interference Alignment for UWB-MIMO Communication Systems 137

� . (17)

� , (18)

� . (21)

. (22)

(23)

**R***Uk* , (19)

**R***Vk* (20)

*<sup>k</sup>* **<sup>H</sup>***<sup>H</sup> kk***U***<sup>k</sup>* � � �

*<sup>k</sup>* **<sup>H</sup>***<sup>H</sup> kk***U***<sup>k</sup>* � � �

*<sup>k</sup>* **<sup>H</sup>***kk***V***k***V***<sup>H</sup>*

*<sup>k</sup>* **<sup>H</sup>***kk***V***k***V***<sup>H</sup>*

, **Q***Vk* are orthonormal *N* × *ds* matrix and **R***Uk* , **R***Vk* are *ds* × *ds* upper triangle matrix it

� �**Q***Vk*

**R***Vk* �*<sup>H</sup>* � **Q***Uk* **R***Uk* � � � �

� <sup>=</sup> 1 it furthermore follows

**U***<sup>k</sup>* = **Q***Uk*

**H***kk***V***<sup>k</sup>* = **Q***Vk*

� � � **<sup>R</sup>***Uk***R***<sup>H</sup> Uk* � �

�<sup>2</sup> � � �**Q***H Uk* **Q***Vk* � � � � � �**Q***H Vk* **Q***Uk* � � � � � � **<sup>R</sup>***Vk***R***<sup>H</sup> Vk* � � � �

�<sup>2</sup> � � � **P***k*1Λ**P***<sup>H</sup> k*2 � � � � � � **P***k*2Λ**P***<sup>H</sup> k*1 � � � � � � **<sup>R</sup>***Vk***R***<sup>H</sup> Vk* � � � �

Since **Q***Uk* and **Q***Vk* are the orthonormal basis of the two subspaces **U***<sup>k</sup>* and **H***kk***V***<sup>k</sup>* respectively, (22) can be linked to the principal angles between the two subspace using (11). Therefore, (22)

**<sup>Q</sup>***Vk* = **<sup>P</sup>***k*1Λ**P***<sup>H</sup>*

where **P***k*<sup>1</sup> and **P***k*<sup>2</sup> are *ds* × *ds* unitary matrices and Λ is *ds* × *ds* diagonal matrix equals

�<sup>2</sup> � *ds* ∏ *i*=1 *αi* �<sup>2</sup> � � � **<sup>R</sup>***Vk***R***<sup>H</sup> Vk* � � �

where (*α*1, ...., *αds* ) are the canonical correlations between subspace **U***<sup>k</sup>* and subspace **H***kk***V***k*. From (23) it is shown that maximizing the canonical correlations increases *C*, but still does

� � � **<sup>R</sup>***Vk***R***<sup>H</sup> Vk* � � �

*k*2,

⎞

⎠ , (24)

is linked to the matrix of coefficients

*σ*2

*σ*2

**Q***<sup>H</sup> Uk*

where **P**<sup>1</sup> is a *d*<sup>1</sup> × *d*<sup>1</sup> unitary matrix and **P**<sup>2</sup> is a *d*<sup>2</sup> × *d*<sup>2</sup> unitary matrix, Λ is is *d*<sup>1</sup> × *d*<sup>2</sup> diagonal matrix with nonnegative real numbers on the diagonal. Therefore, Λ = diag(*α*1, .., *αd*<sup>1</sup> ) and *α*1, .., *αd*<sup>1</sup> are the canonical correlations of the subspaces.

Observe that the principal angles are given by

$$\theta\_{\bar{i}} = \cos^{-1}(\mathfrak{a}\_{\bar{i}}), \qquad \bar{i} = 1, \ldots, d\_1. \tag{12}$$

In order to maximize the SNR at the receiver, the antenna selection criterion relies on maximizing the canonical correlations between **U***<sup>k</sup>* and **H***kk***V***<sup>k</sup>* as follows

$$\mathbf{S}\_{k} = \arg\min \angle \,\mathrm{(U\_{k}; \mathbf{H}\_{kk} \mathbf{V}\_{k})} \qquad ; k = 1 \,, \dots \,\mathrm{K} \tag{13}$$

$$\mathbf{S}\_{k} = \arg\max \cos \left( \angle \left( \mathbf{U}\_{k}; \mathbf{H}\_{kk} \mathbf{V}\_{k} \right) \right) \quad ; k = 1, \dots, K \tag{14}$$

$$S\_k = \arg\max \ (\mathfrak{a}\_1, \dots, \mathfrak{a}\_{d\_s}) \qquad ; k = 1, 2, \dots, K,\tag{15}$$

where (*α*1, ...., *αds* ) are the canonical correlations between subspace **U***<sup>k</sup>* and subspace **H***kk***V***k*.

### **3.2. Relation between sum-rate and canonical correlations**

The impact of the canonical correlations on the sum rate of a *K*-user MIMO system is given by [6]:

$$\mathbf{C} = \sum\_{k=1}^{K} \log \left| \mathbf{I}\_{N} + \left( \sigma^{2} \mathbf{I}\_{N} + \sum\_{l \neq k} \mathbf{W}\_{kl} \right)^{-1} \mathbf{W}\_{kk} \right| \tag{16}$$

where *C* is the sum-rate, **W***kl* = **H***kl***V***l***V***<sup>H</sup> <sup>l</sup>* **<sup>H</sup>***<sup>H</sup> kl* denotes the *N* × *N* interference covariance matrix of the signal from the *l*-th transmitter to the *k*-th receiver, *σ*<sup>2</sup> is the variance of the additive white Gaussian noise, and **W***kk* = **H***kk***V***k***V***<sup>H</sup> <sup>k</sup>* **<sup>H</sup>***<sup>H</sup> kk* denotes the *N* × *N* covariance matrix of the desired signal. While perfect IA is assumed according to (3) and (4), the interference channel is equivalent to a set of parallel Gaussian MIMO channels, where the MIMO channel transfer function is given by **H***<sup>k</sup>* = **U***<sup>H</sup> <sup>k</sup>* **H***kk***V***k*, for *k* = 1, .., *K*. Then the sumrate equation in (16) reduces to

$$\mathbf{C} = \sum\_{k=1}^{K} \log \left| \mathbf{I}\_{N} + \frac{1}{\sigma^2} \mathbf{U}\_{k}^{H} \mathbf{H}\_{kk} \mathbf{V}\_{k} \mathbf{V}\_{k}^{H} \mathbf{H}\_{kk}^{H} \mathbf{U}\_{k} \right|. \tag{17}$$

Note that at high SNR, (17) can be approximated as

$$\mathbf{C} \simeq \sum\_{k=1}^{K} \log \left| \frac{1}{\sigma^2} \mathbf{U}\_k^H \mathbf{H}\_{kk} \mathbf{V}\_k \mathbf{V}\_k^H \mathbf{H}\_{kk}^H \mathbf{U}\_k \right| \tag{18}$$

and by applying a thin QR decomposition

$$\mathbf{U}\_k = \mathbf{Q}\_{U\_k} \mathbf{R}\_{U\_{k'}} \tag{19}$$

where

4 Will-be-set-by-IN-TECH

the interference-free subspace. Minimizing the principal angles is equivalent to maximizing

Let *ζ*<sup>1</sup> and *ζ*<sup>2</sup> be subspaces in the complex plane **C**. Considering the dimension of *ζ*<sup>1</sup> is smaller than or equal to the dimension of *ζ*<sup>2</sup> (dim *ζ*<sup>1</sup> = *d*<sup>1</sup> � dim *ζ*<sup>2</sup> = *d*2). The canonical correlations are defined as the cosines of the principal angles between any two linear subspaces, which can

If **Q**<sup>1</sup> and **Q**<sup>2</sup> are orthonormal bases of the two subspaces *ζ*<sup>1</sup> and *ζ*<sup>2</sup> respectively, the canonical

<sup>1</sup> **<sup>Q</sup>**<sup>2</sup> <sup>=</sup> **<sup>P</sup>**1Λ**P***<sup>H</sup>*

where **P**<sup>1</sup> is a *d*<sup>1</sup> × *d*<sup>1</sup> unitary matrix and **P**<sup>2</sup> is a *d*<sup>2</sup> × *d*<sup>2</sup> unitary matrix, Λ is is *d*<sup>1</sup> × *d*<sup>2</sup> diagonal matrix with nonnegative real numbers on the diagonal. Therefore, Λ = diag(*α*1, .., *αd*<sup>1</sup> ) and

In order to maximize the SNR at the receiver, the antenna selection criterion relies on

*α*1, ...., *αds*

where (*α*1, ...., *αds* ) are the canonical correlations between subspace **U***<sup>k</sup>* and subspace **H***kk***V***k*.

The impact of the canonical correlations on the sum rate of a *K*-user MIMO system is given by

*<sup>l</sup>* **<sup>H</sup>***<sup>H</sup>*

matrix of the signal from the *l*-th transmitter to the *k*-th receiver, *σ*<sup>2</sup> is the variance of the

of the desired signal. While perfect IA is assumed according to (3) and (4), the interference channel is equivalent to a set of parallel Gaussian MIMO channels, where the MIMO channel

*<sup>σ</sup>*2**I***<sup>N</sup>* <sup>+</sup> ∑

*l*�=*k* **W***kl*

*<sup>k</sup>* **<sup>H</sup>***<sup>H</sup>*

*<sup>i</sup>* **b***i*, *i* = 1, ..., *d*1, (10)

<sup>2</sup> , (11)

; *k* = 1, 2, .., *K*, (15)

, (16)

<sup>1</sup> **<sup>Q</sup>**<sup>2</sup> <sup>∈</sup> **<sup>C</sup>***d*1×*d*<sup>2</sup> as follows [3]

*θ<sup>i</sup>* = cos−<sup>1</sup> (*αi*), *i* = 1, .., *d*1. (12)

*Sk* = arg min ∠ (**U***k*; **H***kk***V***k*) ; *k* = 1, .., *K* (13)

−<sup>1</sup>

**W***kk* 

*kl* denotes the *N* × *N* interference covariance

*kk* denotes the *N* × *N* covariance matrix

*Sk* = arg max cos (∠ (**U***k*; **H***kk***V***k*)) ; *k* = 1, .., *K* (14)

*<sup>i</sup>* · **<sup>a</sup>***<sup>i</sup>* <sup>=</sup> **<sup>b</sup>***<sup>H</sup>*

*<sup>i</sup>* · **b***<sup>i</sup>* =

max **b***i*∈*ζ*<sup>2</sup> **a***H*

where **a***<sup>i</sup>* and **b***<sup>i</sup>* are principal vectors of *ζ*<sup>1</sup> and *ζ*<sup>2</sup> respectively, subject to **a***<sup>H</sup>*

**Q***<sup>H</sup>*

maximizing the canonical correlations between **U***<sup>k</sup>* and **H***kk***V***<sup>k</sup>* as follows

*Sk* = arg max

**3.2. Relation between sum-rate and canonical correlations**

*C* =

additive white Gaussian noise, and **W***kk* = **H***kk***V***k***V***<sup>H</sup>*

where *C* is the sum-rate, **W***kl* = **H***kl***V***l***V***<sup>H</sup>*

*K* ∑ *k*=1 log **I***<sup>N</sup>* + 

cos *θ<sup>i</sup>* = max **a***i*∈*ζ*<sup>1</sup>

*<sup>i</sup>* · **b***<sup>j</sup>* = 0, *i* �= *j*.

correlations are obtained as singular values of **Q***<sup>H</sup>*

*α*1, .., *αd*<sup>1</sup> are the canonical correlations of the subspaces.

Observe that the principal angles are given by

so-called *canonical correlations*.

uniquely defined as [3]

*<sup>i</sup>* · **<sup>a</sup>***<sup>j</sup>* <sup>=</sup> **<sup>b</sup>***<sup>H</sup>*

1 and **a***<sup>H</sup>*

[6]:

$$\mathbf{H}\_{kk}\mathbf{V}\_k = \mathbf{Q}\_{V\_k}\mathbf{R}\_{V\_k} \tag{20}$$

and **Q***Uk* , **Q***Vk* are orthonormal *N* × *ds* matrix and **R***Uk* , **R***Vk* are *ds* × *ds* upper triangle matrix it follows

$$\mathbf{C} = \sum\_{k=1}^{K} \log \left| \frac{1}{\sigma^2} \left( \mathbf{Q}\_{\mathrm{UL}} \mathbf{R}\_{\mathrm{UL}\_k} \right)^H \left( \mathbf{Q}\_{V\_k} \mathbf{R}\_{V\_k} \right) \left( \mathbf{Q}\_{V\_k} \mathbf{R}\_{V\_k} \right)^H \left( \mathbf{Q}\_{\mathrm{UL}} \mathbf{R}\_{\mathrm{UL}\_k} \right) \right|. \tag{21}$$

Since **U***<sup>k</sup>* is a unitary matrix, meaning � � � **<sup>R</sup>***Uk***R***<sup>H</sup> Uk* � � � <sup>=</sup> 1 it furthermore follows

$$\mathbf{C} = \sum\_{k=1}^{K} \log \left( \left( \frac{1}{\sigma^2} \right)^2 \left| \mathbf{Q}\_{L\_k}^H \mathbf{Q}\_{V\_k} \right| \left| \mathbf{Q}\_{V\_k}^H \mathbf{Q}\_{Ll\_k} \right| \left| \mathbf{R}\_{V\_k} \mathbf{R}\_{V\_k}^H \right| \right). \tag{22}$$

Since **Q***Uk* and **Q***Vk* are the orthonormal basis of the two subspaces **U***<sup>k</sup>* and **H***kk***V***<sup>k</sup>* respectively, (22) can be linked to the principal angles between the two subspace using (11). Therefore, (22) can be written as

$$\mathbf{C} = \sum\_{k=1}^{K} \log \left( \left( \frac{1}{\sigma^2} \right)^2 \left| \mathbf{P}\_{k1} \boldsymbol{\Lambda} \mathbf{P}\_{k2}^H \right| \left| \mathbf{P}\_{k2} \boldsymbol{\Lambda} \mathbf{P}\_{k1}^H \right| \left| \mathbf{R}\_{V\_k} \mathbf{R}\_{V\_k}^H \right| \right) \tag{23}$$

such that

$$\mathbf{Q}\_{L\_k}^H \mathbf{Q}\_{V\_k} = \mathbf{P}\_{k1} \Lambda \mathbf{P}\_{k2'}^H$$

where **P***k*<sup>1</sup> and **P***k*<sup>2</sup> are *ds* × *ds* unitary matrices and Λ is *ds* × *ds* diagonal matrix equals diag(*α*1, .., *αds* ).

Thereafter, (22) can be re-formulated as

$$\mathbf{C} = \sum\_{k=1}^{K} \log \left( \left( \frac{1}{\sigma^2} \right)^2 \left( \prod\_{i=1}^{d\_s} \alpha\_i \right)^2 \left| \mathbf{R}\_{V\_k} \mathbf{R}\_{V\_k}^H \right| \right), \tag{24}$$

where (*α*1, ...., *αds* ) are the canonical correlations between subspace **U***<sup>k</sup>* and subspace **H***kk***V***k*. From (23) it is shown that maximizing the canonical correlations increases *C*, but still does not result in the maximum *C* because the term � � � **<sup>R</sup>***Vk***R***<sup>H</sup> Vk* � � � is linked to the matrix of coefficients participating in the linear combinations yielding the columns of **H***kk*.

#### 6 Will-be-set-by-IN-TECH 138 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Interference Alignment for UWB-MIMO Communication Systems <sup>7</sup>

### **3.3. Simulation results analysis**

All the following simulation results have been obtained based on real-world deterministic scenarios. The deterministic hybrid EM ray-tracing channel model was considered for the MIMO UWB channel [9]. This model considers the spatial channel and the environmental effects such as path-loss, frequency dependence, reflections, transmissions, and also diffractions. It considers as well the characteristics of the antennas as part of the effective channel such as directional gain, matching and polarization. A fair comparison has been carried out among three systems: the first uses two omnidirectional Half-Wave Dipole (HWD) antennas at each node, the second uses three directional antennas (horn antenna) at each transmitter node and two directional antennas (horn antenna) at each receiver node, in this system antenna selection (AS) technique is applied to select two antennas from the three at each transmitter. The third system uses two directional antennas at each node without AS (we choose the worst case in this manner). Fig. 2 shows a comparison between the three systems using the *average* BER vs. *Eb*/*No* for the whole multiuser system. As shown in Fig. 2 the artificial diversity technique improves the performance of the system significantly compared to the system that uses a HWD omnidirectional antennas and the other which uses the directional antennas without using AS. This figure proves that AS is a powerful technique to improve the BER performance of the system.

systematic framework to fulfill the above 2 conditions, that aids in reducing the number of real world antennas to be used. The goal is to synthesize an optimal UWB antenna system which can be used by both the transmitter and receiver at any location within the investigated scenario. This synthesis method is based on [24] and has been used to design antennas for narrowband systems in [25]. The method is now extended to the design of UWB antenna systems. In the following sections, the theory and concepts behind this systematic synthesis

Interference Alignment for UWB-MIMO Communication Systems 139

The concept of 'spatial sampling' is presented in greater detail in [24, 30], but can be simplified to this: 'Given a predefined overall antenna aperture confined to a limited volume *V*, there exists a maximum spatial capacity limit with transmission system parameters i.e. antenna aperture size and element spacing. This can be determined by sampling the transmit and receive volumes with a set of ideal sampling antennas'. In a realistic case sampling antennas possess an overall aperture size and occupy a certain spatial volume. Therefore three parameters have to be considered for their design. First, the antenna aperture size, which will approximate the size of the real world antennas. The larger the aperture size, the higher the capacity, but this size is limited by the physical size of the desired real world antennas. Second, the minimum distance between the antenna elements in order to decrease their correlation [[18, 22]]. This parameter also influences the number of sampling antenna elements within the selected aperture size. It should be noted that the more elements the aperture contains, the more time is needed for the synthesis algorithm. Third, the frequency dependency of the

**Figure 3.** Sampling antenna configurations for spatial sampling, (left) minimum elements of 2×2×2,

Here, a 5×5×5 cm<sup>3</sup> aperture based on the configuration in Fig. 3 has been chosen to illustrate the sampling antenna design and antenna synthesis algorithm in section 4.1. This configuration is then used in the subsequent antenna synthesis steps and the changes to the resulting radiation pattern over frequency is noted. If the resulting radiation pattern varies too much over frequency, the number of sampling antenna elements is then increased and the synthesis algorithm is repeated. This procedure is iterated until the resulting radiation pattern

For a time-invariant system, the transfer function is a transfer coefficient for all transmit and receive antenna pairs and can be expressed in the form of an *N* × *M*-dimentional matrix **H**. Letting **y**out be an *N* × 1 matrix and **x**in be an *M* × 1 matrix with **n** as the noise vector, the received signal vector of a communication system can then be described in the frequency

will be given, followed by the methodology of realization and the results.

**4. Spatial sampling with sampling antennas**

previously mentioned parameters.

(right) more elements added, 3×3×3.

over frequency appear similar.

**4.1. Channel diagonalization**

**Figure 2.** Comparison of average BER vs. Eb/No for a 3-user system by applying antenna selection (best and worst scenarios) and using omni-directional HWD antennas.

MIMO systems typically use antenna arrays and beamforming and spatial multiplexing. These beamforming methods do not result in (1) orthogonal channels nor (2) guarantee optimum data rates within the predefined environment e.g. indoors, outdoors, urban, suburban, etc. In designing optimal antenna systems, which fulfill the previously stated conditions, up to now, only heuristic methods have been employed. Presented here is a systematic framework to fulfill the above 2 conditions, that aids in reducing the number of real world antennas to be used. The goal is to synthesize an optimal UWB antenna system which can be used by both the transmitter and receiver at any location within the investigated scenario. This synthesis method is based on [24] and has been used to design antennas for narrowband systems in [25]. The method is now extended to the design of UWB antenna systems. In the following sections, the theory and concepts behind this systematic synthesis will be given, followed by the methodology of realization and the results.
