**4. Spatial sampling with sampling antennas**

6 Will-be-set-by-IN-TECH

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

**Figure 2.** Comparison of average BER vs. Eb/No for a 3-user system by applying antenna selection (best

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

and worst scenarios) and using omni-directional HWD antennas.

**3.3. Simulation results analysis**

to improve the BER performance of the system.

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 previously mentioned parameters.

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

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 over frequency appear similar.

### **4.1. Channel diagonalization**

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 domain as **y**out = **Hx**in + **n**. Using singular value decomposition (SVD) **H** can be decomposed into **H** = **USV**†, where **S** = **U**†**HV** is a diagonal matrix whose elements are non-negative square roots of the eigenvalues *λ<sup>i</sup>* of the matrix **HH**†. **U** and **V** are unitary matrices, which fulfill the condition (**X**−**1**)† = **X**. Multiply the input vector **x**in and the output vector **y**out with the matrices **U**† and **V** respectively, and the original channel becomes and equivalent channel,

$$\boldsymbol{\mathfrak{y}\_{\text{out}}} = \mathbf{U}^{\dagger} \mathbf{y}\_{\text{out}} = \mathbf{U}^{\dagger} (\mathbf{H} \mathbf{x}\_{\text{in}} + \mathbf{n}) = \mathbf{U}^{\dagger} (\mathbf{H} \mathbf{V} \mathbf{\hat{x}}\_{\text{in}} + \mathbf{n}) = \mathbf{S} \mathbf{\hat{x}}\_{\text{in}} + \mathbf{\hat{n}} \tag{25}$$

where **ˆxin**, **ˆyout** and **ˆn** are the equivalent input, output and noise vectors respectively. The diagonal matrix **S** now becomes the channel matrix of the equivalent channel where each Eigenmode is interpreted as an independent SISO (single-input-single-output) subchannel and the capacity of the system becomes a sum over these SISO capacities as expressed by [12]

$$\mathcal{C} = \sum\_{i=1}^{K} \log\_2 \left( 1 + \frac{p\_i \lambda\_i}{\sigma\_{\text{noise}}^2} \right) \tag{26}$$

**Figure 4.** Indoor scenario for ray-tracing

**5.1. SISO to MIMO extrapolation**

phase difference expressed as

MIMO **H** matrix to only one SISO run.

**5.2. Antenna system simplification**

*E*(*d*, *ϑ*, *ψ*) =

**6. Synthesis results**

this, the system will now comprise only 2 inputs and 2 outputs.

plot of the radiation patterns for two subchannels and is computed using:

*E*single(*d*, *ϑ*, *ψ*) ·

A complete characterization of the MIMO channel matrix with ray-tracing requires *N* × *M* runs taking into account the sampling antenna configuration, which is computationally inefficient. [11] presents a method to reduce the calculation effort by assumming that the same plane wave impinges on all sampling antenna elements. Since the sampling antenna elements spacing is small and fixed, the difference of the incident wave at the origin of the sampling antenna configuration shown in Fig. 3 with the other antenna elements is only the

where *ϑ* and *ψ* are the angles of arrival or departure of the incident wave in elevation and azimuth respectively. Hence the SISO to MIMO extrapolation reduces the computation of the

In order to simplify the system, a plot of the eigenvalues (obtained after the SVD) versus the frequency is used to identify the channels with the strongest power. For instance, if only the first two subchannels were identified as having significant power as compared to the rest, the beamforming matrices **U** and **V** can be modified to contain only those two subchannels. With

The resulting synthesized antenna radiation patterns for both the transmitter and receiver at one point for several different frequencies are as shown in Fig. 5. The figure shows the 3D

> *e*−*jβ<sup>d</sup> d* ·

*N*ant ∑ *i*=1 *ai e*

<sup>−</sup>*j*(*β*(*di*−*d*)+*ζi*) (28)

Δ*ϕ<sup>i</sup>* = −*β* (Δ*xi* sin *ϑ* cos *ψ* + Δ*yi* sin *ϑ* sin *ψ* + Δ*zi* cos *ϑ*) (27)

Interference Alignment for UWB-MIMO Communication Systems 141

with *K* =min(*M*, *N*), which is the rank of the matrix **HRxxH**† with **Rxx** being the covariance matrix of the transmit signal, its Eigenvalues *λi*(*i* = 1, 2, ..., *K*) and power coefficients *pi*(*i* = 1, 2, ..., *K*). From (26) the capacity of a MIMO system can be seen as a sum of independent *K* SISO subchannels (Eigenmodes) represented by the Eigenvalues *λi*, where each Eigenmode corresponds to one orthogonal subchannel. More explanation can be found in [16, 24].

Employing the waterfilling algorithm in the case of channel state information (CSI) known to the transmitter will result in an optimum capacity solution for such a MIMO channel. The **U** and **V** matrices are called the 'beamforming matrices' as they determine the mapping and weighting of all the signals onto the antenna elements.
