**5.1. SISO to MIMO extrapolation**

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 phase difference expressed as

$$
\Delta \varphi\_i = -\beta \left( \Delta x\_i \sin \theta \cos \psi + \Delta y\_i \sin \theta \sin \psi + \Delta z\_i \cos \theta \right) \tag{27}
$$

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

### **5.2. Antenna system simplification**

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

### **6. Synthesis results**

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

$$\vec{E}(d,\theta,\psi) = \vec{E}\_{\text{single}}(d,\theta,\psi) \cdot \frac{e^{-j\beta d}}{d} \cdot \sum\_{i=1}^{N\_{\text{ant}}} a\_i \, e^{-j(\beta(d\_i - d) + \zeta\_i)} \tag{28}$$

#### 10 Will-be-set-by-IN-TECH 142 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Interference Alignment for UWB-MIMO Communication Systems <sup>11</sup>

where *E*single is the electric field of the sampling antenna used (a dipole in this case), *N*ant is the total number of transmitter or receiver antennas (since they both use the same sampling antenna configuration), *β* is the wave number, *d* is the distance from the origin of the sampling antenna to a far-field observation point, *ai*∠*ζ<sup>i</sup>* is the weighting from the **U** and **V** beamforming matrices, *di* − *d* = Δ*ϕ* in (27) and Δ*xi*, Δ*yi*, Δ*zi* are the position of the individual elements in the array according to the Cartesian coordinate system.

**Figure 5.** Resulting synthesized antenna radiation pattern for a transmit-receive pair (horizontal polarization) for subchannel 1 (S1), with line-of-sight propagation, and subchannel 2 (S2), with propagation paths reflected from the ceiling and floor, for the aperture size 5×5×5 cm3 with 5×5×<sup>5</sup> elements, (left) side view and (right) bottom view. (Image taken from [29])

### *Averaging strategy*

Three averaging strategies were used, namely averaging over frequency, over location, and of transmitter and receiver radiation patterns. The first averages all radiation patterns obtained at frequency points between 3.1 to 10.6 GHz to obtain a pattern which is valid for the UWB. The second averages the radiation pattern obtained from random points around the scenario so that the resulting radiation pattern is valid for use in the whole scenario. The third averaging is done if the resulting transmitter and receiver radiation patterns look qualitatively similar, so that both can use the same antennas.

**Figure 6.** Capacity analysis for the aperture size 5×5×5 cm3. (Image taken from [29])

(a) Synthesized antennas, (left) subchannel 1 (right) subchannel 2

Interference Alignment for UWB-MIMO Communication Systems 143

(b) Real world antennas (left) subchannel 1 (right) subchannel 2

**Figure 7.** Synthesized antennas vs. real world antennas: radiation pattern (elevation) over frequency
