**5. Scenario-based MIMO antenna synthesis**

In order to apply the SVD technique to obtain parallel subchannels, the channel matrix **H** of the intended scenario must be provided. One of the most reliable and repeatable way of obtaining the SISO channel matrix **H** is through ray-tracing with the software developed by [10]. The ray-tracing simulations are done in parallel with the design of the sampling antenna configuration.

A typical indoor scenario (with glass windows, furniture, ceiling and floor) was built for simulation as shown in Fig. 4. The size of the room is 10 × 10 × 3 m and simulations were done for transmitters and receivers at randomized positions in the room with an antenna height 1.5 m over the frequency band of 3.1 GHz to 10.6 GHz. Omni-directional antennas (dipoles) are used, along with the option of using both vertical and horizontal polarizations for an added degree of freedom for the design of the real world antenna. Around 45600 random transmitter-receiver points were simulated in order to acquire a synthesis result, that when the averaging strategies in section 6 have been applied, will be applicable from virtually any point in the intended scenario.

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

8 Will-be-set-by-IN-TECH

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,

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

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

In order to apply the SVD technique to obtain parallel subchannels, the channel matrix **H** of the intended scenario must be provided. One of the most reliable and repeatable way of obtaining the SISO channel matrix **H** is through ray-tracing with the software developed by [10]. The ray-tracing simulations are done in parallel with the design of the sampling antenna

A typical indoor scenario (with glass windows, furniture, ceiling and floor) was built for simulation as shown in Fig. 4. The size of the room is 10 × 10 × 3 m and simulations were done for transmitters and receivers at randomized positions in the room with an antenna height 1.5 m over the frequency band of 3.1 GHz to 10.6 GHz. Omni-directional antennas (dipoles) are used, along with the option of using both vertical and horizontal polarizations for an added degree of freedom for the design of the real world antenna. Around 45600 random transmitter-receiver points were simulated in order to acquire a synthesis result, that when the averaging strategies in section 6 have been applied, will be applicable from virtually any

*C* = *K* ∑ *i*=1 log2 

weighting of all the signals onto the antenna elements.

**5. Scenario-based MIMO antenna synthesis**

[12]

configuration.

point in the intended scenario.

**ˆyout** = **U**†**yout** = **U**†(**Hxin** + **n**) = **U**†(**HVˆxin** + **n**) = **S ˆxin** + **ˆn** (25)

<sup>1</sup> <sup>+</sup> *piλ<sup>i</sup> σ*2 noise (26)
