**8. Multi-cluster MIMO-OFDM channel model**

According to previous studies, Eqs. (50) and (54) are two key functions for building the MIMO-OFDM channel model based on a single scattering cluster. Combining these two types of channel models, a multi-cluster MIMO-OFDM channel model is constructed. In this way, a physical propagation environment of radio waves is reconstructed by simulations.

Considering a radio wave propagation environment with *K* distant Cauchy-Rayleigh and Rayleigh scattering clusters, as shown in **Figure 10**, it is assumed that the BS is fixed while the MS is moving with speed *v*, and there is noline of sight (LOS) between the BS and MS, all of the signals transmitted and received are via these *K* uncorrelated scattering clusters. Each cluster is grouped into resolvable multi-path components. Besides, within a cluster, the trigonometric relationship among the BS, scatterers, and MS has been introduced, as shown in **Figure 4**.

For this model, only a single scattering event along each path between the transmit and receive antenna arrays is considered. That is, it is assumed that the contribution to the power due to multiple scattering events is much lower and will be ignored.

In addition, the radio waves contributed from different scattering clusters can be added to obtain the contributions of all.

The power contributed from each cluster is dedicated in a portion to the Doppler power spectrum. From this point of view, under the assumption of uncorrelated scattering clusters, the summation of the radio waves can be regarded as adding up each individual portion of power. These contributions will result in a *K*-cluster MIMO-OFDM channel model if the delay factor is taken into account.

#### **Figure 10.**

*Multiple distant scattering clusters, cluster no. 1 to cluster no. K, in a radio wave propagation environment, in which each cluster is grouped into resolvable multi-path components.*

#### **8.1 Multi-cluster angular-delay Spectrum**

The joint angular-delay spectrum associated with *K* scattering clusters can be written as

$$\begin{array}{rcl}f\_{a,\mathsf{r}}(a,\mathsf{r})&=&\frac{\sum\_{k=1}^{K}P\_{\mathsf{k}}f\_{a\_{k},\mathsf{r}\_{k}}(a\_{k},\mathsf{r}\_{k})}{\sum\_{k=1}^{K}P\_{k}}\\&=&\frac{\sum\_{k=1}^{K}P\_{\mathsf{k}}f\_{a\_{k}}(a\_{k})f\_{\mathsf{r}\_{k}}(\mathsf{r}\_{k})}{\sum\_{k=1}^{K}P\_{k}}\end{array} \tag{56}$$

where *Pk* denotes the power contributed from the *kth* cluster. Taking summation over the angles, the marginal distribution represents the PDP, *f <sup>τ</sup>*ð Þ*τ* , of the clusters. The sum over the delays stands for the angular power distribution, *f <sup>α</sup>*ð Þ *α* , of the clusters.

#### **8.2 Building a multi-cluster MIMO-OFDM Channel model**

Connecting multiple MIMO-OFDM channel model blocks in parallel, a multicluster MIMO-OFDM channel model is constructed, as shown in **Figure 11**, and the number of blocks required depends on *K*.

The connection illustrated in **Figure 11** can be transformed into the following mathematical representation,

$$\begin{aligned} \mathbf{x}\_{k+1} &= \Gamma \mathbf{x}\_k + \Psi \mathbf{w}\_k\\ \mathbf{h}\_k &= \Omega \mathbf{x}\_k \end{aligned} \tag{57}$$

where **Γ**, **Ψ**, and **Ω** are given below,

#### **Figure 11.**

*Block diagram of a* K*-cluster MIMO-OFDM channel model, where the input noise vector* **w***<sup>k</sup>* ∈ℂKMfMrMt , *the output channel vector* **<sup>h</sup>** *<sup>k</sup>*, 0 : *Mf* � <sup>1</sup> � � *means that there are Mf sub-carriers from mf* <sup>¼</sup> <sup>0</sup> *to mf* <sup>¼</sup> *Mf* � <sup>1</sup>*, and the AR(p)-based MIMO-OFDM channel model block is shown in Figure 9.*

*Multi-Cluster-Based MIMO-OFDM Channel Modeling DOI: http://dx.doi.org/10.5772/intechopen.112190*

$$\begin{aligned} \boldsymbol{\Gamma} &= \begin{bmatrix} \Gamma\_1 & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \Gamma\_2 & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \Gamma\_K \end{bmatrix}, \boldsymbol{\Psi} = \begin{bmatrix} \Psi\_1 & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \Psi\_2 & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \Psi\_K \end{bmatrix} \\ \boldsymbol{\Omega} &= \begin{bmatrix} \boldsymbol{\Omega}\_1 & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \boldsymbol{\Omega}\_2 & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \mathbf{\Omega}\_K \end{bmatrix} \end{aligned} \tag{58}$$

where **Γ***i*, **Ψ***i*, and **Ω***<sup>i</sup>* are defined in Eq. (44), which represent the matrices either from the AR(1)-based MIMO-OFDM channel model or from the AR(3)-based MIMO-OFDM channel model.

### **9. Conclusions**

This chapter presents a state-space-based simulation model for MIMO-OFDM channels. Based on this model, a physical propagation environment of radio waves can be reconstructed by simulations.

In this approach, for each distant scattering cluster, the received power renders a narrow peak, which contributes a portion to the Doppler power spectrum. The entire Doppler power spectrum is obtained by summing the contributions of all these uncorrelated scattering clusters.

One of the fundamental assumptions in this chapter is the probability distribution of scattering clusters. The AOD, AOA, and TOA due to distant Cauchy-Rayleigh scattering clusters can be approximately modeled as the Cauchy angular and delay power distribution functions, while distant Rayleigh clusters result in the Gaussian angular and delay power distribution functions.

Another underlying assumption is that more than 90% of the power is within a small angular spread. The narrow distribution enables us to study the CSOS using approximations for small angles. This implies that both the upper and lower limits of the integral of the channel spatial-temporal correlation function can be extended from ½ � �*π*, *π* to ð Þ �∞, ∞ without losing its main features. Meanwhile, the assumption of independence of the AOD and AOA makes the channel correlation function integrable.

One of the main results is the decomposition of the spatial-temporal correlation function caused by a single cluster. The CSOS can be decomposed into disjoint antenna spacing and movement parts using the phase-shift method. Thus, an AR(p) model can be employed to describe the temporal dynamics of the channel.

A major result is that the radio channels can be built modularly. A state-spacebased MIMO-OFDM channel model is another major result. A distant scattering cluster contributed to each antenna at a mobile receiver is associated with an AR(1) or AR(3)-based state-space SISO channel model block. The beauty of using state-space representation is that a MIMO-OFDM channel model can be constructed using multiple SISO channel blocks. Meanwhile, a correlated innovation process is employed to

adjust the channel spatial correlation within each MIMO block and spectral correlation between the MIMO blocks. Following the same process, it is easy to extend this model to the multi-cluster case.

Therefore, the spatial-temporal-spectral correlation characteristics of the channel are achievable in the simulated channels.
