**1. Introduction**

IN radio communications, from the traditional voice telephony to the current communication multimedia, to the future augmented reality (AR), virtual reality (VR), mixed reality (MR), and Internet of everything (IoE), the historical process shows that the increasing demand for higher data rates is the fundamental factor driving the development of communication technologies and methods.

One of the biggest challenges in radio communications is how to model radio channels. A radio channel refers to the influence of the propagation medium of electromagnetic (EM) waves on the signal from a transmitter to a receiver.

**Figure 1** depicts a typical terrestrial radio propagation environment. The basic characteristic of radio channels is fading, large-scale, and small-scale fading<sup>1</sup> , which can be classified as distance loss, shadowing, and multi-path fading. Among them, multi-path fading is known as small-scale fading, and its characteristics vary with time, and change over frequency and space. That is, due to the multi-path propagation of EM waves, power-limited transmitted signals will be distorted at a receiver over time, frequency, and space simultaneously.

To better understand the mechanism behind the distortion, this physical phenomenon needs to be studied. Geometry-based stochastic multiple-input and multipleoutput (MIMO) radio channel modeling was a hot topic [1–4]. The idea of this approach is to map the spatial location of scatterers in a cluster to an angular distribution of power through the trigonometric relationship among the scatterers, cluster center, and receiver or transmitter. Furthermore, the angular distribution of power has certain statistical properties if the cluster obeys a specific probability distribution [1, 2]. Hence, from an intuitive point of view, this approach is simple and straightforward.

A distant scattering cluster results in small variation in the angle-of-departure/ angle of arrival (AOD/AOA) and produces a narrowband Doppler spectrum both at the base station (BS) and at the mobile station (MS) [5]. This can be used to explore the computation of the channel second-order statistics (CSOS) with a small angular approach, and this approach is suitable for the decomposition of the Doppler power spectrum into small uncorrelated portions.

Radio wave measurements indicate that the power azimuth spectrum typically has sharp, narrow peaks over a small range of angles [3, 4, 6, 7]. Each measurement has been modeled as a Laplace angular distribution [3, 4, 8–10]. Some measurements display smooth peaks [6, 11], which could be modeled by other distributions.

<sup>1</sup> It refers to the concept of distance described in terms of wavelengths.

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

In this chapter, the sharp peaks in the angular spectrum are modeled as Cauchy angular distributions of power and the smooth peaks are modeled as Gaussian angular power distributions. Other distributions can be approximated as weighted sums of Gaussian angular distributions of power or the combination of Gaussian and Cauchy angular distributions of power [5].

Although the Cauchy power distribution function (PDF) has fat tails as compared to the Laplace PDF, it could be used to achieve our goal if most of the power (such as 90% or more) is concentrated in a smaller angular range. Geometrically, it can be interpreted as that most of the scattering objects are located around the center of a cluster, while the rest contributes a much smaller amount of power to the antennas, which can be ignored. This idea can be used for truncated Gaussian angular power distributions as well.

It has been identified that, for a Cauchy angular power distribution function (APDF), the corresponding cluster has the following property: The distance between the cluster center and the scattering objects should obey the Cauchy-Rayleigh distribution [12], and for a Gaussian APDF, the corresponding cluster has the property that the distance between the cluster center and the scatterers should follow the Rayleigh distribution [13]. They are named as the Cauchy-Rayleigh cluster and Rayleigh cluster, respectively (**Figure 2**).

Based on the trigonometric relationship among transmitter, scatterers, and receiver, the APDFs of these two types of scattering clusters can be derived. In addition, the spatial-temporal correlation function is integrable according to the obtained APDF. The analytical solution, or closed-form solution, will be associated with a distant scattering cluster, i.e., the solution will depend on the characteristics of a given geometrical cluster. Furthermore, to be able to model a state-space MIMO channel, the correlation function needs to be separated into disjoint two parts, a temporal term over the movement and a spatial term over the antenna.

#### **Figure 2.**

*Laplace power distribution function (PDF) gx*ð Þ¼ *<sup>x</sup>* <sup>1</sup> <sup>2</sup> *<sup>e</sup>*�∣*x*<sup>∣</sup> *and Cauchy PDF f <sup>x</sup>*ð Þ¼ *<sup>x</sup>* <sup>1</sup> *π η <sup>η</sup>*2þ*x*<sup>2</sup>*, where <sup>η</sup>* <sup>¼</sup> <sup>0</sup>*:*634, *special case of parametrization.*

The beauty of this approach is that the expression of the channel second-order statistics can eventually be integrable. This analytical solution can be broken down into the temporal dynamics and spatial correlation parts of the channel. Depending on the type of cluster, the temporal dynamics part will be approximately modeled as an autoregressive order one (AR(1)) or an autoregressive order three (AR(3)) model, and the spatial correlation part will be described using the Kronecker matrix. An autoregressive order two (AR(2)) model can also be used if the requirement for approximation is acceptable. Therefore, one can construct the state-space MIMO/massive MIMO channel models, as well as the multi-cluster state-space MIMO-OFDM (orthogonal frequency-division multiplexing) channel models.

### **2. MIMO system**

**Figure 3** depicts a *Mr* � *Mt* MIMO system, where *Mr* and *Mt* denote the numbers of receiving and transmitting antennas, respectively. This system has a total of *MtMr* links between the BS and MS, in which each link is referred to as a radio channel.

Without loss of generality, a narrow-band, time-invariant channel model is used to compute the spatial correlation matrices. In this case, the channel matrix can be represented by [14].

$$\mathbf{H} = \begin{bmatrix} h\_{11} & h\_{12} & \cdots & h\_{1M\_t} \\ h\_{21} & h\_{22} & \cdots & h\_{2M\_t} \\ \vdots & \vdots & \ddots & \vdots \\ h\_{M\_t1} & h\_{M\_t2} & \cdots & h\_{M\_tM\_t} \end{bmatrix} \tag{1}$$

where the elements *hij* are the amplitude and phase change over the link between the *ith* MS antenna and the *jth* BS antenna.

To obtain the channel spatial correlation coefficients at the MS, we choose two arbitrary elements from a certain column of **H**, *hik*, *hjk*, here and calculate the expectation value of the product of these two gains, i.e., *E hikh*<sup>∗</sup> *jk* h i.

Usually, the distance between a transmitter and a receiver is quite large, so both transmitter and receiver will only be affected by scatterers in their vicinity. Therefore, the scatterers around the transmitter are uncorrelated with the scatterers around the receiver. That is, the spatial correlation between two arbitrary antennas at the MS does not depend on the transmitter antennas at the BS, but only depends on the antenna pair. Hence, the value, *E hikh*<sup>∗</sup> *jk* h i, can be assumed to be independent of *<sup>k</sup>*.

All coefficients are then defined by

$$r\_{i,j}^{\text{MS}} = E\left[h\_{ik}h\_{jk}^{\*}\right] \tag{2}$$

Obviously, *r*MS *<sup>i</sup>*,*<sup>j</sup>* <sup>¼</sup> *<sup>r</sup>*MS <sup>∗</sup> *<sup>j</sup>*,*<sup>i</sup>* by this definition. Therefore, the corresponding spatial correlation matrix, a square matrix of order *Mr*, is represented by [14].

$$\mathbf{R\_{MS}} = \begin{bmatrix} r\_{1,1}^{\text{MS}} & r\_{1,2}^{\text{MS}} & \cdots & r\_{1,M\_r}^{\text{MS}} \\ r\_{2,1}^{\text{MS}} & r\_{2,2}^{\text{MS}} & \cdots & r\_{2,M\_r}^{\text{MS}} \\ \vdots & \vdots & \ddots & \vdots \\ r\_{M\_r,1}^{\text{MS}} & r\_{M\_r,2}^{\text{MS}} & \cdots & r\_{M\_r,M\_r}^{\text{MS}} \end{bmatrix} \tag{3}$$

Similarly, the channel spatial correlation coefficients at the BS can be given by

$$\mathbf{R\_{BS}} = \begin{bmatrix} r\_{1,1}^{\text{BS}} & r\_{1,2}^{\text{BS}} & \cdots & r\_{1,M\_t}^{\text{BS}} \\ r\_{2,1}^{\text{BS}} & r\_{2,2}^{\text{BS}} & \cdots & r\_{2,M\_t}^{\text{BS}} \\ \vdots & \vdots & \ddots & \vdots \\ r\_{M\_t,1}^{\text{BS}} & r\_{M\_t,2}^{\text{BS}} & \cdots & r\_{M\_t,M\_t}^{\text{BS}} \end{bmatrix} \tag{4}$$

where all elements *r*BS *<sup>m</sup>*,*<sup>n</sup>* are defined by

$$r\_{m,n}^{\text{BS}} = E\left[h\_{lm}h\_{ln}^{\*}\right] \tag{5}$$

in terms of the channel gains *hlm* and *hln*, which are selected from a certain row of **H**, here *m*, *n* ∈f g 1, 2, ⋯, *Mt* , *l* ∈f g 1, 2, ⋯, *Mr* .

Finally, based on the assumptions that Eqs. (2) and (5) are independent of k and l, respectively, the spatial correlation matrix of the MIMO channels is given by [14].

$$\mathbf{R\_{MIMO}} = E\left[\mathbf{vec}(\mathbf{H})\mathbf{vec}(\mathbf{H})^H\right] = \mathbf{R\_{BS}} \otimes \mathbf{R\_{MS}}\tag{6}$$

where ⊗ denotes the Kronecker product, vec ½ �� represents the vectorization of a matrix, which converts all elements of the matrix into a column vector.

### **3. Channel second-order statistics**

Consider a MIMO system in which the BS has *Mt* antennas, the MS has *Mr* antennas, and the BS is fixed, while the MS is moving. Then, the spatial-temporal-spectral correlation function of a MIMO-OFDM channel can be expressed as [15, 16].

$$\begin{split} \mathbf{C}\_{h}(\Delta t, d\_{t}, d\_{r}, d\_{f}) &= \int\_{a, \boldsymbol{\beta}, \tau} f\_{a, \boldsymbol{\beta}, \tau}(a, \boldsymbol{\beta}, \tau) e^{j2\pi d\_{t} \cos(\boldsymbol{\beta} + \boldsymbol{\beta}\_{0})} \\ &\times e^{j2\pi \left(d, \cos(a + a\_{0}) + f\_{D} \Delta t \cos(a + a\_{0} - \gamma)\right)} e^{-j2\pi d\_{f} \tau} da d\boldsymbol{\beta} d\tau \end{split} \tag{7}$$

where *f <sup>D</sup>* is the Doppler frequency, Δ*t* is total time separation, and *f <sup>D</sup>*Δ*t* is the MS moving distance. *β* is the AOD and *β*<sup>0</sup> is its mean, *α* is the AOA and *α*<sup>0</sup> is its mean, and *γ* is the angle between the moving direction and the antenna array, as shown in **Figure 4**. *dt* ¼ *mt*Δ*dt* is the antenna spacing at the BS, *mt* ∈ f g 0, 1, ⋯, *Mt* � 1 , Δ*dt* is the spacing between two adjacent antenna sensors. *dr* ¼ *mr*Δ*dr* is the antenna spacing at the MS, *mr* ∈f g 0, 1, ⋯, *Mr* � 1 , Δ*dr* is the spacing between two adjacent antenna sensors. *df* ¼ *mf* Δ*f* is frequency separation, Δ*f* denotes the frequency difference between two adjacent sub-carriers, and the sub-carrier frequencies are defined by *fi* <sup>¼</sup> *<sup>f</sup> <sup>c</sup>* <sup>þ</sup> *<sup>i</sup>*Δ*f*, for all *<sup>i</sup>*<sup>∈</sup> 0,1,2, <sup>⋯</sup>, *Mf* � <sup>1</sup> � �, here *f <sup>c</sup>* is the frequency range and *Mf* denotes the number of sub-carrier frequencies required for transmission. The difference between two frequencies *fi* and *fj* is denoted by *mf* <sup>¼</sup> *<sup>j</sup>* � *<sup>i</sup>:* Hence, *mf* <sup>∈</sup> 0,1,2, <sup>⋯</sup>, *Mf* � <sup>1</sup> � �*:* When *mf* <sup>¼</sup> 0, it represents a single-carrier modulation system, the so-called MIMO system. *f <sup>α</sup>*,*β*,*<sup>τ</sup>*ð Þ *α*, *β*, *τ* denotes the joint angular-delay power distribution function, here, *τ* denotes the time delay.

By assuming the independence of *α*, *β*, and *τ*, the joint angular-delay PDF *f <sup>α</sup>*,*β*,*<sup>τ</sup>*ð Þ *α*, *β*, *τ* is separated into *f <sup>α</sup>*,*β*,*<sup>τ</sup>*ð Þ¼ *α*, *β*, *τ f <sup>α</sup>*ð Þ *α f <sup>β</sup>*ð Þ *β f <sup>τ</sup>*ð Þ*τ* , here *f <sup>α</sup>*ð Þ *α* is the APDF of

**Figure 4.**

*A distant cluster with an Mr* � *Mt MIMO antenna array, lk is the distance between the scattering object* Sk *and the cluster center O.*

the AOA, *f <sup>β</sup>*ð Þ *β* denotes the APDF of the AOD, and *f <sup>τ</sup>*ð Þ*τ* is the delay power distribution function (DPDF) of the time-of-arrival (TOA).

This assumption is reasonable because usually radio signals pass through more than one scatterer in a cluster from a transmitter to a receiver, which means that *α*, *β* are independent. *τ* denotes the TOA, which is independent of *α* and *β:* Therefore, Eq. (7) becomes,

$$\begin{split} \mathbf{C}\_{h}(\Delta t, d\_{t}, d\_{r}, d\_{f}) &= \int\_{a, \beta, \mathbf{r}} f\_{a}(a) f\_{\beta}(\beta) f\_{\mathbf{r}}(\mathbf{r}) e^{j2\pi d\_{t}\cos(\beta + \beta\_{0})} \\ &\times e^{j2\pi \left(d\_{r}\cos(a + a\_{0}) + f\_{D}\Delta t \cos(a + a\_{0} - \gamma)\right)} e^{-j2\pi d\_{\mathbf{f}}\mathbf{r}} da d\beta d\mathbf{r} \end{split} \tag{8}$$

which two special cases are highlighted below,

• the channel temporal dynamic function is denoted by *Rh*ð Þ Δ*t*

$$R\_h(\Delta t) = \mathcal{C}\_h(\Delta t, \mathbf{0}, \mathbf{0}, \mathbf{0}) = \int\_a f\_a(a) e^{j2\pi f\_D \Delta t \cos(a+a\varrho - \eta)} da\tag{9}$$

• the spatial-temporal correlation function is denoted by *Ch* Δ*t*, *dt*, *dr* ð Þ , 0

$$\mathbf{C}\_{\hbar}(\Delta t, d\_{t}, d\_{r}, \mathbf{0}) = \int\_{a, \beta} f\_{a}(a) f\_{\beta}(\beta) e^{j2\pi d\_{t}\cos(\beta + \beta\_{0})} e^{j2\pi \left(d\_{r}\cos(a + a\_{0}) + f\_{D}\Delta t \cos(a + a\_{0} - \gamma)\right)} da d\beta \tag{10}$$

A distant scattering cluster causes the AOA and AOD to vary over a small angular range. This motivates us to approximate the CSOS in Eq. (8) and allows us to study its characteristics in a small angular range. Using the Taylor expansion<sup>2</sup> , for all angles *α* and *β* close to zero, the following approximate trigonometric identities are obtained,

$$\begin{aligned} \cos\left(a+a\_0\right) &= \cos\left(a\right)\cos\left(a\_0\right) - \sin\left(a\right)\sin\left(a\_0\right) \\ &\approx \cos\left(a\_0\right) - a\sin\left(a\_0\right) \\ \cos\left(\beta+\beta\_0\right) &\approx \cos\left(\beta\_0\right) - \beta\sin\left(\beta\_0\right) \\ \cos\left(a+a\_0-\gamma\right) &\approx \cos\left(a\_0-\gamma\right) - a\sin\left(a\_0-\gamma\right) \end{aligned} \tag{11}$$

Substituting Eq. (11) into Eq. (8), an approximate channel spatial-temporal-spectral correlation function is obtained. This is the first time to approximate this expression. The notation *Ch* Δ*t*, *dt*, *dr*, *df* � � is used to represent this approximation,

$$\begin{split} \mathbf{C}\_{h}(\Delta t, d\_{t}, d\_{r}, d\_{f}) &\quad \approx \quad \overline{\mathbf{C}}\_{h}(\Delta t, d\_{t}, d\_{r}, d\_{f}) \\ &= \quad \int\_{a, \theta, \tau} f\_{a}(a) f\_{\beta}(\beta) f\_{\tau}(\tau) e^{j2\pi d\_{t} \cos(\beta\_{0})} \\ &\quad \times e^{-j2\pi d\_{t} \sin(\beta\_{0})\beta} e^{j2\pi \left(d\_{t}\cos(\alpha\_{0}) + f\_{D}\Delta t \cos(\alpha\_{0} - \gamma)\right)} \\ &\quad \times e^{-j2\pi \left(d\_{t}\sin(\alpha\_{0}) + f\_{D}\Delta t \sin(\alpha\_{0} - \gamma)\right)a} e^{-j2\pi d\_{\tilde{f}}\tau} d\alpha d\beta d\tau \end{split} \tag{12}$$

<sup>2</sup> For small *<sup>ϵ</sup>*, cosð Þ¼ *<sup>ϵ</sup>* <sup>1</sup> <sup>þ</sup> *<sup>O</sup> <sup>ϵ</sup>*<sup>2</sup> ð Þ and sinð Þ¼ *<sup>ϵ</sup> <sup>ϵ</sup>* <sup>þ</sup> *<sup>O</sup> <sup>ϵ</sup>*<sup>3</sup> ð Þ.

Considering the channel spatial-temporal correlation function *Ch* Δ*t*, *dt*, *dr* ð Þ , 0 , the separation of antenna spacing and motion into disjoint parts is an essential step to model a MIMO channel with a state-space representation.

However, the Cauchy angular distribution-based analytical solution contains an absolute sum of terms in the exponent related to antenna spacing and movement, in which the sign of the absolute value needs to be removed, while the Gaussian angular distribution-based solution has a cross-term that is related to the antenna spacing and motion, which can neither be classified as channel temporal dynamics nor as spatial correlation [12, 13].

To separate antenna spacing and motion (channel dynamics) while avoiding errors caused by unnecessary further approximations [12, 13], a linear transformation is introduced to handle this separation. Since this linear transformation eventually affects the phase of the CSOS, it is called the phase-shift method.

### **4. Linear transformation**

Mathematically, the linear transformation approach implies converting the current Cartesian system to another system. In the new system, the antenna spacing and movement can be separated into error-free disjoint parts, and the channel characteristics can then be modeled using a state-space representation. Finally, an inverse linear transformation is performed to convert the channel properties back to and represent them in the original system.

To study the channel correlation properties caused by distant scattering clusters, the correlation related to the MS was approximated by the AOA near zero degrees around the angles *α*<sup>0</sup> and *α*<sup>0</sup> � *γ*, as expressed in Eq. (12).

As depicted in **Figure 5**, this approximation means decomposing the movement and antenna spacing into a phase change on OA and a damping change

#### **Figure 5.**

*The motion vector is decomposed into a phase change on OA and a damping change on AW, where AB and EF denote the antenna arrays, and a, B, E, F are antenna sensors.*

on AW in accordance with the moving direction, in which the lines OA and AW are orthogonal.

$$\begin{aligned} \text{Let } \mathbf{AG} = d\_{\mathbf{AG}}, \mathbf{GH} = d\_{\mathbf{GH}}, \mathbf{AU} = d\_{\mathbf{AU}}, \text{ and } \mathbf{UW} = d\_{\mathbf{UW}}, \text{ then,} \\ d\_{\mathbf{AG}} + d\_{\mathbf{GH}} &= \mathfrak{s} \cos(a\_0 - \chi) + d\_r \cos(a\_0) \\ d\_{\mathbf{AU}} + d\_{\mathbf{UW}} &= \mathfrak{s} \sin(a\_0 - \chi) + d\_r \sin(a\_0) \end{aligned} \tag{13}$$

Geometrically, Eq. (13) interprets the meaning of the approximate expression in Eq. (12). Alternatively, AW can be considered as the result of AD projection.

Let AD ¼ *d*AD ¼ *κ*, then the right triangle relationship shows that,

$$\kappa = \frac{s\sin(a\_0 - \chi) + d\_r\sin(a\_0)}{\cos(90^\circ - a\_0 + \chi)} = s + d\_r \frac{\sin(a\_0)}{\sin(a\_0 - \chi)}\tag{14}$$

This can be regarded as that antenna A moves to D, but its real position is at E. Hence, the changed phase will cause Eq. (12) to become

$$\begin{split} \overline{C}\_{h}^{\*}(\Delta t\_{\kappa}, d\_{t}, d\_{r}, \mathbf{0}) &= \ \prescript{-j2\pi d\_{t} \sin(\gamma)/\sin(a\_{0}-\gamma)}{\operatorname{\mathbf{e}}^{j}} \mathbf{e}^{j2\pi d\_{t} \cos(\beta\_{0})} \mathbf{e}^{j2\pi f\_{D} \Delta t\_{\kappa} \cos(a\_{0}-\gamma)} \\ &\times \int\_{a,\beta} f\_{a}(\alpha) f\_{\beta}(\beta) \mathbf{e}^{-j2\pi d\_{t} \sin(\beta\_{0}) \theta} \mathbf{e}^{-j2\pi f\_{D} \Delta t\_{\kappa} \sin(a\_{0}-\gamma)a} da d\beta \end{split} \tag{15}$$

in the new system.

Obviously, in the new system, the spatial correlation of the MS-related channels is represented by a phase rotation. In this way, we do separate the movement and antenna spacing into disjoint parts.

Moreover, this phase rotation is not related to the antennas at the BS. Thus, the Kronecker product can be used to construct the state-space MIMO channels [17].

The phase-shift approach provides an alternative way to study the same problem. By changing variables, an error-free and simple method is found to separate the movement and antenna spacing, in which the channel spatial-temporal correlation function can be regarded as the product of the phase rotation and channel temporal dynamics shifted by a value along the moving direction.<sup>3</sup>

### **5. Geometry-based approach**

In this section, two APDFs are presented. They are obtained based on Cauchy-Rayleigh and Rayleigh clusters. This geometric approach provides an intuitive way to map scattering objects in a cluster as an angular distribution of power.

#### **5.1 Cauchy-Rayleigh cluster**

Given a distant Cauchy-Rayleigh cluster, the Cauchy APDFs are obtained according to the geometric relations shown in **Figure 4** [12],

$$\,\_1f\_a(a) \approx \mathfrak{f}\_a^c(a) = \frac{1}{\pi} \frac{\eta\_r}{\eta\_r^2 + a^2}, \; f\_\beta(\beta) \approx \mathfrak{f}\_\beta^c(\beta) = \frac{1}{\pi} \frac{\eta\_t}{\eta\_t^2 + \beta^2} \tag{16}$$

<sup>3</sup> It can also be considered as the time delay of the channel dynamics.

where ½ �� *<sup>c</sup>* indicates that the APDF is obtained in terms of the Cauchy-Rayleigh cluster, the parameters, *η<sup>t</sup>* ¼ *ζ=d*OB1 and *η<sup>r</sup>* ¼ *ζ=d*OM1 , *d*OB1 ¼ OB1, *d*OM1 ¼ OM1, are used to control the angular width of these two distributions, respectively. *ζ* >0 is the dispersion of the Cauchy-Rayleigh distribution.

Obviously, both *f c <sup>α</sup>*ð Þ *α* and *f c <sup>β</sup>*ð Þ *β* in Eq. (16) are defined on ½ � �*π*, *π* , and they are not proper Cauchy angular power density functions because the Cauchy probability density function is defined over the infinite interval ð Þ �∞, ∞ . Hence, it is necessary to extend the integral from ½ � �*π*, *π* to ð Þ �∞, ∞ to use them to participate in the calculation of the integral.

Since *f c <sup>α</sup>*ð Þ *α* and *f c <sup>β</sup>*ð Þ *β* in Eq. (16) represent the angular powers of the MS and BS, which are similar, the discussion will focus only on the AOD. The same conclusions can be obtained for AOA.

Clearly, *f c <sup>β</sup>*ð Þ *β* is truncated tails, which lead to

$$\int\_{-\pi}^{\pi} f\_{\beta}^{\epsilon}(\beta) d\beta \lesssim \mathbf{1} \tag{17}$$

However, if the intervals ð Þ �∞, �*π* and ð Þ *π*, ∞ contain much less power, then *f <sup>β</sup>*ð Þ *β* in Eq. (16) is a suitable approximation.

Assuming that in the interval �*βy*%, *β<sup>y</sup>*% h i, *Pc <sup>β</sup>* contains *y*% power, then the following equation describes the relationship among the critical angle, the power, and the width of the distribution,

$$P^{\varepsilon}\_{\beta} = \int\_{-\beta\_{\mathcal{I}^{\rm th}}}^{\beta\_{\mathcal{I}^{\rm th}}} f^{\varepsilon}\_{\beta}(\beta) d\beta = \frac{2}{\pi} \arctan\left(\frac{\beta\_{\mathcal{I}^{\rm th}}}{\eta\_{t}}\right) \tag{18}$$

i.e., *<sup>β</sup><sup>y</sup>*% <sup>¼</sup> tan *<sup>π</sup>P<sup>c</sup> <sup>β</sup>=*2 � �*ηt:* Assuming *Pc <sup>β</sup>* ¼ 90%, then *β*90% ¼ 6*:*3138*ηt:* Similarly, *<sup>α</sup>*90% <sup>¼</sup> <sup>6</sup>*:*3138*η<sup>r</sup>* for *<sup>P</sup><sup>c</sup> <sup>α</sup>* ¼ 90%.

Moreover, 90% of power within the angular interval 2*β*90% means that the intervals ð Þ �∞, �*β*90% and ð Þ *β*90%, ∞ contain at most 10% of the transmitted power. With this in mind, the possibility of extending the angular interval from ½ � �*π*, *π* to ð Þ �∞, ∞ is explored next.

Based on the formula,

$$P^{\varepsilon}(\eta\_t) = \int\_{-\pi}^{\pi} f\_{\beta}^{\varepsilon}(\beta) d\beta = \frac{2}{\pi} \arctan\left(\frac{\pi}{\eta\_t}\right) \tag{19}$$

and *β*90% ¼ 6*:*3138*ηt*, the following table is obtained.

**Table 1** indicates that for each critical angle *β*90%, the interval ½ � �*π*, *π* contains almost all the power contributed from the cluster. Therefore,

$$\frac{2}{\pi}\arctan\left(\frac{\pi}{\eta\_t}\right) = \int\_{-\pi}^{\pi} f\_\beta^c(\beta)d\beta \approx \int\_{-\infty}^{\infty} f\_\beta^c(\beta)d\beta = 1\tag{20}$$

Moreover, the assumption of a small angular range with most of the power will help one to obtain the following characteristic function as well,

$$\Phi\_{\beta}^{\epsilon}(w) = \int\_{-\infty}^{\infty} f\_{\beta}^{c}(\beta) e^{-j\alpha\beta} d\beta \approx \int\_{-\pi}^{\pi} f\_{\beta}^{c}(\beta) e^{-j\alpha\beta} d\beta \tag{21}$$

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


#### **Table 1.**

*The widths η<sup>t</sup> and the corresponding powers.*

Eq. (21) indicates that if some power is left out in one domain, then the same small amount will be missing in the other.

Therefore, Eq. (21) can be used to solve the integrals in Eq. (15) as

$$\Phi^c\_{\beta}(\mathsf{2}\pi d\_t \sin(\beta\_0)) \approx \int\_{\beta} f^c\_{\beta}(\beta) e^{-j2\pi d\_t \sin(\beta\_0)\beta} d\beta \tag{22}$$

which is known,

$$\begin{split} \tilde{C}^{\varepsilon\_{\mathbf{r}}}\_{h}(\Delta t\_{\kappa}, d\_{t}, d\_{r}, \mathbf{0}) &= \, \_{\mathbf{r}}e^{-2\pi\eta\_{d}d\_{t}|\sin(\beta\_{0})|} \, \_{\mathbf{r}}\mathbf{e}^{j2\pi d\_{t}\cos(\beta\_{0})} \, \_{\mathbf{r}}\mathbf{e}^{-j2\pi d\_{r}\sin(\gamma)/\sin(a\_{0}-\gamma)} \\ & \times \, \_{\mathbf{r}}e^{-2\pi\eta\_{d}f\_{D}\Delta t\_{\kappa}|\sin(a\_{0}-\gamma)|} \, \_{\mathbf{r}}\mathbf{e}^{j2\pi f\_{D}\Delta t\_{\kappa}\cos(a\_{0}-\gamma)} \end{split} \tag{23}$$

and the channel dynamic function,

$$\tilde{R}\_h^{\varepsilon\_\kappa}(\Delta t\_\kappa) = \tilde{C}\_h^{\varepsilon\_\kappa}(\Delta t\_\kappa, \mathbf{0}, \mathbf{0}, \mathbf{0}) = e^{-2\pi\eta f\_D \Delta t\_\kappa |\sin(a\_0 - \gamma)|} \mathbf{e}^{j2\pi f\_D \Delta t\_\kappa \cos(a\_0 - \gamma)} \tag{24}$$

According to Eq. (23), a specific expression of each element of **R**BS in Eq. (4) is assigned,

$$r\_{m,n}^{\epsilon, \text{BS}}(d\_t) = e^{-2\pi\eta\_t d\_t |\sin(\beta\_0)|} \mathcal{e}^{j2\pi d\_t \cos(\beta\_0)}\tag{25}$$

and the notation **R***<sup>c</sup>* BS is to replace **R**BS*:* Furthermore, all elements of **R**MS in Eq. (3) will have the following specific expression,

$$r\_{i,j}^{\text{MS}}(d\_r) = e^{-j2\pi d\_r \sin(\gamma)/\sin(a\_0 - \gamma)}\tag{26}$$

Eq. (26) indicates that the spatial correlation between MS channels will depend only on the antenna spacing *dr* but not on the cluster type. Thus, the notation **R**MS will be kept.

#### **5.2 Rayleigh cluster**

Similarly, given a distant Rayleigh cluster, the following approximate Gaussian APDFs are derived [13],

$$f\_a''(a) = \frac{\mathbf{1}}{\sqrt{2\pi}\sigma\_r} e^{-\frac{\rho^2}{2\sigma\_r^2}}, \quad f\_\beta''(\beta) = \frac{\mathbf{1}}{\sqrt{2\pi}\sigma\_l} e^{-\frac{\rho^2}{2\sigma\_l^2}}\tag{27}$$

where ½ �� *<sup>r</sup>* indicates that the APDF is obtained from the Rayleigh cluster, *σ<sup>t</sup>* ¼ *σ=d*OB1 , *σ<sup>r</sup>* ¼ *σ=d*OM1 , and *σ* is obtained from the Rayleigh distribution.

As described in the previous section, the truncated Gaussian APDFs can also be extended from *π* to infinity, and the analytical solution of the CSOS, denoted by the notation *<sup>C</sup>*~*<sup>r</sup><sup>κ</sup> <sup>h</sup>* Δ*tκ*, *dt*, *dr* ð Þ , 0 , is obtained by substituting Eq. (27) into Eq. (15) [13],

$$\begin{split} \tilde{C}^{r\_{\kappa}}\_{h}(\Delta t\_{\kappa}, d\_{t}, d\_{r}, \mathbf{0}) &= \, \mathrm{e}^{-2\pi^{2}\sigma\_{t}^{2}d\_{t}^{2}\sin^{2}(\theta\_{0})} \mathrm{e}^{j2\pi d\_{t}\cos(\theta\_{0})} \mathrm{e}^{-j2\pi d\_{r}\sin(\gamma)/\sin(a\_{0}-\gamma)} \\ &\times \, \mathrm{e}^{-2\pi^{2}\sigma\_{r}^{2}\sin^{2}(a\_{0}-\gamma)\underline{f}\_{D}^{2}\Delta t\_{\kappa}^{2}j2\pi\cos(a\_{0}-\gamma)\underline{f}\_{D}\Delta t\_{\kappa}} \end{split} \tag{28}$$

and the channel temporal dynamic function is

$$\tilde{R}\_h^{\mathcal{T}\_\kappa}(\Delta t\_\kappa) = e^{-2\pi^2 \sigma\_r^2 \sin^2(a\_0 - \gamma) f\_D^2 \Delta t\_\kappa^2} \mathcal{C}^{2\pi \cos(a\_0 - \gamma) f\_D \Delta t\_\kappa} \tag{29}$$

Thus, each element of **R***<sup>r</sup>* BS can be given by

$$\sigma\_{m,n}^{r, \text{BS}}(d\_t) = e^{-2\pi^2 \sigma\_t^2 d\_t^2 \sin^2(\beta\_0)} e^{j2\pi d\_t \cos(\beta\_0)}\tag{30}$$

and all elements of **R***<sup>r</sup>* MS are also given by Eq. (26).

### **6. AR-based state-space channel model**

In the previous sections, two types of scattering clusters were introduced to obtain the analytical solutions of the CSOS. These two analytical solutions were decomposed into the product of channel temporal dynamics and spatial correlation. In this section, the channel temporal dynamics will be approximated as an AR(p) model, by which, a state-space MIMO channel model can be constructed for fitting the channel spatialtemporal correlation function.

A state-space model describes a dynamic system associated with the input, state variables, and output. The system input and output are linked by a state vector which is determined by a state transition matrix, and the last variable in the state vector will be the contribution from the cluster to the channel.

That is, for each scattering cluster, an AR(p) model is used to describe the MIMO radio channel temporal dynamics, and the Kronecker correlation matrix is employed to characterize the channel spatial correlation.

A coloring matrix is used to drive input Gaussian noise innovations to create channel spatial correlations, and the coloring matrix is determined by the channel correlation properties of the BS and MS described by the Kronecker product.

#### **6.1 AR(p) model**

An AR(p) model specifies that the output variable depends linearly on its previous values. It is a very ordinary model and has a wide variety of applications in time series. One of the significant features of an AR(p) model is that it can be transformed into a state-space representation. Therefore, a large number of approaches in the control domain can potentially be applied to MIMO channel modeling and be used to study radio channels.

An AR(p) model can be represented by [18].

$$\mathbf{x}\_{k} = \sum\_{i=1}^{p} \phi\_{i} \mathbf{x}\_{k-i} + w\_{k} \tag{31}$$

where *ϕ*1, ⋯, *ϕ<sup>p</sup> ϕ<sup>p</sup>* �¼ 0 � � are complex coefficients and *wk* is a complex Gaussian sequence N 0; *σ*<sup>2</sup> *w* � �. That is, the stochastic variable *xk* is defined as a linear combination of its previous p values of the series plus an innovation noise.

In this section, Eq. (24) will be described by an AR(1) model, while Eq. (29) will be approximated by an AR(3) model. Since Eq. (24) is itself an AR(1) model, its coefficient *ϕ* and standard variance *σ*AR 1ð Þ can be obtained directly from equations [12, 19]. The coefficient *ϕ<sup>i</sup>* and standard variance *σ*AR 3ð Þ of an AR(3) model can be estimated using the least-squares (LS) method [20] or computed using the spectral-equivalent (SE) method [13].

In this way, a single peak on the Doppler spectrum corresponding to the contribution from a distant scattering cluster is modeled by an AR(p) model.<sup>4</sup> The advantage of using an AR(p) model is that it can be directly parameterized by the properties of the cluster, and it allows changing the angles in the simulation, which corresponds to changing the directions of the mobile receiver.

#### **6.2 SISO channel model**

The AR(p) model given by Eq. (31) can be transformed into the controllable canonical form [18, 21, 22] to obtain a state-space representation,

$$\begin{aligned} \mathbf{x}\_{k+1} &= \mathbf{A}\mathbf{x}\_k + \mathbf{B}w\_k \\ h\_k &= \mathbf{C}\mathbf{x}\_k \end{aligned} \tag{32}$$

where **<sup>B</sup>** <sup>¼</sup> ½ � 0 0<sup>⋯</sup> 0 1 *<sup>T</sup>* is a *<sup>p</sup>* � 1, **<sup>C</sup>** <sup>¼</sup> ½ � 0 0 <sup>⋯</sup> 0 1 is a 1 � *<sup>p</sup>* vector, the output *hk* ¼ *xk* is a scalar, the channel, and

$$\begin{array}{rclcrcl}\mathbf{x}\_{k+1} &=& \begin{bmatrix} \mathbf{x}\_{k-p+1} \\ \mathbf{x}\_{k-p+2} \\ \vdots \\ \vdots \\ \mathbf{x}\_{k} \\ \mathbf{x}\_{k+1} \end{bmatrix}, \mathbf{A} = \begin{bmatrix} \mathbf{0} & \mathbf{1} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{1} & \cdots & \mathbf{0} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{1} \\ \boldsymbol{\phi}\_{p} & \boldsymbol{\phi}\_{p-1} & \boldsymbol{\phi}\_{p-2} & \cdots & \boldsymbol{\phi}\_{1} \end{bmatrix} \\ & \mathbf{x}\_{k} &=& \begin{bmatrix} \boldsymbol{\chi}\_{k-p} & \boldsymbol{\chi}\_{k-p+1} & \cdots & \boldsymbol{\chi}\_{k-1} & \boldsymbol{\chi}\_{k} \end{bmatrix}^{T} \end{array} \tag{33}$$

The input vector **<sup>B</sup>** <sup>¼</sup> 0 0 <sup>⋯</sup> <sup>0</sup> *<sup>σ</sup>*AR pð Þ � �*<sup>T</sup>* is redefined, i.e., the noise input will be scaled by *<sup>σ</sup>*AR pð Þ, then the variance of *xk* is 1, i.e., *<sup>σ</sup>*<sup>2</sup> *<sup>x</sup>* ¼ 1 [13]. It makes a lot of sense to let *xk* have unit variance before **C** and let **C** scale be the contribution from a cluster, including path loss to *hk*.

It must be noted that, in reality, the matrices **A** and **B** are time-variant because both of them have angle-dependent elements. The angle *α*<sup>0</sup> � *γ* is used to describe the moving direction to the cluster center, which may change all the time during the movement.

However, these matrices are assumed to be time-invariant due to small movements compared with the distance from the MS to the center of a scattering cluster. That is, within some time slots, all matrices are approximately constant. This implies that

<sup>4</sup> The channel dynamics due to the Cauchy-Rayleigh clusters is modeled as an AR(1) model and it approximately represents the channel dynamics due to the Rayleigh clusters by an AR(3) model.

**Figure 6.** *Block diagram of the AR(p)-based state-space SISO channel model.*

constant angles toward clusters, constant speed during the movement, and hence a time-invariant environment is satisfied. This assumption is related to the stationarity of **x***<sup>k</sup>* and **h***<sup>k</sup>* sequences as well.

The block diagram corresponding to the singe-input and single-output (SISO) channel model in Eq. (32) is shown below,

**Figure 6** is also known as the AR(p)-based state-space SISO channel model block. In the block diagram, the inputs and outputs are scalars, described by a single line, and double lines are used to represent vectors.

This is the simplest state-space model used to describe the channel temporal dynamics and will be employed to construct state-space single-input and multipleoutput (SIMO) and MIMO channel models.

### **6.3 SIMO channel model**

Based on the SISO channel model block shown in **Figure 6**, a state-space SIMO channel model is constructed by connecting multiple SISO channel model blocks in parallel, in which a correlated innovation process is employed to adjust the spatial correlation between these SISO channel blocks, the SIMO channels, as shown in **Figure 7**. This can be done by introducing **Φ**Mr , an *Mr Mr* coloring matrix. The number of SISO channel model blocks required for the SIMO channels will depend on the number of receiving antenna elements *Mr*.

Mathematically, this parallel connection can be interpreted as the following state-space representation,

**Figure 7.** *Block diagram of the AR(p)-based state-space SIMO channel model.*

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

$$\begin{array}{rcl} \mathbf{x}\_{k+1} &=& \mathbf{I}^{simo} \mathbf{x}\_k + \mathbf{V}^{simo} \mathbf{w}\_k\\ \mathbf{h}\_k^{simo} &=& \mathbf{\Omega}^{simo} \mathbf{x}\_k \end{array} \tag{34}$$

where the state vector **<sup>x</sup>***<sup>k</sup>* <sup>∈</sup> pMr , **<sup>w</sup>***<sup>k</sup>* � <sup>N</sup> **<sup>0</sup>**; *<sup>σ</sup>*<sup>2</sup> *<sup>w</sup>***I**Mr � �∈ Mr are independent and identically distributed (i.i.d), the channel vector and the driving noise vector are expressed as

$$\begin{array}{rcl} \mathbf{h}\_{k}^{simo} &=& \begin{bmatrix} h\_{k}[\mathbf{1}] & h\_{k}[\mathbf{2}] & \cdots & h\_{k}[\mathbf{M}\_{r}] \end{bmatrix}^{T} \in \mathbb{C}^{\mathbf{M}\_{r}} \\\ \mathbf{w}\_{k} &=& \begin{bmatrix} w\_{k}[\mathbf{1}] & w\_{k}[\mathbf{2}] & \cdots & w\_{k}[\mathbf{M}\_{r}] \end{bmatrix}^{T} \in \mathbb{C}^{\mathbf{M}\_{r}} \end{array} \tag{35}$$

Moreover, the matrices **Γ***simo*, **Ψ***simo*, and **Ω***simo* are defined by

$$\begin{array}{rcl} \mathbf{I}^{imo} &=& \mathbf{I}\_{\mathbf{M}\_{\mathrm{r}}} \otimes \mathbf{A} = \begin{bmatrix} \mathbf{A} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{A} & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \mathbf{A} \end{bmatrix}\_{\mathrm{pM}\_{\mathrm{r}} \times \mathrm{pM}\_{\mathrm{r}}} \\\\ \mathbf{V}^{imo} &=& (\mathbf{I}\_{\mathbf{M}\_{\mathrm{r}}} \otimes \mathbf{B}) \Phi\_{\mathrm{M}\_{\mathrm{r}}} = \begin{bmatrix} \mathbf{B} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{B} & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \mathbf{B} \end{bmatrix}\_{\mathrm{pM}\_{\mathrm{r}} \times \mathrm{M}\_{\mathrm{r}}} \\\\ \boldsymbol{\Omega}^{imo} &=& \mathbf{I}\_{\mathbf{M}\_{\mathrm{r}}} \otimes \mathbf{C} = \begin{bmatrix} \mathbf{C} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{C} & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \mathbf{C} \end{bmatrix}\_{\mathrm{M}\_{\mathrm{r}} \times \mathrm{pM}\_{\mathrm{r}}} \end{array} \tag{36}$$

where **<sup>I</sup>**Mr is the identity matrix of size *Mr*, **<sup>Ψ</sup>***simo* is a *pMr* � *Mr* matrix, and **<sup>Φ</sup>**Mr is the coloring matrix employed to control the spatial correlation properties between the channels.

To acquire **Φ**Mr, we need to study the system output in Eq. (34). By definition, the auto-covariance matrix of channels **R***<sup>h</sup>* equals *E* **h***simo <sup>k</sup>* **<sup>h</sup>***simo<sup>H</sup> k* h i*:* Simple algebra gives **<sup>R</sup>***<sup>h</sup>* <sup>¼</sup> **<sup>Φ</sup>**Mr**Φ***<sup>H</sup>* Mr ¼ **R**MS.

The Cholesky decomposition method can be employed to solve the equation **Φ**Mr**Φ***<sup>H</sup>* Mr ¼ **R**MS numerically. This results in a lower triangular matrix with strictly positive diagonal entries. For small *Mr*, however, a lower triangular matrix **Φ**Mr can be found analytically [12]. Therefore, it significantly reduces the computational complexity of getting all the elements in a closed-form representation.

The key idea of modeling the SIMO channel using the state-space representation is the modular approach, i.e., just add the required number of SISO channel blocks to form another bigger block called an AR(p)-based state-space SIMO channel model block. This constructed block has an i.i.d. Gaussian noise input vector and a correlated output vector, i.e., the SIMO channels.

#### **6.4 MIMO channel model**

To build an AR(p)-based state-space MIMO channel model, the spatial correlation properties at the BS will be added. Thus, based on the Kronecker matrix given in Eq. (6), a correlated innovation matrix, a coloring matrix, is employed to characterize the spatial correlation of the channels.

Similar to modeling the state-space SIMO channel model, a state-space MIMO channel model is constructed by connecting multiple SIMO channel blocks in parallel, as **Figure 8** illustrates, in which **Φ**MrMt is the coloring matrix, and the number of SIMO channel blocks needed for the MIMO channels will depend on the number of transmitting antenna elements *Mt*.

Mathematically, this block diagram can be implemented as the following statespace representation,

$$\begin{array}{rcl} \mathbf{x}\_{k+1} &=& \mathbf{I}^{mimo} \mathbf{x}\_k + \mathbf{Y}^{mimo} \mathbf{w}\_k \\\\ \mathbf{h}\_k^{mimo} &=& \mathbf{\Omega}^{mimo} \mathbf{x}\_k \end{array} \tag{37}$$

where **x***<sup>k</sup>* ∈ pMrMt , **<sup>w</sup>***<sup>k</sup>* � <sup>N</sup> ð Þ 0; 1 <sup>∈</sup> MrMt and **<sup>h</sup>***mimo <sup>k</sup>* ¼ vec ð Þ **H***<sup>k</sup>* , here

$$\mathbf{H}\_{k} = \begin{bmatrix} h\_{k}[\mathbf{1}, \mathbf{1}] & h\_{k}[\mathbf{1}, \mathbf{2}] & \cdots & h\_{k}[\mathbf{1}, M\_{l}] \\ \\ h\_{k}[\mathbf{2}, \mathbf{1}] & h\_{k}[\mathbf{2}, \mathbf{2}] & \cdots & h\_{k}[\mathbf{2}, M\_{l}] \\ \vdots & \vdots & \ddots & \vdots \\ h\_{k}[\mathbf{M}\_{r}, \mathbf{1}] & h\_{k}[\mathbf{M}\_{r}, \mathbf{2}] & \cdots & h\_{k}[\mathbf{M}\_{r}, M\_{l}] \end{bmatrix} \tag{38}$$

and **Ψ***mimo*, **Ω***mimo*, **Γ***mimo* are defined by

**Figure 8.** *Block diagram of the AR(p)-based state-space MIMO channel model.*

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

$$\begin{aligned} \mathbf{I}^{\text{emimo}} &= \mathbf{I}\_{\text{M},\text{M}\_{i}} \otimes \mathbf{A} = \begin{bmatrix} \mathbf{A} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{A} & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \mathbf{A} \end{bmatrix}\_{\text{pM}\_{i}\text{M}\_{i}\times\text{pM}\_{i}\text{M}\_{i}} \\ \mathbf{A}^{\text{emino}} &= (\mathbf{I}\_{\text{M},\text{M}\_{i}} \otimes \mathbf{B})\boldsymbol{\Phi}\_{\text{M},\text{M}\_{i}} = \begin{bmatrix} \mathbf{B} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{B} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{B} & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \mathbf{B} \end{bmatrix} \boldsymbol{\Phi}\_{\text{M},\text{M}\_{i}} \\ \boldsymbol{\Phi}\_{\text{M}} &= \begin{bmatrix} \mathbf{C} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{C} & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \mathbf{C} \end{bmatrix}\_{\text{M}\_{i}\text{M}\_{i}\times\text{pM}\_{i}\text{M}\_{i}} \end{aligned} \tag{39}$$

where **Φ**MrMt is defined as a lower triangular matrix, which fulfills the condition,

$$\mathbf{^0 \Phi\_{\rm M\_tM\_t} \Phi\_{\rm M\_rM\_t}^H} = \mathbf{R\_{MIMO}} = \mathbf{R\_{BS}} \otimes \mathbf{R\_{MS}} \tag{40}$$

Similarly, the Cholesky decomposition method can be used to solve Eq. (40) numerically. However, for a small size matrix **Φ**MrMt , like a 2 � 2 MIMO channel model, an analytical solution of a lower triangular matrix **Φ**<sup>4</sup> is obtained [13, 19].

### **7. MIMO-OFDM channel model**

The demand for multimedia services requires high data rates for communications. However, in a single-carrier modulation system, this is limited by inter-symbol interference, which occurs due to time dispersion of channel caused by multi-path propagation [23, 24]. A multi-carrier modulation technique, OFDM, is proposed to overcome this problem. That is, OFDM is employed to the channels that exhibit a time delay spread, or equivalently, have the characteristic of frequency selectivity.

Notice that the MIMO channel model presented earlier is used for narrow-band and single-carrier frequency. In this section, as a promising strategy, a combination of MIMO and OFDM technology is proposed to deal with the frequency-selective fading channels, i.e., a wide-band MIMO channel model and a MIMO-OFDM channel model.

To this end, the so-called time delay factor is introduced to describe the delay spread due to the two-dimensional (2D) scattering clusters, which will focus on the spatial-temporal-spectral correlation properties of the channel and not only on the spatial-temporal correlation characteristics of the channel.

#### **7.1 Spectral correlation matrix**

Let us define the elements of the channel spectral correlation matrix below,

$$r\_f(d\_f) = \int\_{\mathbb{T}} f\_{\mathfrak{r}}(\mathfrak{r}) e^{-j2\pi d\_f \mathfrak{r}} d\mathfrak{r} \tag{41}$$

then the spectral correlation matrix of size *Mf* � *Mf* can be represented by the sequence *rf mf* � �,

$$\mathbf{C}\_{f} = \begin{bmatrix} r\_{f}[\mathbf{0}] & r\_{f}[\mathbf{1}] & \cdots & r\_{f}[\mathbf{M}\_{f} - \mathbf{1}] \\ r\_{f}^{\*}[\mathbf{1}] & r\_{f}[\mathbf{0}] & \cdots & r\_{f}[\mathbf{M}\_{f} - \mathbf{2}] \\ \vdots & \vdots & \ddots & \vdots \\ r\_{f}^{\*}[\mathbf{M}\_{f} - \mathbf{1}] & r\_{f}^{\*}[\mathbf{M}\_{f} - \mathbf{2}] & \cdots & r\_{f}[\mathbf{0}] \end{bmatrix} \tag{42}$$

where the diagonal element *rf* ½ �¼ <sup>0</sup> <sup>Ð</sup> *<sup>τ</sup> f <sup>τ</sup>*ð Þ*τ dτ* ¼ 1*:* The above matrix will be used to derive the coloring matrix for the MIMO-OFDM channels.

#### **7.2 Building a MIMO-OFDM channel model**

Similarly, based on the MIMO channel model block, a MIMO-OFDM channel model can be constructed. This time, however, a colored input noise vector for the MIMO-OFDM channels is generated using the spectral correlation matrix **C***<sup>f</sup>* .

The vector **h** *t*, *f* <sup>0</sup> � �*<sup>T</sup>* **<sup>h</sup>**ð*t*, *<sup>f</sup>* <sup>1</sup>Þ*<sup>T</sup>* <sup>⋯</sup> **<sup>h</sup>** *<sup>t</sup>*, *<sup>f</sup> Mf* �<sup>1</sup>Þ*<sup>T</sup>* � i*<sup>T</sup>* � is used to represent all of the MIMO-OFDM channels. This results in the channels that are characterized by a spatial-temporal-spectral correlation function.

In **Figure 9**, **h**[*k*, *i*] is a discretized representation of the continuous-time channel vector **h** *k*Δ*t*, *fi* � �, each dotted box represents a MIMO channel model, which includes a total of *MrMt* state-space SISO channel blocks and one spatial correlation matrix **Φ**MrMt *:* Moreover, each block involves a single-carrier frequency, and this parallel

**Figure 9.** *Block diagram of the MIMO-OFDM channel model.*

connection will generate *Mf* frequency-selective channels. In addition, the block diagram **D** is a square matrix of order *Mf* obtained from the spectral correlation matrix **C***<sup>f</sup>* in Eq. (42). This matrix is employed to adjust the spectral correlation properties between the MIMO channel blocks.

Mathematically, this state-space MIMO-OFDM channel model can be represented by

$$\begin{array}{rcl} \mathbf{x}\_{k+1} &=& \boldsymbol{\Gamma} \mathbf{x}\_{k} + \boldsymbol{\Psi} \mathbf{w}\_{k} \\ \mathbf{h}\_{k} &=& \boldsymbol{\Omega} \mathbf{x}\_{k} \end{array} \tag{43}$$

where **<sup>h</sup>**½ � *<sup>k</sup>*, 0 *<sup>T</sup>* **<sup>h</sup>**½*k*, 1� *<sup>T</sup>* <sup>⋯</sup> **<sup>h</sup>** *<sup>k</sup>*, *Mf* � <sup>1</sup>� *<sup>T</sup>* � �*<sup>T</sup>* ∈ MfMrMt h is denoted by **h***k*, **x***<sup>k</sup>* ∈ pMfMrMt , **w***<sup>k</sup>* ∈ MfMrMt , **Γ** is a complex square matrix of order *pMfMrMt*, **Ψ** is a *pMfMrMt* � *MfMrMt* complex matrix, and **Ω** is a *MfMrMt* by *pMfMrMt* real matrix. The matrices **Γ**, **Ψ**, and **Ω** are given by

$$\mathbf{I} = \mathbf{I}\_{\mathcal{M}\_f} \otimes \mathbf{I}^{mimo}, \Psi = \mathbf{D} \otimes \Psi^{mimo}, \mathbf{\Omega} = \mathbf{I}\_{\mathcal{M}\_f} \otimes \mathbf{\Omega}^{mimo} \tag{44}$$

where *p* is the order of the AR model, **Γ***mimo*, **Ψ***mimo*, and **Ω***mimo* are given by Eq. (39), and **D** is defined as a lower triangular matrix that satisfies,

$$\mathbf{D} \mathbf{D}^H = \mathbf{C}\_{\circ} \tag{45}$$

As mentioned earlier, the Cholesky decomposition method can also be used to obtain all of the elements of the lower triangular matrix **D**. However, in the case of two sub-carriers, simple algebra will result in the following closed-form solution.

$$\mathbf{D} = \begin{bmatrix} \mathbf{1} & \mathbf{0} \\ r\_f^\* \begin{bmatrix} m\_f \end{bmatrix} & \sqrt{\mathbf{1} - \left| r\_f \begin{bmatrix} m\_f \end{bmatrix} \right|^2} \end{bmatrix} \tag{46}$$

Therefore, given a DPDF, the corresponding spatial-temporal-spectral correlation function can be obtained. This will be presented next.

#### **7.3 Cauchy delay PDF**

Similarly, given a distant Cauchy-Rayleigh cluster, the delay PDF of TOA is approximately equal to Cauchy [25],

$$f\_{\tau}^{\epsilon}(\tau) = f\_{\tau}(\tau) \approx \frac{1}{\pi} \frac{\eta}{\eta^2 + (\tau - \overline{\tau})^2} \tag{47}$$

where *τ* denotes the average time delay,

$$\overline{\tau} = \frac{d\_{\rm OB\_1} + d\_{\rm OM\_1}}{\upsilon\_{\rm c}}, \qquad \eta = \frac{\zeta \sqrt{2 + 2 \cos(\theta\_0)}}{\upsilon\_{\rm c}}, \qquad \cos(\theta\_0) = \frac{d\_{\rm OB\_1}^2 + d\_{\rm OM\_1}^2 - d\_{\rm B\_1M\_1}^2}{2 d\_{\rm OB\_1} d\_{\rm OM\_1}} \tag{48}$$

and *vc* is the speed of light, *θ*<sup>0</sup> is the angle between the two edges B1O and OM1, as illustrated in **Figure 4**. Notice that the time delay *τ* is a non-negative variable. Hence, Eq. (47) is valid only if the main area under the curve is in the positive direction of the delay axis. In other words, the area under the tail in the negative direction of the delay axis is small and can be ignored.

Since *ζ* ¼ *α*90%*d*OM1*=*6*:*3138, from Eq. (48), we get,

$$\eta = \frac{a\_{90\text{\#}}\sqrt{2 + 2\cos(\theta\_0)}}{6.3138} \frac{d\_{\text{OM}\_1}}{v\_c} \tag{49}$$

Notice that the ratio of *d*OM1 and *vc* is very small, the width of the delay power distribution function *η* will be a very small value. Therefore, the integration of Eq. (47) will be approximately equal to 1 over the interval 0, *τϵ* ½ �, and it can thereby be extended to 0, ½ Þ ∞ *:* Here, *τϵ* ≫ *τ*max is a number and *τ*max denotes the maximum delay.

Adding the spectrum *df* to the expression, the following equation is obtained by substituting *f c <sup>α</sup>*ð Þ *α* , *f c <sup>β</sup>*ð Þ *β* . and *f c <sup>τ</sup>*ð Þ*τ* into Eq. (15),

$$\begin{split} \overline{\mathbf{C}}\_{h}^{\epsilon}(\Delta t\_{\kappa}, d\_{t}, d\_{r}, d\_{f}) &\quad \approx \quad \overline{\mathbf{C}}\_{h}^{\epsilon\_{\kappa}}(\Delta t\_{\kappa}, d\_{t}, d\_{r}, d\_{f}) \\ &= \quad \overline{\mathbf{R}}\_{h}^{\epsilon\_{\kappa}}(\Delta t\_{\kappa}) r\_{m,n}^{\epsilon, \operatorname{BS}}(d\_{t}) r\_{i,j}^{\operatorname{MS}}(d\_{r}) r\_{f}^{\epsilon}(d\_{f}) \end{split} \tag{50}$$

where *R*~*<sup>c</sup><sup>κ</sup> <sup>h</sup>* ð Þ <sup>Δ</sup>*t<sup>κ</sup>* is the channel dynamics given in Eq. (24), *<sup>r</sup><sup>c</sup>*,BS *<sup>m</sup>*,*<sup>n</sup>* ð Þ *dt* ,*r*MS *<sup>i</sup>*,*<sup>j</sup>* ð Þ *dr* are spacing correlations given in Eqs. (25) and (26), respectively, and *rc <sup>f</sup> df* � � is the spectral correlation given by

$$r\_f^{\varepsilon}(d\_f) = e^{-2\pi\eta d\_f}e^{-j2\pi d\_f \overline{\tau}} \tag{51}$$

#### **7.4 Gaussian delay PDF**

Given a distant Rayleigh cluster, the approximate Gaussian DPDF of TOA is obtained [26],

$$f\_{\tau}^{\tau}(\tau) = f\_{\tau}(\tau) \approx \frac{1}{\sqrt{2\pi}\sigma\_0} e^{-\frac{(\tau - \overline{\tau})^2}{2\sigma\_0^2}} \tag{52}$$

where

$$
\sigma\_0 = \frac{\sigma \sqrt{2 + 2 \cos(\theta\_0)}}{v\_c} \tag{53}
$$

and cosð Þ *θ*<sup>0</sup> and *τ* are defined in Eq. (48). Therefore, for Gaussian distributed TOA, we have,

$$\begin{split} \overline{\mathbf{C}}\_{h}^{\boldsymbol{\kappa}}(\Delta t\_{\kappa}, d\_{t}, d\_{r}, d\_{f}) &\quad \approx \quad \overline{\mathbf{C}}\_{h}^{r\_{\kappa}}(\Delta t\_{\kappa}, d\_{t}, d\_{r}, d\_{f}) \\ &= \quad \quad \quad \overline{\boldsymbol{R}}\_{h}^{r\_{\kappa}}(\Delta t\_{\kappa}) r\_{m,n}^{r, \operatorname{BS}}(d\_{t}) r\_{i,j}^{\operatorname{MS}}(d\_{r}) r\_{f}^{r}(d\_{f}) \end{split} \tag{54}$$

where *R*~*<sup>r</sup><sup>κ</sup> <sup>h</sup>* ð Þ <sup>Δ</sup>*t<sup>κ</sup>* is defined in Eq. (29), *<sup>r</sup><sup>r</sup>*,BS *<sup>m</sup>*,*<sup>n</sup>* ð Þ *dt* ,*r*MS *<sup>i</sup>*,*<sup>j</sup>* ð Þ *dr* are spacing correlations defined in Eqs. (30) and (26), respectively, and *r<sup>r</sup> <sup>f</sup> df* � � is the spectral correlation given by

$$r\_f^r(d\_f) = e^{-2\pi^2 \sigma\_0^2 d\_f^2} e^{-j2\pi \overline{\pi} d\_f} \tag{55}$$

Thus, an AR(p)-based state-space MIMO-OFDM channel model has been constructed. However, this approach is only applicable to a single scattering cluster. Next, the method for constructing a multi-cluster MIMO-OFDM channel model is described.
