**3.1 Planar massive MIMO**

From an architectural point of view, a massive MIMO is structured depending on the geometry pattern that is able to form. There exist several design configurations that usually are function of the kind of application to which these systems are destined. Anyway, in this chapter, we consider three different types of planar antenna arrays: the uniform rectangular planar array (URPA), the hexagonal planar array (HPA), and the circular planar array (CPA). Substantially, the term uniform means that the weight parameters *w*1,*w*2,.…,*wM* are all unity, thus it cannot be readjusted as mentioned earlier in Section 2, **Figure 2**. The following subsections synthesize the main feature of the mentioned configurations.

### *3.1.1 Massive MIMO URPA*

The uniform rectangular planar array technology is the most simple planar massive MIMO configuration presenting a 2D (two-dimensional) element plane disposition. The geometry pattern, in this case, can be considered as a two-dimensional matrix within which the antenna elements are placed.

**Figure 4** illustrates an example of massive MIMO URPA configuration. Basically, a URPA is a two-dimensional matrix filled with a certain number of antenna elements (the circles in the figure) both along the x- and y-axis; these antenna elements are equally spaced between any successive pair of elements and this spacing is usually expressed in wavelengths. If we denote the number of elements placed on the x-axis with *M* (the rows of the matrix) and with *N* the number of antennas lying in the y-axis (the columns of the matrix), the total number of elements of the URPA is given by [22]:

$$NumElem \quad = \ M \times N\tag{1}$$

where *M* and *N* are arbitrary integers typically higher than 1. In the first versions of the URPA, *M* and *N* were identical and limited to 8; in the modern application, *M* and *N* are commonly different and chosen between 8 and 12. In general, the radiation field formed by the antenna elements (known also with the term *element factor*) is expressed as:

$$E\_m(r, \Theta, \phi) \quad = \ A \times f(\Theta, \phi) \frac{e^{-\phi r}}{r} \tag{2}$$

In Eq. (2), *A* is the nominal field amplitude, *f*(*θ*,*ϕ*) is the radiation field pattern of the element, and r is the radial distance between the element and the observation point, which highlights the decrease of the field in function of the distance. According to the pattern multiplication principle, the antenna array total electrical field can be expressed as:

$$E\_{TOT} = \ \ \ E\_m \times AF(\theta, \phi) \tag{3}$$

**93**

*Smart Antenna Systems Model Simulation Design for 5G Wireless Network Systems*

<sup>ψ</sup>*<sup>M</sup>* <sup>=</sup> *kd* sin<sup>θ</sup> cos<sup>ϕ</sup> <sup>+</sup> <sup>β</sup>*M*, <sup>ψ</sup>*<sup>N</sup>* <sup>=</sup> *kd* sin<sup>θ</sup> sin<sup>ϕ</sup> <sup>+</sup> <sup>β</sup>*<sup>N</sup>* <sup>β</sup>*<sup>M</sup>* <sup>=</sup> <sup>−</sup>*kd* sinθ<sup>0</sup> cosϕ0, <sup>β</sup>*<sup>N</sup>* <sup>=</sup> <sup>−</sup>*kd* sinθ<sup>0</sup> sinϕ<sup>0</sup>

*<sup>G</sup>*(θ,ϕ) <sup>=</sup> <sup>4</sup>*π*|*f*(θ,ϕ) *AF*(θ,ϕ))|

isotropic, we have *f*(*θ,ϕ*) = 1 and the gain becomes [23]:

mum gain in case of isotropic antenna elements [23]:

uniformly distributed in the hexagonal side (**Figure 5**).

array factor can be expressed as the following expressions [24]:

2

*AFUHPA* = ∑

*<sup>G</sup>*(θ,ϕ) <sup>=</sup> *<sup>D</sup>*(θ,ϕ) <sup>=</sup> <sup>4</sup>*π*|*AF*(θ,ϕ))|

*GMAX*(θ,ϕ) <sup>=</sup> <sup>4</sup>*<sup>π</sup>* <sup>×</sup> <sup>|</sup>*AFMAX*(θ,ϕ))|

The terms *ψM* and *ψN* indicate the array phase along the x- and y-axis, respectively, while the terms *βM* and *βN* denote the scanning steering factors along x and y in function of the steering angle; finally, *ϕ0* is the elevation angle relative to the steering angle *θ0*<sup>φ</sup>0. Please observe that the array factor expression related to Eq. (6) is not normalized with respect to *M* and *N*. The overall gain of the URPA is

2

2

2

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ ∫ϕ=0 <sup>2</sup>*<sup>π</sup>* ∫θ=0

*<sup>π</sup>* |*AF*(θ,ϕ))|

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ ∫ϕ=0 <sup>2</sup>*<sup>π</sup>* ∫θ=0

is the square modulus related to the maximum value of the

*<sup>π</sup>* |*AFMAX*(θ,ϕ))|

An HPA configuration usually consists of *M* hexagonal rings, each one having a total number of *6m*, where *m* is the *mth* ring of the system; the antenna elements are

In case of isotropic elements, because the excitation amplitude is set to 1, the

*e<sup>j</sup>*[*mvy*−\_\_ *N* <sup>2</sup> *vx*−\_\_ *m* <sup>2</sup> *vx*] ∑ *n*=0 *N*

*N* = 2*M* − |*m*|; *vx* = sinθ cosϕ; *vx* = sinθ sinϕ (10)

Note that in Eq. (9), the dependence on *θ* and *ϕ* is omitted and furthermore the steering factor for beam scanning is not considered, while vx and vy denote the planar vectorial components along the x- and y-axis, respectively. The maximum theoretical gain is the same of the URPA case, except from the array factor term.

*m*=−*M M*

<sup>2</sup> sin<sup>θ</sup> *<sup>d</sup>d* (6)

<sup>2</sup> sin<sup>θ</sup> *<sup>d</sup>d* (7)

<sup>2</sup> sin<sup>θ</sup> *<sup>d</sup>d* (8)

*e<sup>j</sup>nvx* (9)

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ ∫ϕ=0 <sup>2</sup>*<sup>π</sup>* ∫θ=0

*<sup>π</sup>* |*f*(θ,ϕ) *AF*(θ,ϕ))|

Equation (6) is the generic expression of the gain valid for all antenna types and is function of the element factor and the array factor. If the antenna elements are

Equation (7) also expresses the directivity of the antenna; thus, from antenna array theory, it is possible to obtain the expression which corresponds to the maxi-

(5)

*DOI: http://dx.doi.org/10.5772/intechopen.79933*

expressed by the following [21]:

where |*AFMAX*(θ,ϕ))|

*3.1.2 Massive MIMO HPA*

array factor.

With:

In the case of URPA, the array factor equation is very similar to the ULA with the only difference that is designed by considering two dimensions [15, 22]:

$$AF\_{URPA}\{\Theta,\Phi\}\quad = \left[\frac{\sin\left(\frac{M\Psi\Psi}{2}\right)}{\sin\left(\frac{\Psi\Psi}{2}\right)}\right]\left[\frac{\sin\left(\frac{N\Psi\Psi}{2}\right)}{\sin\left(\frac{\Psi\Psi}{2}\right)}\right] \tag{4}$$

**Figure 4.** *Massive MIMO URPA example.*

*Smart Antenna Systems Model Simulation Design for 5G Wireless Network Systems DOI: http://dx.doi.org/10.5772/intechopen.79933*

With:

*Array Pattern Optimization*

*3.1.1 Massive MIMO URPA*

elements of the URPA is given by [22]:

*factor*) is expressed as:

field can be expressed as:

*AFURPA*(θ,ϕ) <sup>=</sup> [

The uniform rectangular planar array technology is the most simple planar massive MIMO configuration presenting a 2D (two-dimensional) element plane disposition. The geometry pattern, in this case, can be considered as a two-dimen-

**Figure 4** illustrates an example of massive MIMO URPA configuration. Basically, a URPA is a two-dimensional matrix filled with a certain number of antenna elements (the circles in the figure) both along the x- and y-axis; these antenna elements are equally spaced between any successive pair of elements and this spacing is usually expressed in wavelengths. If we denote the number of elements placed on the x-axis with *M* (the rows of the matrix) and with *N* the number of antennas lying in the y-axis (the columns of the matrix), the total number of

*NumElem* = *M* × *N* (1)

*Em*(*r*, <sup>θ</sup>,ϕ) <sup>=</sup> *<sup>A</sup>* <sup>×</sup> *<sup>f</sup>*(θ,ϕ) *<sup>e</sup>*<sup>−</sup>*jkr* \_\_\_\_ *<sup>r</sup>* (2)

In Eq. (2), *A* is the nominal field amplitude, *f*(*θ*,*ϕ*) is the radiation field pattern of the element, and r is the radial distance between the element and the observation point, which highlights the decrease of the field in function of the distance. According to the pattern multiplication principle, the antenna array total electrical

*ETOT* = *Em* × *AF*(θ,ϕ) (3)

sin( \_\_\_\_ *M*ψ*<sup>M</sup>* <sup>2</sup> ) \_\_\_\_\_\_\_\_ sin( \_\_\_ ψ*<sup>M</sup>* <sup>2</sup> ) ][

only difference that is designed by considering two dimensions [15, 22]:

In the case of URPA, the array factor equation is very similar to the ULA with the

sin( \_\_\_\_ *N*ψ*<sup>N</sup>* <sup>2</sup> ) \_\_\_\_\_\_\_\_ sin( \_\_\_ ψ*<sup>N</sup>*

<sup>2</sup> ) ] (4)

where *M* and *N* are arbitrary integers typically higher than 1. In the first versions of the URPA, *M* and *N* were identical and limited to 8; in the modern application, *M* and *N* are commonly different and chosen between 8 and 12. In general, the radiation field formed by the antenna elements (known also with the term *element* 

sional matrix within which the antenna elements are placed.

**92**

**Figure 4.**

*Massive MIMO URPA example.*

\*\*With:\*\*

$$\begin{array}{llll} \Psi\_M & = & kd\sin\theta\cos\phi + \beta\_M, & \Psi\_N & = kd\sin\theta\sin\phi + \beta\_N\\ \beta\_M & = & -kd\sin\theta\_0\cos\phi\_0, & \beta\_N & = & -kd\sin\theta\_0\sin\phi\_0 \end{array} \tag{5}$$

The terms *ψM* and *ψN* indicate the array phase along the x- and y-axis, respectively, while the terms *βM* and *βN* denote the scanning steering factors along x and y in function of the steering angle; finally, *ϕ0* is the elevation angle relative to the steering angle *θ0*<sup>φ</sup>0. Please observe that the array factor expression related to Eq. (6) is not normalized with respect to *M* and *N*. The overall gain of the URPA is expressed by the following [21]:

\*\*Fessel\*\*
By the following

 $\lfloor \tau \rfloor$ :

$$G(\emptyset, \phi) = \frac{4\pi \|f(\emptyset, \phi) \ A F(\emptyset, \phi)\|^2}{\int\_{\emptyset=0}^{2\pi} \int\_{\emptyset=0}^{\mu} \left| f(\emptyset, \phi) \ A F(\emptyset, \phi) \right|^2 \sin \theta \, d\theta d\phi} \tag{6}$$

Equation (6) is the generic expression of the gain valid for all antenna types and is function of the element factor and the array factor. If the antenna elements are isotropic, we have *f*(*θ,ϕ*) = 1 and the gain becomes [23]:

$$\text{Propic, we have}\\f(\theta,\phi) = \mathbf{1} \text{ and the gain becomes [23]:}\\\text{s.t.}\\\text{ }G(\theta,\phi) = \frac{4\pi |AF(\theta,\phi)|^2}{\int\_{\theta=0}^{2\pi} \int\_{\theta=0}^{\theta} |AF(\theta,\phi)|^2 \sin\theta \,d\theta d\phi} \\ \tag{7}$$

Equation (7) also expresses the directivity of the antenna; thus, from antenna array theory, it is possible to obtain the expression which corresponds to the maximum gain in case of isotropic antenna elements [23]:

 *GMAX*(θ,ϕ) <sup>=</sup> <sup>4</sup>*<sup>π</sup>* <sup>×</sup> <sup>|</sup>*AFMAX*(θ,ϕ))| 2 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ ∫ϕ=0 <sup>2</sup>*<sup>π</sup>* ∫θ=0 *<sup>π</sup>* |*AFMAX*(θ,ϕ))| <sup>2</sup> sin<sup>θ</sup> *<sup>d</sup>d* (8)

where |*AFMAX*(θ,ϕ))| 2 is the square modulus related to the maximum value of the array factor.

#### *3.1.2 Massive MIMO HPA*

An HPA configuration usually consists of *M* hexagonal rings, each one having a total number of *6m*, where *m* is the *mth* ring of the system; the antenna elements are uniformly distributed in the hexagonal side (**Figure 5**).

In case of isotropic elements, because the excitation amplitude is set to 1, the array factor can be expressed as the following expressions [24]:

$$\mathbf{A}F\_{\text{UHPA}} = \sum\_{m=-M}^{M} \mathbf{e}^{j\pi \left[ m v\_j \cdot \frac{N\_v}{2} v\_x \cdot \frac{m}{2} v\_x \right]} \sum\_{n=0}^{N} \mathbf{e}^{j\pi m v\_x} \tag{9}$$

$$N = \ \beth - \!\!\!M - \!\!\!m\text{)}; \quad \nu\_x = \ \sin\theta\cos\phi; \quad \nu\_x = \ \sin\theta\sin\phi \tag{10}$$

Note that in Eq. (9), the dependence on *θ* and *ϕ* is omitted and furthermore the steering factor for beam scanning is not considered, while vx and vy denote the planar vectorial components along the x- and y-axis, respectively. The maximum theoretical gain is the same of the URPA case, except from the array factor term.

**Figure 5.** *Massive MIMO HPA example.*

**Figure 6.** *Massive MIMO CPA example.*
