2.1 MIMO channel model

It has been studied to enhance the spectral efficiency without increasing antenna space by using multiple radiation patterns of the antennas [4–7]. It was theoretically proposed that six times of spectral efficiency can be obtained compared to a single antenna by using three electric dipoles and three magnetic dipoles allocated at the same position. It means that the multiple antennas can be integrated within a compact size and provide multiplexing gain. Thus, the MIMO system can increase the spectral efficiency by not only using larger antenna array size but also combin-

On the other hand, there have been studies of array optimization technologies for the linear antenna array according to various objective functions [8, 9]. This antenna array could have better interference cancelation using genetic algorithms. Then, it is also possible to enhance the spectral efficiency and diversity gain by combining the array optimization technologies and the integrated antenna-based

In this chapter, we introduce an integrated antenna array which is a type of the MIMO systems. The integrated antenna array consists of multiple array elements, and the array element has multiple antenna elements. Each antenna element of the integrated antenna has different radiation patterns to increase the spectral efficiency in wireless communication area. The purpose of this chapter is to introduce the concept of the integrated antenna array and to show the possibility to apply it practically to wireless communication technology. We will also explain a framework for next-generation technology so that we can provide further works as well. We organize the chapter as follows. In Section 2, we explain channel models for the integrated antenna array. In Section 3, several practical integrated antennas are introduced to verify that the integrated antenna array can be implemented practically. In Section 4, we verify the performance of the integrated antenna array in urban macro-cell environment compared with mono-polarization and dual-

The channel model is dependent on the structure of the MIMO system. To provide the channel model for the integrated antenna array, firstly we provide a channel model for a general case as given in Figure 1 [1]. We develop the channel

ing multiple radiation patterns.

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polarization dipole antenna arrays.

model with considering complex structures.

2. Channel model

Figure 1.

70

Configuration of general MIMO system.

array system.

We consider a MIMO channel with a transmit and receive system, which is equipped with Nt antennas and Nr antennas, respectively. The received signal y can be expressed as

$$\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n},\tag{1}$$

where H is the Nr � Nt MIMO channel, x is an Nt � 1 transmit signal and n is an Nr � 1 complex Gaussian noise. Based on spatial fading correlation, the elements of H can be described as

$$\mathbf{h} = \mathbf{vec}(\mathbf{H}) = \mathbf{R\_h^{1/2}} \mathbf{vec}(\mathbf{H}\_w), \tag{2}$$

where H<sup>w</sup> is an Nr � Nt Rayleigh fading channel with uncorrelated and zero mean entries which follow complex Gaussian distribution, Rh <sup>¼</sup> <sup>E</sup> hh† � � is the NrNt � NrNt covariance matrix and ð Þ� † and <sup>E</sup>f g� are Hermitian and expectation operator, respectively [1]. Here, vecð Þ A is a vectorization operation to produce a vector with the columns of A.

We assume that the sizes of the transmit and receive antenna are negligible compared to the distance between the transmit system and the receive system. Then, the covariance matrix of the MIMO channel is given by

$$\mathbf{R\_h} = \mathbf{R'\_{H\_t}} \otimes \mathbf{R\_{H\_r}} \tag{3}$$

where <sup>R</sup>H<sup>r</sup> <sup>¼</sup> <sup>E</sup> HH† � �, <sup>R</sup>H<sup>t</sup> <sup>¼</sup> <sup>E</sup> <sup>H</sup>†<sup>H</sup> � � and ð Þ� 0 and ⊗ are matrix transpose and Kronecker product operator, respectively. Therefore, the MIMO channel can be described as [1]

$$\mathbf{H} = \mathbf{R}\_{\mathbf{H}\_{\mathbf{r}}}^{1/2} \mathbf{H}\_{w} \mathbf{R}\_{\mathbf{H}\_{\mathbf{t}}}^{1/2}. \tag{4}$$

This channel model is called by the Kronecker model of MIMO system in general case.

## 2.2 Spherical vector wave-based channel model

There are spherical vector wave (SVW) modes which are orthonormal basis functions for arbitrary radiation pattern [10–12]. The radiation pattern of an arbitrary transmitter antenna is described as

$$\begin{split} \mathbf{F}(\hat{\mathbf{r}}) &= k \sqrt{2\eta} \sum\_{\iota=1}^{l\_{\max}} \sum\_{\iota=0}^{l} \sum\_{\iota=1}^{2} \sum\_{\iota=1}^{2} j^{\iota-\iota+2} T\_{\iota \bullet \mathbf{r}} \mathbf{A}\_{\mathsf{u} \bullet \mathbf{r}}(\hat{\mathbf{r}}) \\ &= k \sqrt{2\eta} \sum\_{a=1}^{a\_{\max}} j^{\tau-\iota+2} T\_a \mathbf{A}\_a(\hat{\mathbf{r}}) = k \sqrt{2\eta} \mathbf{A}'(\hat{\mathbf{r}}) \mathbf{t}, \end{split} \tag{5}$$

where ^<sup>r</sup> is the direction of the radiation, <sup>A</sup>ινs<sup>τ</sup>ð Þ¼ ^<sup>r</sup> <sup>A</sup>αð Þ¼ ^<sup>r</sup> <sup>A</sup>θ,αð Þ^<sup>r</sup> <sup>A</sup>ϕ,αð Þ^<sup>r</sup> � � is <sup>α</sup>th SVW mode with the multi-index <sup>α</sup> <sup>¼</sup> <sup>2</sup> ι ιð Þ� <sup>þ</sup> <sup>1</sup> <sup>1</sup> þ �ð Þ<sup>1</sup> <sup>s</sup> <sup>ð</sup> <sup>ν</sup>Þ þ <sup>τ</sup>, <sup>α</sup>max is the maximum number of SVW modes, Að Þ^r is an αmax � 2 matrix containing the row vector j τ�ιþ2 Aαð Þ^r and t is an αmax � 1 vector with the element T<sup>α</sup> which is the transmitting coefficient of the transmitter antenna [10]. The SVW mode Aαð Þ^r is described well in Appendix. In order to decompose the radiation pattern of a receiver antenna, the receiving coefficient <sup>R</sup>ινs<sup>τ</sup> can be given by <sup>R</sup>ιν1<sup>τ</sup> ¼ �ð Þ<sup>1</sup> <sup>ν</sup> <sup>T</sup>ιν2<sup>τ</sup> and <sup>R</sup>ιν2<sup>τ</sup> ¼ �ð Þ<sup>1</sup> <sup>ν</sup> Tιν1<sup>τ</sup> by using the reciprocity relationship [10, 12].

Consider that the transmit and the receive systems have an integrated antenna without array structure, which are composed of Nt and Nr antennas as shown in Figure 2, respectively. From (1), the MIMO channel H may be expressed as

$$\mathbf{y} = \mathbf{RMTx} + \mathbf{n},\tag{6}$$

It is also assumed that the joint probability density function of the PAS in (10) can

^r � �pαβ <sup>k</sup>

kt

<sup>R</sup>MtT,RH<sup>r</sup> <sup>¼</sup> <sup>R</sup>RMrR†

ð A<sup>∗</sup> k^<sup>r</sup> � �<sup>P</sup> <sup>k</sup>^<sup>r</sup>

� � <sup>¼</sup> <sup>P</sup>αp<sup>α</sup> ^

kt

<sup>y</sup> <sup>¼</sup> <sup>R</sup>�M� Tx� <sup>þ</sup> <sup>n</sup>, (14)

^t

� �: (11)

� �A<sup>0</sup> <sup>k</sup>^<sup>r</sup>

� � and <sup>p</sup><sup>α</sup> ^

� �dΩr: (12)

kt � � is the

: (13)

pαβ k ^r; k ^t � � <sup>¼</sup> <sup>p</sup>αβ <sup>k</sup>

� �A<sup>0</sup> <sup>k</sup>^<sup>t</sup>

kt � �;P<sup>ϕ</sup> ^

<sup>R</sup>H<sup>t</sup> <sup>¼</sup> <sup>T</sup>†

From (3) and (7), therefore, we can see that

2.3 Channel model for integrated antenna array

integrated antenna array is illustrated in Figure 3.

Configuration of MIMO system with integrated antenna array.

integrated antenna arrays is given by

Then, the SVW mode channel of the transmitter is obtained by [14].

� �dΩt, and <sup>R</sup><sup>M</sup><sup>r</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup>

PAS of orthogonal polarization α at transmitter side and α stands for θ and ϕ.

We can describe the channel model for the integrated antenna by using the SVW

For 5G communication technology, it is important that the integrated antenna is expandable to array structure. Thus, it is necessary to derive the channel model for the MIMO system equipped with the integrated antenna array. The structure of the

Each antenna element has not only a radiation pattern but also a relative position to the other antenna elements. The received signal of the MIMO system with the

where <sup>M</sup>� is the MrLr � MtLt extended SVW mode channel, which considers antenna positions; <sup>R</sup>� <sup>¼</sup> <sup>I</sup>Lr⊗R, where <sup>R</sup> is an Br � Mr receiving coefficient matrix of the receive integrated antenna; and <sup>T</sup>� <sup>¼</sup> <sup>I</sup>Lt⊗T, where <sup>T</sup> and <sup>I</sup><sup>N</sup> are a Mt � Bt

kt n o � � , <sup>P</sup><sup>α</sup> ^

be decomposed by

ð A<sup>∗</sup> k^<sup>t</sup> � �<sup>P</sup> <sup>k</sup>^<sup>t</sup>

kt

� � <sup>¼</sup> diag <sup>P</sup><sup>θ</sup> ^

Array Pattern Based on Integrated Antenna DOI: http://dx.doi.org/10.5772/intechopen.81087

R<sup>M</sup><sup>t</sup> ¼ 2π

where P ^

modes.

Figure 3.

73

where R is an Nr � Mr matrix with the receiving coefficient Rnr,mr, T is a Mt � Nt matrix with the transmitting coefficient Tmt,nt and M is a Mr � Mt matrix which is the channel between SVW modes [10, 12, 13]. Then, the covariance matrix of the channel H can be derived by [14]

$$\mathbf{R\_h} = \left(\mathbf{T'} \otimes \mathbf{R}\right) \mathbf{R\_m} \left(\mathbf{T'} \otimes \mathbf{R}\right)^\dagger,\tag{7}$$

where <sup>h</sup> <sup>¼</sup> vecð Þ¼ <sup>H</sup> <sup>T</sup><sup>0</sup> ⊗R � �m, <sup>m</sup> <sup>¼</sup> vecð Þ <sup>M</sup> and <sup>R</sup><sup>m</sup> <sup>¼</sup> <sup>E</sup> mm† f g. The covariance matrix of the channel for the SVW modes can be described as

$$\mathbf{R\_m} = 4\pi^2 \iint \ddot{\mathbf{A}}\left(\hat{\mathbf{k}}\_r, \hat{\mathbf{k}}\_t\right) \mathbf{P}\left(\hat{\mathbf{k}}\_r, \hat{\mathbf{k}}\_t\right) \ddot{\mathbf{A}}\dagger\left(\hat{\mathbf{k}}\_r, \hat{\mathbf{k}}\_t\right) d\Omega\_r d\Omega\_t. \tag{8}$$

Here, A~ k^r; k^<sup>t</sup> � � <sup>¼</sup> <sup>A</sup> <sup>k</sup>^<sup>r</sup> � �⊗<sup>A</sup> <sup>k</sup>^<sup>t</sup> � �, and <sup>P</sup> <sup>k</sup>^r; <sup>k</sup>^<sup>t</sup> � � <sup>¼</sup> diag <sup>P</sup>θθ;Pθϕ;Pϕθ;Pϕϕ � � where <sup>P</sup>αβ � <sup>P</sup>αβ <sup>k</sup>^r; <sup>k</sup>^<sup>t</sup> � � are functions of the power angular spread (PAS) which is expressed by

$$\mathcal{P}\_{a\beta}\left(\hat{\mathbf{k}}\_r, \hat{\mathbf{k}}\_t\right) = P\_{a\beta} p\_{a\beta}\left(\hat{\mathbf{k}}\_r, \hat{\mathbf{k}}\_t\right),\tag{9}$$

where α∈f g θ; ϕ , β∈f g θ; ϕ and Pαβ is coupling power from β to α [15]. The joint probability function pαβ k ^r; k^<sup>t</sup> � � is required to be normalized as follows:

$$\iint p\_{a\beta} \left(\hat{\mathbf{k}}\_r, \hat{\mathbf{k}}\_t\right) d\Omega\_r d\Omega\_t = 1. \tag{10}$$

Figure 2. Configuration of MIMO system with integrated antenna.

receiving coefficient <sup>R</sup>ινs<sup>τ</sup> can be given by <sup>R</sup>ιν1<sup>τ</sup> ¼ �ð Þ<sup>1</sup> <sup>ν</sup>

Consider that the transmit and the receive systems have an integrated antenna without array structure, which are composed of Nt and Nr antennas as shown in Figure 2, respectively. From (1), the MIMO channel H may be expressed as

where R is an Nr � Mr matrix with the receiving coefficient Rnr,mr, T is a Mt � Nt matrix with the transmitting coefficient Tmt,nt and M is a Mr � Mt matrix which is the channel between SVW modes [10, 12, 13]. Then, the covariance matrix of the

> ⊗R � �

R<sup>m</sup> T<sup>0</sup>

P k^r; k^<sup>t</sup> � �

, and P k^r; k^<sup>t</sup> � �

where α∈f g θ; ϕ , β∈f g θ; ϕ and Pαβ is coupling power from β to α [15]. The joint

<sup>p</sup>αβ <sup>k</sup>^r; <sup>k</sup>^<sup>t</sup> � �

¼ Pαβpαβ k

⊗R � �†

<sup>R</sup><sup>h</sup> <sup>¼</sup> <sup>T</sup><sup>0</sup>

matrix of the channel for the SVW modes can be described as

⊗A k^<sup>t</sup> � �

Pαβ k ^r; k^<sup>t</sup> � �

^r; k^<sup>t</sup> � �

ðð

A~ k^r; k^<sup>t</sup> � �

⊗R � �

ðð

<sup>R</sup><sup>m</sup> <sup>¼</sup> <sup>4</sup>π<sup>2</sup>

<sup>¼</sup> <sup>A</sup> <sup>k</sup>^<sup>r</sup> � �

� �

Configuration of MIMO system with integrated antenna.

using the reciprocity relationship [10, 12].

channel H can be derived by [14]

where <sup>h</sup> <sup>¼</sup> vecð Þ¼ <sup>H</sup> <sup>T</sup><sup>0</sup>

Array Pattern Optimization

� �

where <sup>P</sup>αβ � <sup>P</sup>αβ <sup>k</sup>^r; <sup>k</sup>^<sup>t</sup>

probability function pαβ k

Here, A~ k^r; k^<sup>t</sup>

expressed by

Figure 2.

72

<sup>T</sup>ιν2<sup>τ</sup> and <sup>R</sup>ιν2<sup>τ</sup> ¼ �ð Þ<sup>1</sup> <sup>ν</sup>

, (7)

dΩrdΩt: (8)

, (9)

¼ diag Pθθ;Pθϕ;Pϕθ;Pϕϕ � �

dΩrdΩ<sup>t</sup> ¼ 1: (10)

y ¼ RMTx þ n, (6)

<sup>m</sup>, <sup>m</sup> <sup>¼</sup> vecð Þ <sup>M</sup> and <sup>R</sup><sup>m</sup> <sup>¼</sup> <sup>E</sup> mm† f g. The covariance

A~ † k^r; k^<sup>t</sup> � �

are functions of the power angular spread (PAS) which is

^r; k^<sup>t</sup> � �

is required to be normalized as follows:

Tιν1<sup>τ</sup> by

It is also assumed that the joint probability density function of the PAS in (10) can be decomposed by

$$p\_{a\beta}(\hat{\mathbf{k}}\_r, \hat{\mathbf{k}}\_t) = p\_{a\beta}(\hat{\mathbf{k}}\_r) p\_{a\beta}(\hat{\mathbf{k}}\_t) \,. \tag{11}$$

Then, the SVW mode channel of the transmitter is obtained by [14].

$$\mathcal{R}\_{\mathbf{M}\_{\mathbf{t}}} = 2\pi \int \mathbf{A}^\*(\hat{\mathbf{k}}\_t) \mathbf{P}\left(\hat{\mathbf{k}}\_t\right) \mathbf{A}^\prime(\hat{\mathbf{k}}\_t) d\Omega\_t,\text{ and }\; \mathbf{R}\_{\mathbf{M}\_{\mathbf{t}}} = 2\pi \int \mathbf{A}^\*\left(\hat{\mathbf{k}}\_r\right) \mathbf{P}\left(\hat{\mathbf{k}}\_r\right) \mathbf{A}^\prime\left(\hat{\mathbf{k}}\_r\right) d\Omega\_r. \tag{12}$$

where P ^ kt � � <sup>¼</sup> diag <sup>P</sup><sup>θ</sup> ^ kt � �;P<sup>ϕ</sup> ^ kt n o � � , <sup>P</sup><sup>α</sup> ^ kt � � <sup>¼</sup> <sup>P</sup>αp<sup>α</sup> ^ kt � � and <sup>p</sup><sup>α</sup> ^ kt � � is the PAS of orthogonal polarization α at transmitter side and α stands for θ and ϕ.

From (3) and (7), therefore, we can see that

$$\mathbf{R\_{H\_{\ell}}} = \mathbf{T}^{\dagger} \mathbf{R\_{M\_{\ell}}} \mathbf{T}, \mathbf{R\_{H\_{r}}} = \mathbf{R} \mathbf{R\_{M\_{r}}} \mathbf{R^{\dagger}}.\tag{13}$$

We can describe the channel model for the integrated antenna by using the SVW modes.

## 2.3 Channel model for integrated antenna array

For 5G communication technology, it is important that the integrated antenna is expandable to array structure. Thus, it is necessary to derive the channel model for the MIMO system equipped with the integrated antenna array. The structure of the integrated antenna array is illustrated in Figure 3.

Each antenna element has not only a radiation pattern but also a relative position to the other antenna elements. The received signal of the MIMO system with the integrated antenna arrays is given by

$$\mathbf{y} = \breve{\mathbf{R}} \breve{\mathbf{M}} \breve{\mathbf{T}} \mathbf{x} + \mathbf{n},\tag{14}$$

where <sup>M</sup>� is the MrLr � MtLt extended SVW mode channel, which considers antenna positions; <sup>R</sup>� <sup>¼</sup> <sup>I</sup>Lr⊗R, where <sup>R</sup> is an Br � Mr receiving coefficient matrix of the receive integrated antenna; and <sup>T</sup>� <sup>¼</sup> <sup>I</sup>Lt⊗T, where <sup>T</sup> and <sup>I</sup><sup>N</sup> are a Mt � Bt

Configuration of MIMO system with integrated antenna array.

transmitting coefficient matrix of the transmit integrated antenna and an N � N identity matrix. The covariance matrix of the MIMO channel H can be described as

$$\mathbf{R\_{h}} = \left(\check{\mathbf{T}}' \otimes \check{\mathbf{R}}\right) \mathbf{R} \check{\mathbf{m}} \left(\check{\mathbf{T}}' \otimes \check{\mathbf{R}}\right)^{\dagger},\tag{15}$$

direction of the antenna element can be changed according to θrad, because the antenna element radiates perpendicularly to the antenna structure. The parameters are given by L ¼ 70 mm, W ¼ 66 mm, lc ¼ 22 mm, wc ¼ 1:5 mm, ls ¼ 24:5 mm, ws ¼ 1:5 mm, lf ¼ 6:2 mm, wp ¼ 8 mm and lp ¼ 32 mm. It is possible to control the impedance matching by modifying l<sup>7</sup> and w6. The integrated antenna is produced on a CER-10 substrate with a permittivity of 10 and a thickness of 0.64 mm. The

Array Pattern Based on Integrated Antenna DOI: http://dx.doi.org/10.5772/intechopen.81087

Configuration of practical integrated antennas. (a) 6-port integrated antenna and (b) 16-port integrated

Configuration of practical integrated antenna array. (a) Antenna element for 4-port integrated antenna, (b)

4-port integrated antenna and (c) 4-port integrated antenna based 1 � 4 array.

Figure 4.

antenna.

Figure 5.

75

where <sup>h</sup> <sup>¼</sup> <sup>T</sup>� <sup>0</sup> <sup>⊗</sup>R� � �m� , <sup>m</sup>� <sup>¼</sup> vec <sup>M</sup>� � � and <sup>R</sup>m� <sup>¼</sup> <sup>E</sup> <sup>m</sup>� m� † n o. Here, the correlation matrix of the channel of the SVW modes can be given by

$$\mathbf{R\check{m}} = 4\pi^2 \iint \bar{\mathbf{B}}\left(\hat{\mathbf{k}}\_r, \hat{\mathbf{k}}\_t\right) \mathbf{P}\left(\hat{\mathbf{k}}\_r, \hat{\mathbf{k}}\_t\right) \bar{\mathbf{B}}\dagger \left(\hat{\mathbf{k}}\_r, \hat{\mathbf{k}}\_t\right) d\Omega\_r d\Omega\_t,\tag{16}$$

where B~ ^ kr; ^ kt � � <sup>¼</sup> <sup>B</sup> <sup>k</sup>^<sup>r</sup> � �⊗<sup>B</sup> <sup>k</sup>^<sup>t</sup> � �. Here, <sup>B</sup> <sup>k</sup>^<sup>r</sup> � � <sup>¼</sup> <sup>e</sup><sup>r</sup> ^ kr � �⊗<sup>A</sup> <sup>k</sup>^<sup>r</sup> � �, and the <sup>E</sup> <sup>k</sup>^ � � is an array element response matrix with the elements of el k^ � � <sup>¼</sup> ej<sup>β</sup>ð Þ <sup>r</sup>l�r<sup>o</sup> <sup>k</sup>^ , where r<sup>l</sup> is the position of the lth array element and r<sup>o</sup> is the position of the antenna array system.

We explained the channel models for the MIMO system with the integrated antenna arrays mathematically. In the following section, we will verify the performance of the integrated antenna array based on the channel model we discussed.
