A. Appendix

Figure 11.

Figure 12.

80

Channel capacities of various antenna arrays.

Configuration of antenna arrays.

Array Pattern Optimization

In a source-free region V filled with a homogeneous medium, the electric field and magnetic field, with assumption of time dependence of exp ð Þ �jωt , satisfy

$$
\nabla \times \nabla \times \mathbf{A}(\mathbf{r}) - k^2 \mathbf{A}(\mathbf{r}) = \mathbf{0}. \tag{21}
$$

A rð Þ is derivable from the scalar potential Ψð Þr which satisfies

$$(\nabla^2 + k^2)\Psi(\mathbf{r}) = \mathbf{0}.\tag{22}$$

Consider vector wave functions M rð Þ, N rð Þ and L rð Þ given by

$$\mathbf{M}(\mathbf{r}) = \nabla \times \mathbf{c} \Psi(\mathbf{r}), \ \mathbf{N}(\mathbf{r}) = \frac{1}{k} \nabla \times \mathbf{M}(\mathbf{r}), \ \mathbf{L}(\mathbf{r}) = \ \Psi(\mathbf{r}), \tag{23}$$

where c is a pilot vector. Here, M rð Þ and N rð Þ are divergence-free and orthogonal. L rð Þ is curl-free and orthogonal with M rð Þ and N rð Þ. Then, an arbitrary electromagnetic field can be described with M rð Þ and N rð Þ and L rð Þ [21].

In spherical coordinates, we can obtain the solution to (22) for unbounded media as

$$
\Psi(k, \mathbf{r}) = \mathbf{z}\_n^{(c)}(kr) Y\_{nm}(\theta, \phi), \tag{24}
$$

where z ð Þc <sup>n</sup> is the radial function and Ynmð Þ θ; ϕ is the complex spherical harmonic function. The radial function z ð Þc <sup>n</sup> is determined by an upper index ð Þc as one of the following functions:

$$z\_n^{(1)}(kr) = j\_n(kr) \tag{8\text{ spherical Bessel function}} \tag{25}$$

E rð Þ¼ ∑ <sup>σ</sup>¼e, <sup>o</sup> ∑ ∞ n¼1 ∑ m m¼0

approximately as

respectively.

Author details

83

Daehee Park\* and Dong-Ho Cho

provided the original work is properly cited.

\*Address all correspondence to: indeener@gmail.com

ð<sup>∞</sup> 0 dk k<sup>2</sup>

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

> 8 < :

<sup>ψ</sup>τσnmð Þ¼ � <sup>θ</sup>; <sup>ϕ</sup> ð Þ<sup>j</sup> <sup>n</sup>þ2�<sup>τ</sup> exp jkr

can be decomposed approximately as

½ � a<sup>σ</sup>nmð Þk M<sup>σ</sup>nmð Þþ k; r b<sup>σ</sup>nmð Þk N<sup>σ</sup>nmð Þþ k; r c<sup>σ</sup>nmð Þk L<sup>σ</sup>nmð Þ k; r :

<sup>n</sup> ð Þ kr Y<sup>σ</sup>nmð Þ θ; ϕ ,

kr <sup>A</sup>τσnmð Þþ ^<sup>r</sup> o kr ð Þ�<sup>1</sup> � �, <sup>τ</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>: (35)

f <sup>α</sup>Aαð Þ^r , (36)

<sup>n</sup> ð Þ kr Y<sup>σ</sup>nmð Þ θ; ϕ ,

From (25) and (32), the outgoing vector waves ψτσnm are expressed as

<sup>∇</sup> � <sup>∇</sup> � <sup>r</sup>hð Þ<sup>1</sup>

These spherical vector waves may become the solutions to the vector wave functions M<sup>σ</sup>nm and N<sup>σ</sup>nm. The outgoing spherical vector waves can be described

The far field is the outgoing spherical vector waves when the r is a very large value relatively to the wavelength. Therefore, the general expression of the far field

ð Þ �<sup>j</sup> <sup>n</sup>þ2�<sup>τ</sup>

where multi-index <sup>α</sup> <sup>¼</sup> <sup>2</sup> n nð Þ� <sup>þ</sup> <sup>1</sup> <sup>1</sup> þ �ð Þ<sup>1</sup> <sup>s</sup> <sup>ð</sup> <sup>m</sup>Þ þ <sup>τ</sup> and <sup>s</sup> <sup>¼</sup> <sup>1</sup>, 2 for <sup>σ</sup> <sup>¼</sup> e, o,

Korea Advanced Institute of Science and Technology (KAIST), Republic of Korea

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

<sup>ψ</sup>1σnmð Þ¼ <sup>θ</sup>; <sup>ϕ</sup> <sup>∇</sup> � <sup>r</sup>hð Þ<sup>1</sup>

Fð Þ¼ ^r ∑ α

k

<sup>ψ</sup>2σnmð Þ¼ <sup>θ</sup>; <sup>ϕ</sup> <sup>1</sup>

(33)

(34)

$$z\_n^{(2)}(kr) = n\_n(kr) \qquad\qquad\text{(spherical Neumann function)}\tag{26}$$

zð Þ<sup>3</sup> <sup>n</sup> ð Þ¼ kr <sup>h</sup>ð Þ<sup>1</sup> <sup>n</sup> ð Þ¼ kr j <sup>n</sup>ð Þþ kr jnnð Þ kr ð Þ spherical Hankel function of the first kind (27)

zð Þ <sup>4</sup> <sup>n</sup> ð Þ¼ kr <sup>h</sup>ð Þ<sup>2</sup> <sup>n</sup> ð Þ¼ kr j <sup>n</sup>ð Þ� kr jnnð Þ kr ð Þ spherical Hankel function of the second kind , (28)

where c ¼ 1 and c ¼ 2 indicate standing waves and c ¼ 3 and c ¼ 4 indicate an outgoing wave and incoming wave, respectively. The complex spherical harmonic function Ynmð Þ θ; ϕ is expressed as

$$Y\_{nm}(\theta,\phi) = \mathcal{C}\_{nm} P\_n^m(\cos\theta) \varepsilon^{jm\phi} = \sqrt{\frac{(n-m)!(2n+1)}{(n+m)!4\pi}} P\_n^m(\cos\theta) \varepsilon^{jm\phi},\tag{29}$$

where P<sup>m</sup> <sup>n</sup> ð Þ cos θ is the associated Legendre polynomial, Cnm is a normalization factor, n ¼ 0, 1, … and m ¼ �n, � n þ 1, :::0; 1, …, n � 1, n [10]. The complex spherical harmonic function Ynmð Þ θ; ϕ can be defined by the real combinations of the azimuth functions as follows [22]:

$$Y\_{\sigma m}(\theta,\phi) = \sqrt{2}C\_{nm}P\_n^m(\cos\theta)\begin{Bmatrix}\cos m\phi\\ \sin m\phi\end{Bmatrix},\tag{30}$$

where m ¼ 0, 1, …, n � 1, n and σ ¼ e, o (even or odd in ϕ).

The normalized vector spherical harmonics Aτσnmð Þ^r can be defined with using the real spherical harmonic function Yσnmð Þ¼ ^r ^rYσnmð Þ θ; ϕ as follows [11]:

$$\begin{cases} \mathbf{A}\_{1\sigma mm}(\hat{\mathbf{r}}) = \frac{\mathbf{1}}{\sqrt{n(n+1)}} \nabla \times \mathbf{r} Y\_{\sigma mm}(\hat{\mathbf{r}}), \\ \mathbf{A}\_{2\sigma mm}(\hat{\mathbf{r}}) = \frac{\mathbf{1}}{\sqrt{n(n+1)}} \frac{1}{k} r \nabla Y\_{\sigma mm}(\hat{\mathbf{r}}), \\ \mathbf{A}\_{3\sigma mm}(\hat{\mathbf{r}}) = \hat{\mathbf{r}} \nabla Y\_{\sigma mm}(\hat{\mathbf{r}}). \end{cases} \tag{31}$$

where n ¼ 0, 1, …, and m ¼ 0, 1, …, n � 1, n and σ ¼ e, o. In this case, (23) may be modified by

$$\mathbf{M}\_{\sigma mm}(k,\mathbf{r}) = \nabla \times \mathbf{r} \mathbf{z}\_n^{(c)}(kr) Y\_{\sigma mm}(\theta,\phi),$$

$$\mathbf{N}\_{\sigma mm}(k,\mathbf{r}) = \frac{1}{k} \nabla \times \nabla \times \mathbf{r} \mathbf{z}\_n^{(c)}(kr) Y\_{\sigma mm}(\theta,\phi), \tag{32}$$

$$\mathbf{L}\_{\sigma mm}(k,\mathbf{r}) = \frac{1}{k} \nabla \mathbf{z}\_n^{(c)}(kr) Y\_{\sigma mm}(\theta,\phi).$$

Because of the orthogonal relationship, furthermore, an arbitrary field E rð Þ can be represented as

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

where z

zð Þ<sup>3</sup>

zð Þ <sup>4</sup>

ð Þc

Array Pattern Optimization

following functions:

zð Þ<sup>1</sup> <sup>n</sup> ð Þ¼ kr j

<sup>n</sup> ð Þ¼ kr <sup>h</sup>ð Þ<sup>1</sup>

zð Þ<sup>2</sup>

<sup>n</sup> ð Þ¼ kr <sup>h</sup>ð Þ<sup>2</sup>

where P<sup>m</sup>

modified by

be represented as

82

function. The radial function z

<sup>n</sup> ð Þ¼ kr j

<sup>n</sup> ð Þ¼ kr j

function Ynmð Þ θ; ϕ is expressed as

the azimuth functions as follows [22]:

<sup>n</sup> ð Þ cos θ e

<sup>Y</sup>σnmð Þ¼ <sup>θ</sup>; <sup>ϕ</sup> ffiffi

where m ¼ 0, 1, …, n � 1, n and σ ¼ e, o (even or odd in ϕ).

A1σnmð Þ¼ ^r

8 >>>>><

>>>>>:

A2σnmð Þ¼ ^r

<sup>M</sup><sup>σ</sup>nmð Þ¼ <sup>k</sup>; <sup>r</sup> <sup>∇</sup> � <sup>r</sup>zð Þ<sup>c</sup>

1 k

L<sup>σ</sup>nmð Þ¼ k; r

N<sup>σ</sup>nmð Þ¼ k; r

Ynmð Þ¼ <sup>θ</sup>; <sup>ϕ</sup> CnmPm

<sup>n</sup> is the radial function and Ynmð Þ θ; ϕ is the complex spherical harmonic

<sup>n</sup> ð Þ¼ kr nnð Þ kr ð Þ spherical Neumann function (26)

where c ¼ 1 and c ¼ 2 indicate standing waves and c ¼ 3 and c ¼ 4 indicate an outgoing wave and incoming wave, respectively. The complex spherical harmonic

s

factor, n ¼ 0, 1, … and m ¼ �n, � n þ 1, :::0; 1, …, n � 1, n [10]. The complex spherical harmonic function Ynmð Þ θ; ϕ can be defined by the real combinations of

> 2 <sup>p</sup> CnmPm

the real spherical harmonic function Yσnmð Þ¼ ^r ^rYσnmð Þ θ; ϕ as follows [11]:

A3σnmð Þ¼ ^r ^r∇Y<sup>σ</sup>nmð Þ^r :

<sup>n</sup> ð Þ cos θ is the associated Legendre polynomial, Cnm is a normalization

<sup>n</sup> ð Þ cos θ

n nð Þ <sup>þ</sup> <sup>1</sup> <sup>p</sup> <sup>∇</sup> � <sup>r</sup>Yσnmð Þ^<sup>r</sup> ,

<sup>n</sup> ð Þ kr Y<sup>σ</sup>nmð Þ θ; ϕ ,

r∇Y<sup>σ</sup>nmð Þ^r ,

<sup>n</sup> ð Þ kr Y<sup>σ</sup>nmð Þ θ; ϕ ,

<sup>n</sup> ð Þ kr Y<sup>σ</sup>nmð Þ θ; ϕ :

1 k

The normalized vector spherical harmonics Aτσnmð Þ^r can be defined with using

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n nð Þ <sup>þ</sup> <sup>1</sup> <sup>p</sup>

where n ¼ 0, 1, …, and m ¼ 0, 1, …, n � 1, n and σ ¼ e, o. In this case, (23) may be

<sup>∇</sup> � <sup>∇</sup> � <sup>r</sup>zð Þ<sup>c</sup>

1 k ∇zð Þ<sup>c</sup>

Because of the orthogonal relationship, furthermore, an arbitrary field E rð Þ can

jm<sup>ϕ</sup> <sup>¼</sup>

<sup>n</sup> is determined by an upper index ð Þc as one of the

(27)

(28)

jmϕ, (29)

, (30)

(31)

(32)

<sup>n</sup>ð Þ kr ð Þ spherical Bessel function (25)

<sup>n</sup>ð Þþ kr jnnð Þ kr ð Þ spherical Hankel function of the first kind

<sup>n</sup>ð Þ� kr jnnð Þ kr ð Þ spherical Hankel function of the second kind ,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ n � m !ð Þ 2n þ 1 ð Þ n þ m !4π

Pm

cos mϕ sin mϕ � �

<sup>n</sup> ð Þ cos θ e

ð Þc

$$\mathbf{E}(\mathbf{r}) = \sum\_{\sigma=\epsilon\_{2}} \sum\_{n=1}^{\infty} \sum\_{m=0}^{m} \int\_{0}^{\infty} dk \, k^{2} \left[ a\_{\sigma m}(k) \mathbf{M}\_{\sigma mn}(k, \mathbf{r}) + b\_{\sigma m}(k) \mathbf{N}\_{\sigma mn}(k, \mathbf{r}) + c\_{\sigma m}(k) \mathbf{L}\_{\sigma mn}(k, \mathbf{r}) \right]. \tag{33}$$

From (25) and (32), the outgoing vector waves ψτσnm are expressed as

$$\begin{cases} \boldsymbol{\Psi}\_{1\sigma m}(\boldsymbol{\theta},\boldsymbol{\phi}) = \nabla \times \mathbf{r} h\_{n}^{(1)}(kr) Y\_{\sigma m n}(\boldsymbol{\theta},\boldsymbol{\phi}),\\ \boldsymbol{\Psi}\_{2\sigma m n}(\boldsymbol{\theta},\boldsymbol{\phi}) = \frac{1}{k} \nabla \times \nabla \times \mathbf{r} h\_{n}^{(1)}(kr) Y\_{\sigma m n}(\boldsymbol{\theta},\boldsymbol{\phi}), \end{cases} \tag{34}$$

These spherical vector waves may become the solutions to the vector wave functions M<sup>σ</sup>nm and N<sup>σ</sup>nm. The outgoing spherical vector waves can be described approximately as

$$\Psi\_{\mathbf{r}\sigma\mathbf{m}}(\theta,\phi) = (-j)^{n+2-\tau} \frac{\exp jkr}{kr} \mathbf{A}\_{\sigma\mathbf{m}m}(\hat{\mathbf{r}}) + o\left((kr)^{-1}\right), \ \tau = 1,2. \tag{35}$$

The far field is the outgoing spherical vector waves when the r is a very large value relatively to the wavelength. Therefore, the general expression of the far field can be decomposed approximately as

$$F(\hat{\mathbf{r}}) = \sum\_{a} (-j)^{n+2-\tau} f\_{a} \mathbf{A}\_{a}(\hat{\mathbf{r}}),\tag{36}$$

where multi-index <sup>α</sup> <sup>¼</sup> <sup>2</sup> n nð Þ� <sup>þ</sup> <sup>1</sup> <sup>1</sup> þ �ð Þ<sup>1</sup> <sup>s</sup> <sup>ð</sup> <sup>m</sup>Þ þ <sup>τ</sup> and <sup>s</sup> <sup>¼</sup> <sup>1</sup>, 2 for <sup>σ</sup> <sup>¼</sup> e, o, respectively.
