2. Sierpiński Fractal

The Sierpiński iteration process of isosceles triangle is shown in **Figure 17** [114]. The coordinates for the next iteration can be obtained by

$$v\_1(\mathbf{x}, \mathbf{y}) = \begin{pmatrix} \frac{1}{2} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} \\ \mathbf{0} & \mathbf{2} \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix} + \begin{pmatrix} \frac{1}{2} \\ \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} \end{pmatrix} \tag{1}$$

$$v\_2(\mathbf{x}, \mathbf{y}) = \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix} + \begin{pmatrix} -\frac{1}{2} \\ \frac{\sqrt{3}}{2} \\ \frac{1}{2} \end{pmatrix} \tag{2}$$

$$w\_3(\mathbf{x}, \mathbf{y}) = \begin{pmatrix} \mathbf{1} \\ \mathbf{2} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix} + \begin{pmatrix} \mathbf{0} \\ \mathbf{0} \end{pmatrix} \tag{3}$$

This iteration process described by formulas (1)–(3) can be summarized as, find the midpoint of each segment and connect them to form four same isosceles triangles, then remove the middle triangle, we will get the final iteration shape.

#### **3.3 Characteristic mode analysis**

The CMA permits the researcher to know the antenna performance at the initial stage without considering the specific excitations so that the user can have an insight into the radiation essence, which is extensively used in the application of mobile handset antenna where the large metal chassis exists [116]. And the MIMO antennas are also extensively investigated for the mobile terminal application, there are a lot of works using the CMA [47, 117, 118].

In [47], a meandered dipole working at 3.5 GHz was studied by using the CMA. The first 10 modes are compared and the symmetrical reverse modal currents are

**Figure 17.** *Iteration process of Sierpiński fractal [114]: left is initiator and right is the 1st iteration.*

**Figure 18.** *First four modes at 2.2 GHz of [118].*

excited to get the low SAR, and the isolation of 16 dB is achieved. On the contrary, the CMA is implemented on both the ground plane and the antenna in [117], but only on the chassis ground plane in [118]. The first four modes of [118] are shown in **Figure 18**. With the help of the current distributions, the antennas are put intensityweak position of the corners for the MIMO antenna design.

Depending on the extensive application in MIMO antenna designs, we simply introduce the corresponding theory of the electric-field boundary conditions as follows [116, 117].

The related scattering field expression satisfy:

$$[L(J)]\_{\text{tan}} = E^i\_{\text{tan}}(r), r \in \mathbb{S} \tag{4}$$

where *L*ð Þ∙ is the intetro-differential operator.

Owing to *L*ð Þ∙ features with the impedance property, thus the formula (4) is rewritten as:

$$Z(\mathbf{J}) = [L(\mathbf{J})]\_{\text{tan}} \tag{5}$$

where *Z*ð Þ∙ represent the tangential component of electric field related with *J*.

The impedance matrix *Z* has the real and imaginary parts, we get the following related formulas:

$$\mathbf{Z} = \mathbf{R} + j\mathbf{X} \tag{6}$$

$$\mathbf{R} = (\mathbf{Z} + \mathbf{Z}^\*)/2\tag{7}$$

$$\mathbf{X} = (\mathbf{Z} - \mathbf{Z}^\*) / 2\mathbf{j} \tag{8}$$

By means of Poynting's theorem, we get the straightforward relation for *R* and *X*, which has clear physical meaning, as

$$\mathbf{X} \mathbf{J}\_n = \lambda\_n \mathbf{R} \mathbf{J}\_n \tag{9}$$

where *J<sup>n</sup>* and *λ<sup>n</sup>* are the real eigenvector and eigenvalue of *n*th mode, respectively. The orthogonality of modal currents is defined by:

$$<\!f\_m, \mathbf{R} \bullet \!f\_n> = \! <\!f\_m^\*, \mathbf{R} \bullet \!f\_n> = \delta\_{mn} \tag{10}$$

$$<\mathbf{J}\_m, \mathbf{X} \bullet \mathbf{J}\_n> = <\mathbf{J}\_m^\*, \mathbf{X} \bullet \mathbf{J}\_n> = \lambda\_n \delta\_{mn} \tag{11}$$

*Techniques for Compact Planar MIMO Antennas DOI: http://dx.doi.org/10.5772/intechopen.112040*

$$<\mathbf{J}\_m, \mathbf{Z} \bullet \mathbf{J}\_n> = <\mathbf{J}\_m^\*, \mathbf{Z} \bullet \mathbf{J}\_n> = (1 + j\lambda\_n)\delta\_{mn} \tag{12}$$

where *δmn* ¼ 1, *m* ¼ *n* or *δmn* ¼ 0, *m* 6¼ *n*.

Depending on the theory of characteristic mode, we get the induced currents on the PEC body and the resultant fields:

$$J = \sum\_{n} a\_{n} \mathbf{J}\_{n} \tag{13}$$

By using the Z impedance operator, formula (13) becomes:

$$\sum\_{n} a\_{n} Z(\mathbf{J}\_{n}) = \mathbf{E}\_{\tan}^{i}(\mathbf{r}) \tag{14}$$

Taking the inner product by the current *J<sup>m</sup>* for (14), we have:

$$\sum\_{n} a\_{n} < Z(\mathbf{J}\_{n}), \mathbf{J}\_{m}> = \prec E\_{\tan}^{i}(r), \mathbf{J}\_{m}>\tag{15}$$

Applying the orthogonality of currents, we will get under the condition of m = n:

$$a\_n(1+j\lambda\_n) = \prec E\_{\tan}^i(r), f\_m > \tag{16}$$

so we know:

$$a\_n = \frac{\prec E\_{\tan}^i(r), I\_m >}{1 + j\lambda\_n} \tag{17}$$

where *< E<sup>i</sup>* tanð Þ*r* ,*J<sup>m</sup> >* is called the modal excitation coefficient and the modal significance MS is defined as:

$$\mathbf{MS} = \left| \frac{\mathbf{1}}{\mathbf{1} + j\lambda\_n} \right| \tag{18}$$

#### **3.4 Optimization algorithms**

The optimization algorithms are often used to reduce the antenna size, but for the MIMO antenna it becomes a multi-objective optimization problem due to we have to consider other parameters, like the effect of mutual coupling and the related position change etc. of antenna element. If we use the single optimization algorithm to optimize the MIMO antenna, it will take more time, so the hybrid algorithms are employed to process the complicated discrete and continuous mixed parameters [119–121].

In [119], both the antenna shape and the decoupling structure were considered, thus the hybrid algorithm of both the multiobjective evolutionary algorithm based on decomposition combined with differential evolution (MOEA/D-DE) and MOEA/D combined with genetic operator (MOEA/D-GO) were used, where the MOEA/D-DE is adopted to optimize the radiator while the MOEA/D-GO optimizes the isolated area shown in **Figure 19**. They divided the circle into 8 areas of the same size, each has the 45 degrees angle and divided into several small pieces. In the optimization process, "1" represents the existence of metal while "0" not. Literature [120] Integrates the particle

**Figure 19.** *Fragment-type isolation of [119]: (a) split method, (b) discretization and assignment of "0" or "1", and (c) frqgment-type structure.*

swarm optimization (PSO) and binary PSO into multi-objective evolutionary algorithm based on decomposition (MOEA/D) to realize the optimization of antenna size and isolation, while the surrogate-based optimization was employed in [121].

Though different algorithms have been proposed to improve the optimization result, the corresponding optimization process is similar as in [119].

The optimized problem can be expressed as:

$$\min \; F(X) = \left( f\_1(X), f\_2(X), \dots, f\_n(X) \right) \\ \text{s.t.} \\ X \in \mathfrak{U} \tag{19}$$

where *fi* ð Þ *X* ð Þ *i* ¼ 1, 2, … , *n* indicates the corresponding optimized objective, X is a decision variable, and *Ω* is the design space.

The authors of [119] presented three optimized objectives due to the four-port MIMO antenna so that they have to consider the mutual coupling of different ports. We can simplify the optimized objectives by two ports, they are:

$$f\_1(X) = \max\left(Q\_1 - \min\left| (\mathbf{S}\_{11})\_{dB} \right|, \mathbf{0} \right) \\ o \in [o\_1, o\_2] \\ \tag{20}$$

$$f\_2(X) = \max\left(Q\_2 - \min\left| (\mathbf{S}\_{12})\_{dB} \right|, \mathbf{0} \right) \ o \in [o\_1, o\_2] \tag{21}$$

where ½ � *ω*1,*ω*<sup>2</sup> indicates the frequency band, *Q*<sup>1</sup> is the desired minimum of return loss, which is set to be 10 dB, and *Q*<sup>2</sup> desired minimum isolation.

With these optimized objectives and the constraints, the iteration process can be implemented by the hybrid utilization of EM simulator and the proposed algorithm.

### **4. Conclusions**

Depending on the development trend of 5G+/6G, we focused on the summaries of the planar MIMO antennas and the related compact techniques in this chapter owing to they are easily fabricated and integrated into a system. These planar antennas contain several common antenna types, including the patch, dipole/monopole, slot etc. Even so, they still can be designed into the 3D structure and there are specific applications like in the mobile terminal. The compact techniques implicit in the designs are dug up and summarized into seven categories, including the no-decoupling and the decoupling designs, multiple antenna structure, meander line technique, co-radiator design, and the fractal and radiator-cutting antennas. Then, in Section 3, we discussed the related fundamentals for the compact designs though the antenna types are conventional, and

showed the corresponding simple design methods, they are mode analyses, fractal techniques, characteristic analysis, and the optimization algorithms.
