Organization …

This work will be organized as follows: In Chapter 2, the overview of multiple antenna systems is introduced, and the two analysis strategies of multiple antenna systems including radiation efficiency and spatial correlation are reviewed for the further investigation in the following chapters. In Chapter 3, the newly-defined eigenvalue based reflection coefficient and eigenvalue based radiation efficiency are first proposed, and then the investigations on the impact of termination networks are further presented in the following sections of this chapter.

Chapter 4 first describes two new approximate formulations of antenna spatial correlation without mutual coupling, and then applies them for the calculation of parameterized spatial correlation formulations incorporating antenna mutual coupling, both for 2-D and 3-D cases, respectively.

Chapter 2

Fundamental Theory of Multiple Antenna

4

Systems

Wireless communication systems are becoming more important in our daily lives. The need for more data rates, wider signal coverage, larger channel capacity are several challenges in communication technologies .Multiple antenna systems have great potential in extending the signal coverage of wireless networks, increasing channel capacity, and reaching high information throughput by exploiting the spatial domain. In this chapter, we will first review the multiple antenna systems and especially focus on the detailed classification of different multiple antenna system schemes. Based on these classifications of multiple antenna systems, we further introduce two parameters which have great importance on the performance judgments of multiple antenna systems. The first one is the antenna radiation efficiency where the general definitions of single antenna and dual antennas cases will be discussed and shown why it plays an important role in multiple antenna systems. The second parameter is the antenna spatial correlation where we will review several definitions of antenna spatial correlations. Finally, because the studies we provide in the whole thesis are simulated using dipole antenna, the theory of dipole antenna is briefly introduced as well.

2.1 Overview of Multiple Antenna Systems

The multiple antenna technologies have been researched and developed for more than a decade which is considered substantially beneficial for the wireless communication systems.

Multiple antenna systems have been implemented by several strategies, and we will introduce all of them briefly and summarize their benefits respectively as follow:

．Beamforming: Beamforming strategies originate from phased array system. The total radiation pattern of the phased antenna array system can be controlled by feeding signals with

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different phase delays and antenna element spacing [2]. With a specific feeding network, the total pattern of the array can be directed to the desired direction. Recent researches focus on adaptive beamforming strategies more because it can be implemented simply by using intelligent algorithm method to steer beams toward desired signals and nulls toward interfering signals [3].One of the adaptive beamforming method is called optimal beamforming method. Current research about this topic not only takes all electromagnetic characteristics like mutual coupling into account but also minimizes the total power radiated by the antenna array using optimization method while the response in a desired direction is maintained [4]. Beamforming offers interference rejection and antenna gain which have the equivalent effects of improving signal-interference-noise ratio (SINR) as well.

．Diversity: In telecommunications, a diversity scheme refers to a method for improving the reliability of a message signal by utilizing two or more communication channels with different characteristics. Multiple antenna systems are proposed to create the diversified channels including polarization, spatial and pattern diversity. Polarization diversity combines pairs of antenna with orthogonal polarizations. By pairing two complementary polarizations, this scheme can immunize a system from polarization mismatches that would cause signal fading. Spatial diversity systems are designed such that the signals at the different antennas of the receiver have low cross correlation with maximum gain achieved for uncorrelated signals.

Antenna pattern diversity consists of two or more co-located antennas with different radiation patterns. This type of diversity makes use of directive antennas that are physically separated by some short distance. Collectively they are capable of discriminating a large portion of angle space and can provide a higher gain versus a single omnidirectional radiator.

．Spatial Multiplexing: Spatial multiplexing is a transmission technique in MIMO wireless communication to transmit independent and separately encoded data signals, so called streams, from each of the multiple transmit antennas. Therefore, the space dimension is

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reused, or multiplexed, more than one time. Spatial multiplexing can reach the goal of higher data rate compared to the single-antenna communication systems and it is considered very powerful for increasing channel capacity.

The above three multiple antenna strategies can be called the family of multiple-input multiple-output (MIMO) antenna technologies and sometimes have the same characteristics of multiple antennas. Furthermore, a combination of MIMO with orthogonal frequency division multiplexing (OFDM) is promising to use the spectrum much more efficiently by spacing the channels much closer together, which is achieved by making all the carriers orthogonal to one another, preventing interference between the closely spaced carriers.

No matter what kind of MIMO technology is implemented, antenna radiation efficiency and spatial correlation have always been two very important parameters for MIMO systems.

For a multi-polarization antenna system, radiation efficiency is an important issue because how much power will radiate with respect to different polarization states are concerned topics on this kind of system. Antenna spatial correlation is another issue we want to take care. For beamfoming technology, we always want to design the spatial correlation as high as possible, while for diversity and spatial multiplexing techniques demand low correlation antenna setup.

In the following two sections, we will review the definitions of these two parameters.

2.2 Radiation Efficiency

For single antenna case, the radiation efficiency is defined and computed by implementing the equivalent circuit shown in Figure 2.1[2].We can see that input impedance is composed of real part and imaginary parts:

A A A

Z =R +jX (2.1) The input resistance RArepresents dissipation, which occurs in two ways. Power that leaves

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the antenna and never returns (i.e.,radiation) is a form of dissipation. There is also ohmic loss associated with heating on the antenna structure. The input reactance XA represents the power stored in the near field of the antenna.

Figure 2.1 The equivalent circuit of single antenna in transmitting mode.

The average power dissipated in an antenna is:

1 2

in 2 A A

P  R I (2.2) where IA is the current at input terminals. Separating the dissipated power into radiative and ohmic losses gives: The radiation efficiency is defined as the ratio of total radiated power to the net power accept by the antenna, so The total radiation efficiency must take input mismatch effect into account. Therefore, the full expression of radiation efficiency on single antenna case is:

1 1 where Γ is the voltage reflection coefficient and Z0 is the characteristic impedance of the

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transmission line.

Figure 2.2 The equivalent circuit of an antenna pair in transmitting mode.

We further introduce the general definition of radiation efficiency in multiple antenna systems [5]. The radiation efficiency is defined mostly convenient in the transmit mode as the equivalent circuit shown in Figure 2.2. The voltage source V and source impedance Z_S1 show the excitation of the antenna port 1, and the load impedance ZL2 is the termination at the second antenna port. Z₁₂ is the mutual impedance which can describe the mutual coupling effect between two antennas. For simplicity this equivalent circuit is constructed based on the antenna pair with identical structure, which means that Z₁₁=Z₂₂ and Z₁₂=Z₂₁.

The equivalent circuit in Figure 2.2 can be used to calculate the input impedance Zin, which can further calculate the voltage reflection coefficient Γ_in. We can determine the input impedance Zin as

1 2 12

11 I

I Z Z

Z_in   (2.7)

Equation (2.7) can be further expressed using the circuit loop theory which represents the relation between I1 and I2 as:

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As a result, we can finally determine the input impedance Z_in as:

The total power leaving antenna 1 is shown as PZin=(1/2)Real{Zin}|I1|², and the power which will be absorbed by Z_L2 via mutual coupling effect and cause reduction of radiation power is PZL2=(1/2)Real{ZL2}|I2|². The difference between PZin and PZL2is called the radiation power Pr, The radiation efficiency is the composite power efficiency representation for it includes not only the reflection caused by input mismatch of the excitation port but also the power absorption resulting from the termination at the other unexcited antenna branch.

2.3 Antenna Spatial Correlation

Under multiple scattering environments, signal fading is the dominant impairment existing in the wireless communication. To overcome this problem multiple antennas are typically employed to provide diversity and the performance of the multiple antennas is determined by the spatial correlation between the antennas [6]. Antenna spatial correlation was first proposed by W. C. Jakes [7]. If a signal of interest arriving at an array can be

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described by the summation of plane waves arriving from azimuth angle Φ relative to the normal between two sources a distance d apart, the spatial correlation can be determined as

)

where λ is the wavelength and pΦ(Φ) is the azimuth angular probability distribution function.

The most special case is when pΦ(Φ)=1/2π which is called the Clarke’s model [8] and the antenna spatial correlation has a closed form well-known as the Bessel function. Based on equation (2.13), several works on spatial correlation has relied on numerical integration or series expansion to evaluate the correlation coefficient between two sources based on different azimuth angular probability distribution functions [9-11]. The author in [12]

especially discussed and derived a simple formula for spatial correlation and showed that it provided a good approximation for spatial correlation of small angular spread angular distributions.

The above definitions of antenna spatial correlation only take the signal phase and the angular PDF of the incoming waves in azimuth plane. Therefore, the antenna spatial correlation including full antenna patterns and mutual coupling effect was further proposed in the literatures. There were two main formulations proposed for the antenna spatial correlation including antenna patterns and mutual coupling effect. The first is direct Hermitian product of the far-field patterns between two antenna elements. The second is parameterized correlation formulation described by scattering matrix.

．Pattern Multiplication: This is the most direct but also the most complex definition. R.

G. Vaughan and J. B. Andersen proposed in [6] that the spatial correlation is given by

 

where ‧ denotes the Hermitian product and F means the normalize antenna pattern.

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．Parameterized Formulation: The authors in [13] proposed exact representation of antenna envelope correlation in terms of scattering parameter description under the assumption of uniformly incoming waves. The approach has the advantage that it is not necessary to known the radiation pattern of the antenna system and that the explicit influence of mutual coupling and input match is revealed. The formulation is given by

 

21²



Compared with the pattern multiplication and the correlation represented in S-parameter manner, both of them do not take AoA distribution of arriving signals into account. Therefore, the spatial correlation in equation (2.17) is considered the most complete and general correlation formulation so far because it takes all the possible factors into consideration.

2.4 Dipole Antenna

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All the case studies we provide in the whole thesis are simulated using two or more half-wave dipole antennas. Therefore, the basic theory of half-wave dipole antenna is introduced in this section. The half-wave dipole antenna is the most general antenna structure, and the current distribution on the dipole usually assumes that the antenna is center-fed and the current vanishes at the end points. Moreover, to reduce the mathematical complexities, the diameter of the dipole is ideally much thinner than the wavelength of the operating frequency.

With the above assumptions, the current distribution is placed along the z-axis and for the half-sine wave current on the half-wave dipole. It is written as:

 

^{sin , z} where I_m is the maximum current occurring at the center-fed point, andβis the phase constant in the free space. After the cumbersome mathematical integration, the far-field Eθ pattern is

In the similar manner, the total HΦ component can be written as

cos cos

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(a) (b)

Figure 2.3 (a) The λ/2 dipole and (b) the Eθ pattern in theta plane (Φ=0°).

The current distribution of the half-wavelength dipole and the theta-plane E-field pattern is plotted in Figure 2.3, and Zin=73+42j [2] .We need to notice that we assume the diameter of the dipole is much thinner than the wavelength of the operating frequency, and there exists only Eθ and HΦ fields. However, in the chapter 4, EΦ and Hθ fields also exist in the simulation results since the diameter of the dipole is not thin enough compared to the wavelength of the operating frequency.

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Chapter 3

Composite Analysis of Radiation Efficiency

Antenna arrays play a crucial role in wireless communication over multipath fading channels. When using multiple antenna elements for implementation on small personal communications devices, the resulting closely spaced antenna elements exhibits well-known mutual coupling, which alters radiation pattern characteristics and is obviously impact the performance of multiple antenna systems. Radiation efficiency is considered an important factor to measure the performance of multiple antenna systems including mutual coupling. In this chapter, the analysis of radiation efficiency is in transmitting mode. The general definition of radiation efficiency for dual antenna systems has been introduced in Chapter 2 and we continue this concept for the further investigations on the composite analysis of reflection coefficient and radiation efficiency. In Section 3.1, the power representation using microwave network theory and TARC are first introduced as well as the concept of [16] . The composite analysis of how different kinds of termination network impact on the reflection coefficient and radiation efficiency are conducted in Section 3.2.

3.1 Introduction of The Eigenvalue based TARC and Radiation Efficiency

3.1.1 Multi-port Antennas and Total Active Reflection

Coefficient

15 The total powers incident on the n-port network is given by

2 2 and the total power reflected from the n-port network is

The total input and reflected power can be represented as the summation of individual power incident on and reflected from the port i, respectively. The multiport antennas discount antenna ohmic loss for simplicity to analysis. First, we give an alternative expression to total reflected power and then we turn to calculate the radiated power. By substituting equation

,where R we call it reflection power matrix . This expression relates total power generated by excitations and the total reflected power. We now want to represent equation (3.4) in an alternative way for convenience to later analysis. Since the reflection power matrix is a Hermitian matrix, we can perform unitary similarity transformation [16] on R and U is unitary (i.e, UU^H=I ) as below

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H

R = U D US with D_S diag



_s1,_s2,K ,_sn



whereD_S  R^{n n} (3.5) This transformation is also called eigenvalue decomposition (EVD). Substituting equation (3.5) into equation (3.4), an alternative representation of total reflected power is shown as

 

fact of UU^H=I, the total input power can also be derived as

   

The power radiated by the antenna neglecting antenna ohmic loss is the difference between Pin

and P_refl, further substitution yields as below like [16]

 

The behavior of this microwave network is primarily defined by λsi. For a lossless network, P_rad=0 so that λsi=1 for all i. For a lossy network like multiport antennas, P_rad>0 so that 0≤ λsi

<1 for all i. As a result, EVD for reflection power matrix facilitates analysis and provides a useful interpretation of circuits’ fundamental behavior.

Now we turn to introduce the concept of TARC briefly. For a desired port excitation, the

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total active reflection coefficient (TARC) [17] is defined as the square root of the available power generated from all excitations minus radiated power, divided by the available power as

in rad

in

P P

TARC P

   (3.10)

For example, if an N-port antenna is excited at ith port and the other ports are connected to the matched load, the TARC can be calculated as

2 For multiport excitation, the TARC is therefore in the form of

2 reflects back or goes to the other ports. This parameter is developed to describe the properties of multi-port antennas like frequency bandwidth and radiation performance, while all ports simultaneously excite signals with their own port impedances. In this manner, one is able to assess the true bandwidth of the antenna for a desired port excitation. This bandwidth information should give the multi-port antenna designer a much better understanding of the antenna bandwidth.

3.1.2 Eigenvalue Representation of TARC and Radiation Efficiency

A general definition of radiation efficiency in multiple antenna systems is introduced in

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Section 2.2.Now the alternative definition of radiation efficiency is discussed in [16]. Before the derivation of radiation efficiency, we first give an alternative representation for TARC.

Substituting equation (3.6) and equation (3.8) into equation (3.12), a new representation of TARC is defined as below

2

Since this expression is based on the eigenvalues of the reflection power matrix, we redefine it as eigenvalue based reflection coefficient (EVRC) for convenient to analysis and further derive an alternative representation of radiation efficiency as below which is similar with [16]and[20] We rename this radiation efficiency as eigenvalue based radiation efficiency (EVRE) for simplicity of the following writing. It is wondered what is the difference between equation (3.14) and equation (2.10). Equation (2.10) is considered the composite power efficiency representation. It includes not only the reflection caused by input mismatch of the excitation port but also the power absorption resulting from the termination at the other unexcited antenna branch. Based on this definition, we may find equation (2.10) is actually a special case of equation (3.14). Taking a dual-antenna system for example, equation (2.10) will let one branch of the dual antenna system excite signals and the other terminated with impedance load. While in equation (3.14), two ports of the antenna system simultaneously excite signal with their own port impedances. That exactly means if we determine the radiation efficiency

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using equation (3.14) but with one branch feeding signals of zero amplitude, the analysis result will be the same as that using equation (2.10).

One of the advantages of the EVRE is it takes into account the effect when ports of the multiple antennas system are fed with signals of different phases. EVRC (TARC) is originally developed for signals with various phase delays for multi-polarization operations, and this concept can be further extended to the multiple antennas system [18]. It is well known that mutual coupling causes some portion of signal power within each element to be radiated and absorbed by the other elements. The combination of each antenna port’s primary reflected signal with the coupled signals can be constructive or destructive depending on the phase of the component signals. EVRE of multiple antennas can therefore represent the effect of this constructive or destructive signal combination. Another way to show the effect of input excite signal phase difference on EVRE is based on the perspective of mathematical formulation as below: For dual antennas case and based on equation (3.13), the transformed input excitation signal is shown as

, where θ is the phase difference of input excitation signals and the more explicit EVRC and EVRE are shown as

As a result, it can be obviously viewed from equation (3.16) that EVRC and EVRE are indeed a function of input excitation signal phase difference.

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The most important advantage of EVRC and EVRE are that they provide a simple way to estimate the minimum and maximum values of reflection coefficient and radiation efficiency quickly, no matter how many number of antennas will be gauged [16]. These advantages are revealed by further deriving equation (3.13) as an inequality and it is shown below

2

It is interesting that the minimum and maximum values of EVRC are just the square root of

It is interesting that the minimum and maximum values of EVRC are just the square root of

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