Organization …

This thesis is organized as follows. In Chapter 2, the overview of multiple antenna systems is introduced, and the two analysis strategies of multiple antenna systems including complex antenna spatial correlation and radiation efficiency are reviewed for the further investigation in the following chapters. In Chapter 3, the two-dimensional (2-D) approximate antenna spatial correlation without mutual coupling is first proposed, and then the 2-D and 3-D antenna spatial correlation formulation incorporating antenna mutual coupling is further presented in the following sections in this chapter. Chapter 4 describes the new proposed TARC-based radiation efficiency, and we also provide investigations of impact of different termination networks on antenna spatial correlation and radiation efficiency based on the newly-proposed analysis strategy. Finally, we draw concluding remarks in Chapter 5.

Chapter 2

Fundamental Theory of Multiple Antenna Systems

Wireless communication systems are becoming more complex to cope with the growing demand for more data rates, wider coverage, larger capacity objectives, as well as exciting new wireless applications. Multiple antenna systems have great potential in reaching these specifications and overcoming the impairments of these systems by exploiting the spatial domain to reduce the interference of the undesired signals, extend the coverage of wireless networks, increase capacity, and reach high information throughput. 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 the overall introduction of multiple antenna systems, we further introduce two important parameters for the gauge of the performance of multiple antenna systems. The first one is the antenna spatial correlation where we will review several definitions of antenna spatial correlation in the second section of this chapter. The second parameter is the radiation efficiency which will be fully discussed based on the general definition and shown why it plays an important role in multiple antenna systems. Finally, because the case studies we provide in the whole thesis are simulated using a dipole pair, the dipole antenna is briefly introduced as well in the final section of the chapter.

2.1 Overview of Multiple Antenna Systems

It is a truth that current technologies have maximized the employment of temporal and spectral techniques to improve capacity and data speeds. There is still an additional degree of freedom left for full utilization, namely space [2]. Making use of space means multiple antenna elements are arranged together in the required manner.

The concepts of multiple antenna technology that originate from decade ago are substantially beneficial in the wireless communication systems. Multiple antenna systems have developed into several appearances for implementation, and we further introduce all of them briefly and summarize their benefits respectively as follows.

．Beamforming: The concept of this multiple antenna system originates from the conventional phased antenna array. The radiation pattern of the phased antenna array system can be controlled by feeding different signal phase delays and antenna element spacing [3]. With a specific feeding network, the total pattern of the array can be directed to the desired direction. On basis of this concept, beamforming is developed as one main multiple antenna strategy. There are two general types of beamforming, namely, fixed beamforming and adaptive beamforming. The main advantage of adaptive beamforming antenna systems over fixed beamforming antenna systems is the ability to steer beams toward desired signals and nulls toward interfering signals while the fixed beamforming can only radiate/receive signals at specific directions [4].

Beamforming offers interference rejection, antenna gain and spatial filtering, which have the equivalent effects of improving signal-interference-noise ratio (SINR) as well.

．Diversity: The concept of diversity comes from the fact that when multiple replicas or multipath effects of the transmitted signal fade independently as they go through channels, the probability of a deep fade happening in all propagating routes

are greatly reduced. Diversity techniques provide a diversity gain or a reduction in the margin required to overcome fading. Several antenna schemes are proposed to create the diversified channels to achieve the diversity gain, including polarization and spatial diversity. Polarization diversity exploits antenna with orthogonal polarizations to achieve the performance of high diversity gain. 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. Moreover, transmit diversity such as Alamouti’s space time block coding (STBC) [5] can improve the quality of signals by proving data with multiple independent coded streams. Spatial diversity is supposed to provide diversity gain and prevent fade margin.

．Spatial Multiplexing: Spatial multiplexing is supported to urge forward the data rates and throughput to an even higher level. Multiple data streams are transmitted to multiple antennas with this spatial strategy. Moreover, if the receiver end is also set up with multiple antennas and signals are with sufficiently different spatial signatures, it can separate data streams to reach the goal of high data rate compared to the single-antenna communication systems. Spatial multiplexing is therefore considered very powerful for increasing channel capacity. One thing to emphasize is this spatial technique works under 1) multiple scattering rich environments and 2) enough good signal-to-noise-ratio (SNR).

The above three multiple antenna systems all involve complex vectors and matrix operation on signals, and sometimes can be generally called the family of multiple-input multiple-output (MIMO) antenna technology for these three systems share the same characteristic of multiple antennas. MIMO used to only indicate the diversity and spatial multiplexing techniques. However, with the advance of combining all three techniques into a total communication solution, MIMO now represents the systems which exploit many antennas. For example, spatial

multiplexing or diversity can also be combined with beamforming when the channel is known at the transmitter, and the definition of MIMO can be broadened in an extensive manner as a result. Furthermore, a combination of MIMO with orthogonal frequency division multiplexing (OFDM) is promising for frequency selective channels, high spectral efficiency, and reduction of circuit complexity.

No matter what kind of MIMO technology is implemented, antenna spatial correlation and radiation efficiency have always been very important parameters for evaluating the MIMO systems. Different multiple antenna transmission methods set different requirements on the antenna set up in addition to the number of antenna elements. Generally, the beamforming technique usually needs the antenna setups with spatial correlation as high as possible, while the diversity and spatial multiplexing techniques on the contrary demand uncorrelated antenna setups.

Radiation efficiency is another issue we need to take care of because power consumption and how much power will radiate are concerned topics especially in small terminals like mobile phones. In the following two sections, we will review the definitions of these parameters.

2.2 Antenna Spatial Correlation

Signal fading due to multiple scattering effect is the dominant drawback happening in the wireless communication. Therefore, multiple antennas are proved to provide diversity, and the performance of the multiple antennas is determined by the spatial correlation between antennas. The first discussed spatial antenna correlation was proposed by W. C. Jakes [6]. Consider a plane wave arriving at an array from azimuth angle Φ with respect to the normal bisecting two sources a distance d apart, and the spatial correlation between two sources can be determined as

) ( )) sin(

2 exp(

) (

= ∫

−^π

π φ φ φ φ

π λ

ρ d p d

j

d (2.1)

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 scenario [7], the antenna spatial correlation has a closed form well-known as the Bessel function [8]. Based on (2.1), several works on spatial correlation has relied on numerical integration to evaluate the correlation coefficient between two sources based on different azimuth angular probability distribution functions [9-10]. The author in [11] especially discussed and derived simple generalized formula for spatial correlation and showed a good approximation for spatial correlation for small angular-spread (AS) 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 antenna patterns and mutual coupling effect was further proposed in the literature. There were two main categories for the antenna spatial correlation including antenna patterns and mutual coupling effect. The first is the parameterized correlation formulation which describes the correlation in impedance or scattering matrix, and the second is the correlation formulation generally defined as the Hermitian product of the far-field patterns of two antenna elements.

．Parameterized Formulation: In [12], W. Wasylkiwskyj and W. K. Kahn suggested the antenna spatial correlation as

22 11

12 12( )

R R d = R

ρ (2.2)

where Rii is the self impedances of the i-th antenna, and Rij is the mutual impedance between the i-th and j-th antennas. What needs to be noticed is this formulation is suitable for the minimum scattering antenna theory only. Moreover, the authors in [13]

proposed exact representation of antenna envelope correlation in terms of scattering parameter description under the assumption of uniformly incoming waves as listed in Equation (2.3).

( )(

21 ²

)

．Pattern Multiplication: This is the most direct but also the most complex definition. R. G. Vaughan and J. B. Andersen proposed in [8] that the spatial correlation is given by

[ ]

where ‧ denotes the Hermitian product and P means the antenna pattern. Moreover, C. Waldschmidt and W. Wiesbeck further suggested a more general spatial correlation as [14]

2

E is the far-field E antenna patterns, p(Φ,θ) means the AoA distribution of interest, and the subscript Φ/θ denotes the field polarization for both AoA distribution and antenna patterns.

Compared with the pattern multiplication, the correlation represented in parameter manner encloses the total radiation field of antennas and extracts the

correlation from the principle of energy conservation, which will in turn lose some important information originally existing in the integral equations. As a consequence, the spatial correlation in Equation (2.5) is considered the most general correlation formulation so far because it takes all the possible factors into consideration to calculate the correlation coefficient.

2.3 Radiation Efficiency

For single antenna case, the total antenna radiation efficiency is used to take into account losses at the input terminals and within the antenna configurations. Such losses are listed as follows and refer to Figure 2.1 [3].

In general, the overall efficiency can be written as

Figure 2.1 Reflection, conduction, and dielectric losses.

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

We in turn introduce the general definition of radiation efficiency in multiple antenna systems [15]. The radiation efficiency is most conveniently defined and computed in the transmit mode by implementing the equivalent circuit shown in Figure 2.2. The voltage source V and source impedance ZS1 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. One thing to be mentioned is this equivalent circuit is constructed based on the antenna pair with identical structure, and Z11=Z22 and Z12=Z21 accordingly.

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

1 2 12

11 I

I Z Z

Z_in

= +

(2.8)

Equation (2.8) can be transformed using the circuit loop which represents the equivalent circuit of antenna 2 by

₁

2 22

12

2 I

Z Z I Z

+

L

= −

(2.9)

Then, we can finally determine the input impedance Zin as

2 22

2 12 11

L

in Z Z

Z Z

Z = − + (2.10) The total power leaving antenna 1 is shown as PZin=Real{Zin}|I1|², and the power which will be absorbed by ZL2 via mutual coupling effect and cause reduction of radiation power is PZL2=Real{ZL2}|I2|². The difference between PZin and PZL2is called the radiation power Prad, i.e., Prad= PZin- PZL2. Therefore, the radiation efficiency erad

which is also defined in [3] can be derived as

e

_rad

= e

_refl

e

_ZL2 (2.11) where

0 2 0

and

1

Z Z

Z e Z

in in

refl +

= − Γ Γ

−

= (2.12)

2 1

2 2 2

2

Real { }

} { 1 Real

I Z

I e Z

in L

ZL

= −

(2.13) 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.

2.4 Dipole Antenna

Because the case studies we provide in the whole thesis are simulated using a

dipole pair, the dipole antenna is briefly introduced in this section. The dipole antenna is the most general antenna structure, and the current distribution on the dipole usually assumes 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 can be approximately written as

⎪⎪

⎩

⎪⎪

⎨

⎧

≤

⎥⎦ ≤

⎢⎣ ⎤

⎡ ⎟

⎠

⎜ ⎞

⎝⎛ +

≤

⎥⎦ ≤

⎢⎣ ⎤

⎡ ⎟

⎠

⎜ ⎞

⎝⎛ −

=

0 2 z

2 ,

sin

z 2 0 2 ,

sin )

(

0

^ 0

^

z l k l

I

z l k l

I z

I

z z

a a

(2.14)

where Io is the maximum current occurring at the center-fed point, and k is the phase constant in the free space. After the far-field approximations and integration of all

(a) (b)

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

the infinitesimal elements, the far-field Eθ pattern takes the form of

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

⎥⎥

One of the most commonly used type is the half-wavelength (l=λ/2) dipole for its matching to the transmission line is simplified especially at resonance. By letting l=λ/2, Equations (2.15) and (2.16) can reduce to

⎥⎥

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 computed from the induced EMF method [16]. We need to notice that we assume the diameter of the dipole is ideally much thinner than the wavelength of the operating frequency, and only Eθ and HΦ

fields exist. However, in the following chapters, EΦ and Hθ fields also exist in the simulation results since the diameter of the dipole cannot be in ideally thin manner.

Chapter 3

A New Spatial Correlation Formulation of Arbitrary AoA Scenarios

Signal fading due to multiple scattering effect is the dominant drawback happening in the wireless communication. Therefore, multiple antennas are proved to provide diversity, and the performance of the multiple antennas is determined by the spatial correlation between antennas. In this chapter, we will discuss this parameter including the AoA distributions of the spatial channel and the physical configurations of multiple antenna structures. We first introduce the 2-D approximate spatial correlation formulation and our proposed approximate spatial correlation formulation of arbitrary AoA scenarios in Section 3.1. Based on the proposed formulation, Section 3.2 further introduces the 2-D spatial correlation combining with antenna patterns distorted by the mutual coupling effect based on the spatial correlation proposed in [17]. Finally, in Section 3.3, we derive a 3-D spatial correlation incorporating antenna mutual coupling in the parameterized manner. The new antenna spatial correlation formulation not only effectively reduces computation complexity without sacrificing accuracy but also offers a more detailed analysis presented in the parameterized manner. Again, we emphasize all the simulation results are provided using a dipole pair as the benchmark.

3.1 2-D Approximate Spatial Correlation Formulation of Arbitrary AoA Scenarios

3.1.1 Spatial Correlation of Small Angular Spread AoA Scenarios

The spatial channel model is different from the traditional propagation model which does not take into consideration the spatial angular distribution. A channel model that simultaneously characterizes the AoAs of multipath components is called the spatial channel model [18], and different phi-plane AoA PDFs have been proposed in the literature [19].

With a given AoA scenario, we may substitute it into the spatial correlation. In [11], the author presented the approximate spatial correlation which is suitable for small angular-spread AoA distribution. Consider a plane wave arriving at an array from azimuth angle Φ with respect to the normal bisecting two sources a distance d apart, and the spatial correlation between two sources can be determined as

) ( )) sin(

2 exp(

) (

= ∫

−^π

π φ φ φ φ

π λ

ρ d p d

j

d (3.1) where λ is the wavelength and pΦ(Φ) is the azimuth angular PDF. If the angular energy is a Gaussian distribution, AoA distribution pΦ(Φ) can be represented as

( ) ( )

⎭⎬

⎫

⎩⎨

⎧− −

= ₂

2 0

exp 2 2

1

σ φ φ σ

φ π

pφ (3.2) where σ is the standard deviation of the distribution and Φ0 is the mean angle of the AoA distribution. Substitute Equation (3.2) into Equation (3.1) and make a change of variables, and the spatial correlation is given by

( ) _∫

₋

( )

Under the assumption of small σz over the integration range, Equation (3.3) can be approximated as

( ) ( ) ( )

Similarly, if the AoA distribution is small-∆ uniform distribution which ranges from -∆ to ∆ with mean angle Φ0, the spatial correlation can be derived as

( ) ( ) ( )

⎟

Equations (3.4) and (3.5) have provided simple generalized formula for spatial correlation and shown that both give good approximations. Another advantage of Equations (3.4) and (3.5) is they have reduced the computation time where the calculation of the spatial correlation originally relies on numerical integration or infinite series expansion. However, as shown in [11], Equations (3.4) and (3.5) cannot approximate well when the mean angle Φ0 is much larger than 0°, or the standard deviation of the distribution σ becomes higher (or ∆ becomes larger in the uniform distribution). That is therefore not favorable because the angular spread of the AoA distribution may become larger in the multiple scattering-rich environment, especially in the indoor environment where MIMO systems can make the best use of their advantages. As a result, in Section 3.1.2, we will further propose the spatial correlation formulation which is suitable for large angular-spread AoA distribution and even arbitrary AoA scenarios.

3.1.2 Spatial Correlation of Arbitrary AoA Scenarios

Uniform distribution is suitable to describe large angular spread AoA in multiple scattering rich environments, so it is an appropriate candidate when we analyze the parameter of the multipath channel model. As mentioned in the previous section, the approximate form of the spatial correlation is presented based on Gaussian distribution which is suitable for small angular spread AoA distribution. However, the approximation may be distorted when the angular spread becomes large, and that is the reason why we would like to suggest a good approximate spatial correlation under the condition of large angular spread. For the uniform distribution, its probability density function is presented as

( )

−Δ≤ ≤ +Δ

= Δ _{0 0} 2

1 φ φ φ

φ φ

p (3.6)

where Φ0 is the mean of the given uniform distribution and 2∆ is the range of angles referred to Φ0. If 2∆ is equal to 2π, the spatial correlation has a closed form and is well-known as the Bessel function; however, for the case that 2∆ is smaller than 2π, the spatial correlation is not a closed form formula and thus the time-consuming numerical integration is needed.

Therefore, we propose a new approximate AoA distribution for the uniform distribution as

2 ) exp (

2 1 N

) 1 (

^N

1

n 2

∑

2

=

⎭ ⎬ ⎫

⎩ ⎨

⎧ − −

=

n n

n

p σ

φ φ σ

φ π

φ (3.7)

where N is the number of sampling Gaussian distribution, Φn is the n-th sampling mean AoA and σn is the n-th sampling AoA angular spread. This approximate AoA distribution uses the combination of many small angular spread Gaussian distributions to fit a given large angular spread uniform distribution as shown in Figure 3.1(a).

There are three reasons we choose multiple Gaussian distributions to fit the large

angular spread uniform distribution. First of all, Gaussian distribution is a general distribution to describe a small angular spread AoA scenario. Second, the standard deviation of the Gaussian distribution is actually the angular spread of the Gaussian AoA distribution. Finally, that the spatial correlation of small angular spread Gaussian distribution has a generalized approximation formulation is the most important reason we choose as the fitting function of the uniform distribution.

Substitute Equation (3.7) into Equation (3.1), we may represent the spatial correlation as

2

Using the small angular-spread approximate spatial correlation as shown in Equation (3.4), we may finally represent the spatial correlation as

2 uniform-like distribution. The solid lines are the practical distribution curves while the dash lines are the sampling Gaussian distributions.

The advantage of this approximation is it can be extended to arbitrary AoA

The advantage of this approximation is it can be extended to arbitrary AoA

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