## 1. Introduction

A multiple-input multiple-output (MIMO) system achieves high-frequency utilization efficiency by simultaneously transmitting different signals using multiple antennas at both transmission and reception sides [

1,

2]. MIMO technology is being actively implemented to overcome the shortage of frequency resources caused by the dramatic advent of various kinds of wireless systems.

The performance of MIMO systems is determined by both radio propagation and antenna characteristics. The radio wave propagation characteristics are decided by the spatial relationships of the antennas and the surrounding environment such as buildings, making it difficult to realize the desired characteristics. Fortunately, the multiplicity of antennas offers significant design freedom and thus the possibility of improving MIMO channel capacity if the antenna design suits the given environment well.

To design a MIMO antenna that has high channel capacity, it is required to realize high efficiency and low correlation characteristics. While the consideration of efficiency in antenna design is quite general and well understood, the consideration of correlation characteristics is not simple. This is because calculating the correlation requires either channel models [

3,

4,

5,

6,

7] or full two-dimensional (2D) complex patterns (note that ‘three-dimensional’ is incorrect because the pattern is normally defined by two variables, e.g.,

$\theta $ and

$\varphi $).

As described above, the correlation is not as simple to calculate as other antenna characteristics. Hence, the S-parameter-based correlation coefficients proposed by Blanch [

8] have been used in many studies [

9]. Strictly speaking, a similar method was reported before Blanch’s paper, but the different term, ‘beam coupling factor’ was used instead of the correlation coefficient [

10], and the demand for such antenna evaluation at that time was less than it is at present. The correlation coefficient is easy to evaluate through the use of the S-parameter if a multi-port network analyzer is available. However, this approach is not applicable to antennas with Joule heat loss because the radiation efficiency of the antenna excluding reflection loss and mutual coupling loss must be close to 100%. Hence, this equation should be carefully used by considering the principle and applicable conditions.

As for S-parameter-based correlation coefficients in lossy MIMO antenna, the uncertainty of the correlation coefficients is discussed in [

11], where the radiation efficiency of the antennas is taken into account. It is mentioned that the uncertainty of the correlation coefficient is

$\pm 1$ when the antenna efficiency of two antenna array is 50%, and this means the S-parameter-based correlation coefficient does not provide any information about the performance of MIMO antenna. A very interesting example is demonstrated in [

12], where a single monopole antenna is connected to a Wilkinson divider that extends a single antenna port to two ports. This yields zero-correlation characteristics in the equation [

8] since the divider completely isolates two extended ports. However, two extended ports always observe completely the same signals, i.e., the observed signals at two ports are fully correlated. To resolve the problem of the S-parameter based correlation in lossy MIMO antenna, the correction methods have been presented [

12,

13]. The work [

14] compares the performance of these correction techniques numerically and experimentally, and the accuracy of the correction methods is discussed. It is concluded that the equivalent-circuit-based technique [

13] yields better accuracy, but this requires measuring the radiation efficiency and selecting the adequate circuit model by considering the structure of the MIMO antenna. This alludes to the important relationship among the antenna geometry and characteristics.

Figure 1 conceptualizes the impact of three key antenna factors (impedance, pattern, size) on the correlation characteristics. The impedance characteristics can be conditionally translated into the correlation characteristics [

8]. In addition, the complex pattern is directly translated into the correlation characteristics [

10], and it is well known that low correlation coefficients can be attained if the patterns or polarizations are orthogonal [

15,

16,

17,

18,

19]. It is also empirically known that correlation increases when the array antenna is small [

20].

This paper deals with five correlation coefficients: signal correlation, channel correlation, fading correlation, complex pattern correlation, and S-parameter-based correlation. The main contributions of this paper are listed as follows:

- (a)
Categorizing the correlation coefficients used in MIMO antenna evaluation.

- (b)
Clarifying the commutativity among the correlation characteristics and some of the physical quantities

The remainder of this paper is organized as follows.

Section 2 describes the signal model dealt with in this paper and the five definitions of the correlation coefficient in MIMO antenna, and their categories and relations are mentioned in detail.

Section 3 presents the simulation results and discusses the problem in the S-parameter-based correlation by comparing correlation coefficient errors. Finally, the conclusion is drawn in

Section 4.

## 3. Simulation

As mentioned above, the signal correlation matrix and channel correlation matrix are fundamentally identical if the signal-to-noise ratio is sufficiently high, and their correlation characteristics are easily simulated and measured.

The remaining three correlation characteristics are strongly related, while their similarity is vulnerable to the environment and antenna efficiency. Therefore, the correlation coefficients to be compared here are the fading correlation matrix shown in (

11), the complex pattern correlation matrix as in (

15), and the S-parameter based correlation matrix as in (

20). To assess the impact of the antenna characteristics on these three correlation characteristics, a Rayleigh environment is assumed, where the sufficient numbers of paths distribute uniformly in all directions. In addition, the fading correlation is evaluated only at the receiver side for simplicity.

#### 3.1. Antenna and Channel Models

Two kinds of antenna arrays are dealt with in this paper, and the correlation coefficient is varied by changing the element spacing.

Figure 7 shows the antenna models. Here, (a) is a half-wavelength dipole array and (b) is a square microstrip antenna array (MSAA); both are linear arrays with element intervals of

d in the H-plane. The size of the conductor-backed dielectric substrate of (b) is determined so that the distance between the substrate edge and the center of each antenna element is 0.5 wavelengths. The thickness of the dielectric substrate was 1.6 mm, the relative dielectric constant was 4.2, and the dielectric loss tangent was 0.02. Both the dipole and MSAA had an operating frequency of 2.4 GHz. The moment-method based simulator (HyperLynx 3D EM) was used.

Figure 8 is a conceptual diagram of the model used to compute the propagation channel. This is a geometry-based stochastic channel model (GSCM) [

7], and well explains the physical path distributions around the antennas. The scatterers are arranged sufficiently far from each antenna. Each entry of the MIMO channel matrix is yielded by the sum of all element paths from one of the transmitting antennas to one of the receiving antennas. In this simulation, the statistical conditions of the environment are identical for both transmitting and receiving sides. The number of the scatterers is set to 100 for each side, and the distribution of the scatterers are set to 3D uniform. The propagation channel is generated 1000 times for each antenna condition, where identical antenna arrays are used for both sides. The GSCM model is used only to calculate the fading correlation matrix. This is because both the complex pattern and S-parameter-based correlation matrices are determined by the antenna characteristics only, while the fading correlation matrix needs the propagation channel.

#### 3.2. Results

Figure 9 plots the correlation coefficient versus the element spacing of a dipole array (

$M=2$), where

${\lambda}_{0}$ represents the wavelength in a vacuum. These results show that the correlation coefficients obtained by the three calculation methods almost agree. When the element spacing is wide, the fading correlation calculated by the GSCM diverges from those of the other methods. This is considered to be caused by insufficient resolution (

$\theta $ Direction:

${5}^{\circ}$,

$\varphi $Direction:

${10}^{\circ}$) of the complex pattern used in the scattering ring model.

Figure 10 shows the element spacing characteristics of the correlation coefficient for the dipole array (

$M=4$). Since the structure is bilaterally symmetric, the correlation coefficients, which are identical in principle, are shown jointly. As a result, it was found that all methods yielded almost the same correlation coefficients. However, when S-parameter was used, the estimated correlation coefficient was slightly lower when the element interval was narrow. It is considered that the radiation efficiency (excluding reflection and coupling loss) was degraded by the effect of conductor loss because the finite conductivity (

${\sigma}_{Cu}=5.7\times {10}^{7}$ S/m) of copper was used.

Since the radiation efficiency of the dipole array examined above is high, the correlation coefficients obtained by the three methods agreed well. Next, the accuracy of the correlation coefficients was compared using MSAA with dielectric loss.

Figure 11 shows the result of the correlation coefficient of the 2-element MSAA. From this result, it is found that the complex pattern correlation given by (

15) agrees well with the fading correlation by using GSCM, but the S-parameter-based correlation by (

20) is always underestimated and diverges strongly from the others. This is because, when the loss is large, the radiation power of the antenna decreases, and, as a result, the mutual coupling between the antennas tends to decrease. Thus, the correlation coefficient using S-parameters is inaccurate for lossy antennas. The radiation efficiency, in this case, was about

$33\%$ when the element spacing was

$0.3{\lambda}_{0}$, and about

$39\%$ when the element spacing was greater than

$0.5{\lambda}_{0}$.

Figure 12 shows the correlation coefficient of the 4-element MSAA. The S-parameter based correlation yielded by (

20) diverges significantly from those of the other methods. Similarly, the radiation efficiency is about 27% to 40%, which seriously degrades the accuracy of the S-parameter based correlation coefficient.

To confirm the impact of the radiation efficiency on the accuracy of the S-parameter based correlation coefficient, the lossy dipole model is analyzed; resistive attenuators are intentionally inserted at the feed port.

Figure 13 shows the model of the lossy dipole array, where the dipole antenna elements are the same as those in

Figure 7. The antenna S-parameter matrix, which has a loss term due to attenuator insertion, is expressed as

where

$\eta $ is radiation efficiency (

$0\le \eta \le 1$), and

${\mathit{S}}_{L12}={\mathit{S}}_{L21}={\eta}^{1/2}\mathit{I}$. Equation (

22) is correct only if the antenna element is lossless and the attenuator is fully matched. Since reflection does not occur in the attenuator circuit, its S-parameter matrix is expressed as

Figure 14 shows the simulated relationship between the correlation coefficients and radiation efficiency

$\eta $. It can be seen the S-parameter-based correlation coefficient falls as

$\eta $ becomes low. It is found that the antenna even with the efficiency higher than 97% radiation efficiency yields correlation coefficient errors of up to 10%. This means that the S-parameter based correlation coefficient is quite unreliable for most practical compact antennas.

To circumvent this problem, some methods for correcting the correlation coefficient by considering the radiation efficiency have been studied (for example, [

13]). However, it is necessary to choose the right correction method to suit the location at which the loss occurs in the antenna. This means it is difficult to apply these correction methods to all kinds of antennas without knowing the detailed antenna’s current distributions.