MIMO Channel Model with Propagation Mechanism and the Properties of Correlation and Eigenvalue in Mobile Environments

Abstract

This paper described a spatial correlation and eigenvalue in a multiple-input multiple-output (MIMO) channel. A MIMO channel model with a multipath propagation mechanism was proposed and showed the channel matrix. The spatial correlation coefficient formula 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) between MIMO channel matrix elements was derived for the model and was expressed as a directive wave term added to the product of mobile site correlation 𝜌_{𝑖−𝑖′}(𝑚) and base site correlation 𝜌_{𝑗−𝑗′}(𝑏) without LOS path, which are calculated independently of each other. By using 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚), it is possible to create the channel matrix element with a fixed correlation value estimated by 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) for a given multipath condition and a given antenna configuration. Furthermore, the correlation and the channel matrix eigenvalue were simulated, and the simulated and theoretical correlation values agreed well. The simulated eigenvalue showed that the average of the first eigenvalue λ₁ hardly depends on the correlation 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚), but the others do depend on 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) and approach 𝜆₁ as 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) decreases. Moreover, as the path moves into LOS, the 𝜆₁ state with mobile movement becomes more stable than the 𝜆₁ of NLOS path.

1. Introduction

To support realtime multimedia communication, future mobile communications will require a high-bit-rate transmission system with high utilization of the frequency spectrum in multipath channels with line-of-sight (LOS) and non-line-of-sight (NLOS) paths [1]. Systems capable of fulfilling this requirement, with features such as multiple-input multiple-output (MIMO) [2, 3] and orthogonal frequency division multiplexing (OFDM) [4, 5], have been studied extensively. MIMO is especially advantageous in high utilization of the frequency spectrum. In MIMO, transmission quality and capacity depend on the channel matrix, which consists of the complex transmission coefficients between MIMO antenna elements at a mobile terminal and at the base station. The channel matrix seems to be evaluated by the spatial correlation between matrix elements and the matrix eigenvalue, for which low correlation and a large eigenvalue are better [6]. The source of those properties is in the MIMO channel model composed of multipath propagation and the conditions at the mobile and base sites, and many models have been proposed and studied analytically and experimentally [7–10]. As an analytical model that tries to produce the matrix with a given fixed spatial correlation between MIMO antenna elements, the stochastic MIMO channel model was analyzed on the basis of an independent and identically distributed (i.i.d) random matrix for one-side correlations at the mobile and base sites, and it has also been verified experimentally [11, 12]. However, the analysis method used in the model seems to have trouble interpreting the channel situation directly and visualizing the physical propagation in MIMO transmission studies. Moreover, most models proposed to date have been stationary and NLOS, and there has been little analytical work on the correlation between both sides and LOS, which is still being developed.

With this in mind, we proposed a MIMO channel model with a propagation mechanism composed of multipath propagation and mobile- and base-site antenna configurations and created the channel matrix on the basis of the model. The channel matrix is allowed to consist of matrix elements with a given fixed theoretical correlation between antenna elements at the mobile and base sites because the propagation mechanism is known and we can calculate the correlation. Therefore, the matrix requires a formula for estimating the correlation between each pair of antenna elements at one side and at both sides under various multipath conditions and base- and mobile-site situations. So we derived the correlation formulas by using the matrix for indoors, outdoors, and so forth. Using the matrix, we could also calculate the matrix eigenvalue and we clarified its properties by simulation; moreover, it was possible to study the relation between the correlation and eigenvalue.

This paper is organized as follows. Section 2 covers the theoretical study. First, we describe the MIMO channel model with the propagation mechanism and antenna configurations and then show the channel matrix on the basis of the model. Next, we derive the correlation formulas between antenna elements at one side and between both sides in various site conditions and environments. Section 3 covers simulation. The simulation was done for the correlation and channel matrix eigenvalue with various parameter settings. The simulated and theoretical correlations are discussed and the eigenvalue’s properties are described; moreover, the relation between the correlation and the eigenvalue is studied. Finally, Section 4 summarizes the results.

2. Theory

2.1. MIMO Channel Model

MIMO systems will be used in various areas: the cells are called pico, micro, and macro cells. The MIMO channel model, which consists of a delay profile measured around the base- and mobile-site origins and the antenna configurations with coordinate systems common to the profile’s angle, is shown in Figure 1. The coordinate systems have the origin at the first antenna element center with 𝑖=1 for mobile site and 𝑗=1 for base site, respectively. The delay profile has both horizontal azimuth angles (𝜉_𝑛,𝜁_𝑛) incident to multipath scattering and to the receiving point [13–15], except for path data with an ordinary delay profile. Moreover, each arriving wave expresses a representative wave, which is the peak value in each cluster. The delay profile assumed the following conditions.(i)The number of arriving waves is 𝑁+1, waves are independent of each other, and the 𝑛th-path wave is denoted by subscript 𝑛, where 𝑛=0 means a directive wave and 𝑛≥1 means no directive waves.(ii)The waves have excess delay time 𝜏_𝑛 relative to the shortest path between the two origins and maximum excess delay time 𝜏_max. The 𝜏_𝑛 values are random over range 0≤𝜏_𝑛≤𝜏_max. (iii)The amplitude is ℎ_𝑛 and independent of 𝜉_𝑛 and 𝜁_𝑛, the power ratio of the directive and nondirective waves is denoted by 𝑘 (𝐾 in dB: Rice factor 𝑘), and 𝑘=0 (𝐾=−∞ dB) means NLOS. Furthermore, the nondirective wave’s power is normalized to 1.(iv)Whether a wave’s arriving angle at the mobile or base is also the incident angle to multipath scattering from the mobile or base site depends on whether the site is receiving or transmitting. Here, the mobile-site angle is denoted by 𝜉_𝑛 and the base-site angle is denoted by 𝜁_𝑛. The 𝜉_𝑛 and 𝜁_𝑛 are counterclockwise angles from TN (true north) at the origins on both sites; however, when the mobile station moves, 𝜉_𝑛 is the angle from the mobile’s movement direction. The arriving wave’s initial phase is 𝜙_𝑛 and the values are random over 0≤𝜙_𝑛<2𝜋.

Figure 1 

MIMO channel model.

On the other hand, assuming that all the antennas used have the same pattern with omnidirectionality and no mutual coupling, the antenna coordinates use a polar coordinate system centered at each site’s origin, as shown in Figure 1. For the base site, the coordinates are denoted by ANT_𝑏(𝜆_𝑏𝑗,𝜁_𝑏𝑗), where 𝜆_𝑏𝑗 and 𝜁_𝑏𝑗 mean the radius normalized by wavelength λ and counterclockwise angle from TN for the 𝑗th antenna, respectively. For a mobile site, the coordinates are similarly denoted by ANT_𝑚(𝜆_𝑚𝑖,𝜉_𝑚𝑖); however, when the mobile station moves, 𝜉_𝑚𝑖 is also the angle from the mobile’s movement direction.

In this paper, we also assume that all antenna elements of each station have the same values of 𝜉_𝑛 and 𝜁_𝑛, but strictly 𝜉_𝑛 and 𝜁_𝑛 differ slightly among the elements by value δ. The angular difference δ from 𝜉_𝑛 or 𝜁_𝑛 for the normalized distance (𝑟/𝜆) between the origin and multipath scattering when the antenna element is set at spacing λ away from the origin is shown in Figure 2. This δ is less than 0.01 rad when 𝑟/𝜆=100. As shown later, the spatial correlation is sensitive to 𝜉_𝑛 or 𝜁_𝑛 when the antenna is high and far away and when 𝜉_𝑛 and 𝜁_𝑛 have Gaussian distributions, but not so sensitive to 𝜉_𝑛 or 𝜁_𝑛 when 𝑟/𝜆 is small and 𝜉_𝑛 and 𝜁_𝑛 are spread widely, as in the case with mobile stations or indoor cells.

Figure 2 

Angle difference δ from 𝜉𝑛 or 𝜁𝑛 (for space λ from origin).

2.2. MIMO Channel Matrix

Under the conditions described above and assuming a narrowband system such as OFDM, the MIMO channel, which is composed of the 𝑗th base- and 𝑖th mobile-station antenna elements, is denoted by MIMO 𝑐ℎ(𝑖−𝑗), and the complex received signal level 𝐸_𝑖−𝑗(𝑡,𝑓_𝑐) is given by 𝐸_𝑖−𝑗𝑡,𝑓_𝑐=^𝑁_𝑛=0ℎ_𝑛𝑒^{𝑗𝜃𝑖−𝑗,𝑛},𝜃_{𝑖−𝑗,𝑛}=2𝜋𝑓_𝑐𝜏_𝑛−𝑓_𝑚𝑡cos𝜉_𝑛−𝜆_𝑚𝑖cos𝜉_𝑛−𝜉_𝑚𝑖−𝜆_𝑏𝑗cos𝜁_𝑛−𝜁_𝑏𝑗+𝜙_𝑛, (1)(2) where 𝜃_{𝑖−𝑗,𝑛} is the path phase of MIMO 𝑐ℎ(𝑖−𝑗) for the 𝑛th multipath wave, 𝑓_𝑐 is the radio frequency, and 𝑓_𝑚 is the maximum Doppler frequency. The third and fourth terms of 𝜃_{𝑖−𝑗,𝑛} in brackets depend on the mobile- and base-station antenna configurations and mean phase difference from mobile and base origins, respectively.

When the number of antenna elements at each station is M, the MIMO channel matrix E, which describes the connection between both stations, can be expressed as 𝐸𝐸=¹⁻¹𝐸₁₋₂⋯𝐸_1−𝑀𝐸₂₋₁𝐸₂₋₂⋯𝐸_2−𝑀𝐸⋅⋅⋯⋅_𝑀−1𝐸_𝑀−2⋯𝐸_𝑀−𝑀, (3) where 𝐸_𝑖−𝑗 means 𝐸_𝑖−𝑗(𝑡,𝑓_𝑐) in (1).

2.3. Correlation between MIMO Channel Matrix Elements

2.3.1. General Formula

We start by studying the general spatial correlation between MIMO 𝑐ℎ(𝑖−𝑗) and 𝑐ℎ(𝑖^′−𝑗^′); the complex correlation is denoted by 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚). With variables 𝑥 meaning 𝐸_𝑖−𝑗(𝑡,𝑓_𝑐) and 𝑦 meaning 𝐸_{𝑖′−𝑗′}(𝑡,𝑓_𝑐) obtained by (1), 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) is expressed by 𝜌_{𝑖−𝑗,𝑖′−𝑗′}()=⟨𝑥𝑏𝑚(𝑥−⟩)^∗𝑦−𝑦⟨(𝑥−⟨𝑥⟩)^∗(𝑥−⟨𝑥⟩)⟩𝑦−𝑦^∗𝑦−𝑦^1/2. (4) Here, the symbols ⟨⟩ and * mean ensemble average and conjugate complex, respectively. Under the conditions in Section 2.1, ⟨𝑥⟩ and ⟨𝑦⟩ are zero owing to the independence of ℎ_𝑛 and 𝜃_{𝑖−𝑗,𝑛} (or 𝜃_{𝑖′−𝑗′,𝑛}) with random values from 0 to 2𝜋. Therefore, the denominator in (4) is 𝑘+1, that is, received power. On the other hand, ⟨(𝑥−⟨𝑥⟩)^∗(𝑦−⟨𝑦⟩)⟩ in the numerator of (4) is expressed by (5), where Δ𝜃₀ expresses the directive wave’s path phase difference between MIMO 𝑐ℎ(𝑖^′−𝑗^′) and 𝑐ℎ(𝑖−𝑗) (see Appendix A). (𝑥−⟨𝑥⟩)^∗𝑦=ℎ𝑦−₂₀exp𝑗Δ𝜃₀+^𝑁_𝑛=1ℎ²_𝑛^𝑁_𝑛=1cos2𝜋𝑧_{𝑖−𝑖′}cos𝜉_𝑛−𝜓_{𝑖−𝑖′}−𝑗sin2𝜋𝑧_{𝑖−𝑖′}cos𝜉_𝑛−𝜓_{𝑖−𝑖′}×^𝑁_𝑛=1cos2𝜋𝑧_{𝑗−𝑗′}cos𝜁_𝑛−𝜓_{𝑗−𝑗′}−𝑗sin2𝜋𝑧_{𝑗−𝑗′}cos𝜁_𝑛−𝜓_{𝑗−𝑗′}=𝑘exp𝑗Δ𝜃₀+𝜌_{𝑖−𝑖′}(𝑚)⋅𝜌_{𝑗−𝑗′}(𝑏) (5)(6) Here, 𝑧_{𝑖−𝑖′}, Ψ_{𝑖−𝑖′}, and 𝑧_{𝑗−𝑗′}, Ψ_{𝑗−𝑗′} in (5) are antenna construction parameters for the mobile and base sites, respectively. The 𝑧_{𝑖−𝑖′} and Ψ_{𝑖−𝑖′} are given by (7) and mean, respectively, the spacing between the 𝑖th and 𝑖^′th antenna elements and the counterclockwise angle from TN or the movement direction to a line with both these elements. 𝑧_{𝑖−𝑖′}=𝑎²+𝑏^21/2,𝜓_{𝑖−𝑖′}=tan⁻¹𝑏𝑎𝑎=𝜆_𝑚𝑖′cos𝜉_𝑚𝑖′−𝜆_𝑚𝑖cos𝜉_𝑚𝑖,𝑏=𝜆_𝑚𝑖′sin𝜉_𝑚𝑖′−𝜆_𝑚𝑖sin𝜉_𝑚𝑖. (7)(8)

The first and second terms in (5) are concerned with directive and nondirective waves. Moreover, three ensemble averages in the second term are nondirective wave power, that is, 1, mobile and base station factors, which are calculated independently of each other. Therefore, we denote them as shown 𝜌_{𝑖−𝑖′}(𝑚) and 𝜌_{𝑗−𝑗′}(𝑏) in (6). Furthermore, we expand 𝜌_{𝑖−𝑖′}(𝑚) and 𝜌_{𝑗−𝑗′}(𝑏) to a Neumann expansion because the ensemble averages are integrated with respect to 𝜉_𝑛 and 𝜁_𝑛 [16], and we get (9) for the mobile site (see Appendix B). 𝜌_{𝑖−𝑖′}(𝑚)=^𝑁_𝑛=1^∞_𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑖−𝑖′}cos2𝑙𝜉_𝑛−𝜓_{𝑖−𝑖′}−𝑗^𝑁_𝑛=12^∞_𝑙=0(−1)^𝑙𝐽_2𝑙+12𝜋𝑧_{𝑖−𝑖′})×𝜉cos((2𝑙+1_𝑛−𝜓_{𝑖−𝑖′}. (9) Here, 𝜀_𝑙=1(𝑙=0),𝜀_𝑙=2(𝑙≥1),𝐽_2𝑙(⋅) is Bessel function of the first order. We can also get 𝜌_{𝑗−𝑗′}(𝑏) for the base site in a similar manner to that for 𝜌_{𝑖−𝑖′}(𝑚). From the above description, we can get finally the general formula for MIMO channel spatial correlation 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) by rewriting (4) as 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚)=𝑘exp𝑗Δ𝜃₀+𝜌_{𝑖−𝑖′}(𝑚)⋅𝜌_{𝑗−𝑗′}(𝑏),𝑘+1 (10) where 𝜌_{𝑖−𝑖′}(𝑚) and 𝜌_{𝑗−𝑗′}(𝑏) are mobile and base site correlations without a directive wave. Though the value of the numerator in (10) depends strongly on the first term, that is, 𝑘 and Δ𝜃₀, it becomes 𝜌_{𝑖−𝑖′}(𝑚)𝜌_{𝑗−𝑗′}(𝑏) without LOS, so 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) is the product of 𝜌_{𝑖−𝑖′}(𝑚) and 𝜌_{𝑗−𝑗′}(𝑏). Moreover, (10) shows that 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) is calculated from three items: the mobile- and base-site antenna configurations (𝑧_{𝑖−𝑖′},𝑧_{𝑗−𝑗′},Ψ_{𝑖−𝑖′},Ψ_{𝑗−𝑗′}), the angle distribution for arriving and incident waves to multiple paths (𝜉_𝑛,𝜁_𝑛), and Rice factor 𝑘. By using (10), we can calculate 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) for any MIMO channel matrix since 𝑖,𝑖^′,𝑗,and 𝑗^′ can be chosen freely within 𝑀. Here 𝑖=𝑖^′ and 𝑗=𝑗^′ mean 𝜌_{𝑖−𝑖′}(𝑚)=1 and 𝜌_{𝑗−𝑗′}(𝑏)=1, respectively.

2.3.2. Example of Correlation Coefficient 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚)

The 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) in (10) contains the product of 𝜌_{𝑖−𝑖′}(𝑚) and 𝜌_{𝑗−𝑗′}(𝑏), which depend on the site environments and are independent of each other, though these environments might sometimes be the same. Therefore, to get 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚), it is sufficient to prepare just one side for various situations because we can get 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) by combining them. Therefore, we studied the correlation for two typical distributions with uniform 𝜉_𝑛 and Gaussian 𝜁_𝑛.

(i) For Uniform Distribution of 𝜉_𝑛
We first calculate 𝜌_{𝑖−𝑖′}(𝑚) by (9) when 𝜉_𝑛 has a uniform distribution centered at 𝜉_𝑐 over 𝜉_𝑐−Δ𝜉_𝑛<𝜉_𝑛<𝜉_𝑐+Δ 𝜉_𝑛 with the probability density function as 𝑝𝑑𝑓(𝜉_𝑛)=1/2Δ𝜉_𝑛. Assuming a large 𝑁,we can calculate the ensemble average in (9) by integration with respect to 𝜉_𝑛, and get (11) (see Appendix C). 𝜌_{𝑖−𝑖′}(𝑚)=^∞_𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑖−𝑖′}12𝑙Δ𝜉_𝑛sin2𝑙Δ𝜉_𝑛×cos2𝑙𝜉_𝑐−𝜓_{𝑖−𝑖′}−𝑗2^∞_𝑙=0(−1)^𝑙𝐽_2𝑙+12𝜋𝑧_{𝑖−𝑖′}1(2𝑙+1)Δ𝜉_𝑛×sin(2𝑙+1)Δ𝜉_𝑛cos(2𝑙+1)𝜉_𝑐−𝜓_{𝑖−𝑖′}. (11)

(ii) For Gaussian Distribution of 𝜁_𝑛
Next, we calculate 𝜌_{𝑗−𝑗′}(𝑏) when 𝜁_𝑛 has a Gaussian distribution centered at 𝜁_𝑐 with deviation σ. Similar to the uniform distribution case, we get (12) (see Appendix D). 𝜌_{𝑗−𝑗′}(𝑏)=^∞_𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑗−𝑗′}cos2𝑙𝜁_𝑐−𝜓_{𝑗−𝑗′}×exp−2𝑙²𝜎²−𝑗2^∞_𝑙=0(−1)^𝑙𝐽_2𝑙+12𝜋𝑧_{𝑗−𝑗′}×cos(2𝑙+1)𝜁_𝑐−𝜓_{𝑗−𝑗′}−()exp2𝑙+1²𝜎²2. (12)

3. Simulation

3.1. Simulation Method

A computer simulation was performed to verify (10), (11), and (12) and to study the relation between the MIMO channel matrix correlation and eigenvalue. The simulation parameters are listed in Table 1, assuming pico, micro, and macro cells indoors and outdoors. As suggested by (10), we need to simulate two items for the correlation: mobile or base station one-side channel, that is, 𝜌_{𝑖−𝑖′}(𝑚) or 𝜌_{𝑗−𝑗′}(𝑏), and mobile and base stations both-side channel, that is, 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚). So we simulated the correlation with uniform-in-𝜉_𝑛 and Gaussian-in-𝜁_𝑛 distributions for each item. Concerning the eigenvalue, its dependence on the correlation was simulated while changing the mobile and base site conditions and environments. The radio frequency 𝑓_𝑐 was 3 GHz, and the channel model with the delay profile in Section 2.1 was used. The nondirective wave amplitude ℎ_𝑛 exponentially decreases with increasing 𝜏_𝑛, and the effective amplitude is greater than −25 dB relative to the maximum one. The simulation was performed using (1), (2), (3), and (4); the incident and arriving angles (𝜉_𝑛, 𝜁_𝑛) and antenna parameters (𝜆_𝑚, 𝜉_𝑚,𝜆_𝑏, 𝜁_𝑏) were set as shown in Table 1. Each simulated value was calculated from an ensemble average for more than 10⁶ delay profiles, except for eigenvalue variation with movement in Section 3.3.

Table 1 

Simulation parameters.

3.2. Correlation Coefficient of MIMO Channel

3.2.1. Mobile or Base Station One-Side Channel

Figure 3 shows the absolute value of the simulated correlation for a mobile- or base-station one-side channel assuming a mobile station indoors for Figures 3(a) and 3(b) and a base station outdoors for Figure 3(c), when the antenna’s spacing 𝑧=𝑧_{𝑖−𝑖′}(𝑧_{𝑗−𝑗′}) and setting angle Ψ_{𝑖−𝑖′}(Ψ_{𝑗−𝑗′})=0. Figures 3(a) and 3(b) have arriving angle 𝜉_𝑛 with uniform distribution and the parameter Δ𝜉_𝑛 means the 𝑛th arriving wave uniformly from the direction 𝜉_𝑐−Δ𝜉_𝑛<𝜉_𝑛<𝜉_𝑐+Δ𝜉_𝑛. Figures 3(a) and 3(b) are for NLOS and LOS with 𝐾=5 dB and 𝜉₀=5𝜋/6 paths, respectively, and the correlation of NLOS fluctuates less with increasing 𝑧, but that of LOS is higher than that of NLOS and is close to a fixed value 𝑘/(𝑘+1) fluctuating with increasing 𝑧. Figures 3(a) and 3(b) also show the theoretical value |𝜌_{𝑖−𝑖′}(𝑚)| calculated by (10) with (11); the simulated and theoretical values agree well. The 𝜌_{𝑖−𝑖′}(𝑚) at Δ𝜉_𝑛=𝜋 in Figure 3(a) becomes 𝐽_𝑜(2𝜋𝑧_{𝑖−𝑖′}), as is well known. Figure 3(c) shows the simulated correlation for arriving angle 𝜁_𝑛 with Gaussian distribution centered at 𝜁_𝑐=𝜋/6 with standard deviation σ. The correlation value decreases monotonically with increasing 𝑧 and the simulated values agree well with the theoretical values |𝜌_{𝑗−𝑗′}(𝑏)| obtained by (10) using(12)

(a)

(b)

(c)

Figure 3 

Correlation of one-side channel. (a) NLOS indoors (uniform distribution). (b) LOS indoors (uniform distribution). (c) NLOS outdoors (Gaussian distribution).

Figure 4 shows the dependence of the correlation on centered arriving angle in NLOS for a one-side channel. Figures 4(a) and 4(b) are the cases for angle 𝜉_𝑛 with uniform distribution centered on 𝜉_𝑐 with Δ𝜉_𝑛, and angle 𝜁_𝑛 with Gaussian distribution centered on 𝜁_𝑐 with σ, respectively. The correlation in Figure 4(a) was simulated by changing 𝜉_𝑐 over the range from 0 to 𝜋 while keeping 𝑧_{𝑖−𝑖′}=0.5 and Ψ_{𝑖−𝑖′}=0. The correlation values for Δ𝜉_𝑛=𝜋/4 and 𝜋/2 have minima at 𝜉_𝑐=π/2 and become larger far away from 𝜉_𝑐=𝜋/2, but the value for Δ𝜉_𝑛=𝜋 does not depend on 𝜉_𝑐 since arriving waves arrive from all directions from 0 to 2π. The theoretical value |𝜌_{𝑖−𝑖′}(𝑚)| was calculated using (10), that is, (11); the theoretical and simulated values agree well. The correlation |𝜌_{𝑗−𝑗′}(𝑏)| in Figure 4(b) for the Gaussian distribution was also simulated in a similar way to Figure 4(a), expect for 𝑧_{𝑗−𝑗′}=2. Though |𝜌_{𝑗−𝑗′}(𝑏)| has a minimum at 𝜉_𝑐=𝜋/2 and becomes larger far away from 𝜁_𝑐=𝜋/2 like in Figure 4(a), the values at 𝜁_𝑐=𝜋/2 depend on standard deviation σ, and the minimum value of |𝜌_{𝑗−𝑗′}(𝑏)| is large when σ is small. The theoretical value was calculated using (10), that is, (12); the theoretical and simulated values agree well.

(a)

(b)

Figure 4 

Correlation of dependence on arrival angle for one-side channel. (a) Uniform distribution. (b) Gaussian distribution.

3.2.2. Mobile and Base Station Both-Side Channel

Figure 5 shows the simulated correlation coefficient |𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚)|, that is, 𝑖≠𝑖^′ at the mobile station and 𝑗≠𝑗^′ at the base station for incident 𝜁_𝑛 and arriving 𝜉_𝑛 with the same value of Δ𝜉_𝑛=Δ𝜁_𝑛 in the both-side uniform distribution, assuming that the mobile and base stations are indoors. Figure 5(a) shows the simulated correlation when changing 𝑧=𝑧_{𝑖−𝑖′}=𝑧_{𝑗−𝑗′} and keeping Ψ_{𝑖−𝑖′}=Ψ_{𝑗−𝑗′}=0. The correlation decreases faster than in Figure 3(a). The reason for this can be seen from the theory that the theoretical value 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) from (10) is the product of 𝜌_{𝑖−𝑖′}(𝑚) and 𝜌_{𝑗−𝑗′}(𝑏), or here |𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚)|=|𝜌_{𝑖−𝑖′}(𝑚)|². The simulated and theoretical values agree well. Figure 5(b) is the simulated value when changing 𝑧=𝑧_{𝑗−𝑗′}, like Figure 5(a), except keeping 𝑧_{𝑖−𝑖′}=0.5. So the value of the correlation at 𝑧=0 is less than 1. The theoretical values from (10) at 𝑧_{𝑖−𝑖′}=0.5 are also shown in Figure 5(b); the theoretical and simulated values agree well.

(a)

(b)

Figure 5 

Correlation of both-side channel (uniform-uniform distribution). (a) 𝑍=𝑍𝑖−𝑖′=𝑍𝑗−𝑗′. (b) 𝑍𝑖−𝑖′=0.5.

Figure 6 shows the simulated correlation |𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚)| for incident 𝜉_𝑛 with uniform distribution (𝜉_𝑐=0,Δ𝜉_𝑛=𝜋) and arriving 𝜁_𝑛 with Gaussian distribution (𝜁_𝑐=𝜋/6, σ: parameter), assuming mobile and base sites outdoors. Figure 6(a) was simulated like Figure 5(a), that is, changing 𝑧 and keeping σ fixed; compared with those in Figure 3(c), the simulated values become small rapidly with increasing 𝑧 owing to the 𝜉_𝑛 with uniform distribution at another site. The theoretical value 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) from (10) was calculated as corresponding to (11) for the mobile station 𝜌_{𝑖−𝑖′}(𝑚) and to (12) for the base station 𝜌_{𝑗−𝑗′}(𝑏), respectively. The theoretical and simulated values agree well. Figure 6(b) is the simulated correlation with changing 𝑧=𝑧_{𝑗−𝑗′} as in Figure 6(a), except keeping 𝑧_{𝑖−𝑖′}=0.5. So the simulated value is less than 0.3 whenever 𝑧 is small since we kept 𝑧_{𝑖−𝑖′}=0.5. The theoretical value from (10) was also calculated; the theoretical and simulated values agree well.

(a)

(b)

Figure 6 

Correlation of both-side channel (uniform-Gaussian distribution). (a) 𝑍=𝑍𝑖−𝑖′=𝑍𝑗−𝑗′. (b) 𝑍𝑖−𝑖′=0.5.

3.3. Eigenvalue of MIMO Channel Matrix

All of the eigenvalues were simulated by a MIMO antenna with a 4×4 element configuration with elements placed with equal spacing and on a line in order to obtain the basic to eigenvalue properties.

3.3.1. Eigenvalue Property with Movement

Figure 7 shows an example of eigenvalue variation with movement of the mobile station calculated every 0.05 wavelength on the MIMO channel matrix by (3) in multipath fading. Figures 7(a) and 7(b) were simulated using the same delay profile with both-side uniform distribution with 𝜉_𝑐=0 and Δ𝜉_𝑛=𝜋 in NLOS; the only difference was the antenna radius 𝜆_𝑚𝑖=𝜆_𝑏𝑗=0.25 and 0.07 to make low and high correlations by (11) or the theoretical correlation between one MIMO antenna element and the next one |𝜌_{𝑖−𝑖+1}(𝑚)|=|𝜌_{𝑗−𝑗+1}(𝑏)|=0.45 and 0.95, respectively. Here, ρ in Figure 7 means |𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚)| from (10). Moreover, Figure 7(c) was simulated under the condition in Figure 7(a) with just the addition of a direct wave with 𝐾=5[dB] to the delay profile, when the total power of the profile was normalized to 1, and the ρ by (10) is 0.7. Comparing Figures 7(a) and 7(b), we see that the first eigenvalue 𝜆₁ at 𝜌=0.2 in Figure 7(a) is almost equal to that at 𝜌=0.9 in Figure 7(b) on average but that the other eigenvalues 𝜆₂, 𝜆₃, and 𝜆₄ in Figure 7(a) are larger than that in Figure 7(b) (𝜆₄ is less than 10⁻⁵). Moreover, each eigenvalue variation with movement in Figure 7(a) is smaller than the corresponding one in Figure 7(b). On the other hand, the average value of each eigenvalue in Figure 7(c) in LOS seems to be similar to that in Figure 7(a), but the state is more stable, especially for 𝜆₁, than that in NLOS.

(a)

(b)

(c)

Figure 7 

Example of eigenvalue variation with movement (4×4 MIMO, uniform distribution). (a) NLOS (𝜌=0.2). (b) NLOS (𝜌=0.9). (c) LOS (𝐾=5[dB], 𝜌=0.7).

3.3.2. Dependence of Eigenvalue on Correlation

Figure 8 shows the dependence of the eigenvalue on correlation with the property of antenna space and multipath channel by a cumulative distribution. All the eigenvalue curves in Figure 8(a), that is, at 𝜌=0.2 and 0.9 in NLOS and at 𝜌=0.7 in LOS, were simulated under the same conditions as in Figures 7(a), 7(b), and 7(c), respectively, except for the use of only one delay profile, or at this time the use of more than 10⁶ profiles. As assumed in Figure 7, Figure 8(a) seems to suggest the following: though the 50% cumulative values in 𝜆₁ are weakly dependent on ρ and almost equal, the other eigenvalues 𝜆₂, 𝜆₃, and 𝜆₄ are dependent on ρ and K, and the 𝜆₁ in LOS is the most stable. Figure 8(b) shows the dependence of the eigenvalue on correlation for 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚)=𝜌_{𝑖−𝑖+1}(𝑚)𝜌_{𝑗−𝑗+1}(𝑏) in NLOS, where the simulation was done changing antenna space correspondent to |𝜌_{𝑖−𝑖+1}(𝑚)|=0.2, 0.6, and 0.9 at the mobile station with a uniform distribution keeping a high |𝜌_{𝑗−𝑗+1}(𝑏)|=0.9 at the base station in the Gaussian distribution with 𝜁_𝑐=𝜋/6 and 𝜎=𝜋/36. Figure 8(b) also shows that the 50% cumulative values have little dependence on 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) for 𝜆₁, but are dependent on 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) for the other eigenvalues: the smaller 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) is, the closer the values are to 𝜆₁. All the eigenvalues tend to have a wider distribution with increasing 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚).

(a)

(b)

Figure 8 

Dependence of eigenvalue on correlation (4×4 MIMO). (a) Uniform-uniform distribution. (b) Uniform-Gaussian distribution.

Figure 9 shows the dependence of the eigenvalue on correlation 𝜌_{𝑗−𝑗+1}(𝑏) with the properties of standard deviation σ and centering angle 𝜁_𝑐 in Gaussian distribution, as shown in Figure 4(b). Simulation was done for 𝜌_{𝑗−𝑗+1}(𝑏) with σ and 𝜁_𝑐 as parameters, while the mobile station condition was set to a constant at |𝜌_{𝑖−𝑖+1}(𝑚)|=0.9 in a uniform distribution with 𝜉_𝑐=0 and Δ𝜉_𝑛=𝜋. Figure 9(a) shows the influence of σ on the eigenvalue with 𝜎=𝜋/18, 𝜋/36, and 𝜋/90 as parameters at 𝜁_𝑐=𝜋/6 and 𝑧_{𝑗−𝑗′}=2. These 50% cumulative values are also weakly dependent on σ for 𝜆₁, but do depend on σ for the other eigenvalues, and they become larger when σ becomes large, that is, 𝜌_{𝑗−𝑗+1}(𝑏) is small. Figure 9(b) shows the influence of 𝜁_𝑐 on the eigenvalue, when 𝜁_𝑐=𝜋/2, 𝜋/4, and 0 at 𝜎=𝜋/18 and 𝑧_{𝑗−𝑗′}=2. The dependence of the eigenvalues on 𝜁_𝑐 is similar to σ in Figure 9(a). Figure 9 suggests that the σ and 𝜁_𝑐 are parameters for eigenvalue property.

(a)

(b)

Figure 9 

Dependence of eigenvalue on Gaussian distribution (4×4 MIMO). (a) Effect of standard deviation. (b) Effect of centering arriving angle.

4. Conclusion

To study MIMO channel properties, we proposed a MIMO channel model with a propagating mechanism composed of multipath propagation and antenna configurations and then showed the MIMO channel matrix. Under this model, the spatial correlation formula 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) between MIMO channel matrix elements was derived: the formula 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) was expressed as a directive wave term added to the product of mobile site correlation 𝜌_{𝑖−𝑖′}(𝑚) and base site correlation 𝜌_{𝑗−𝑗′}(𝑏), which are calculated independently of each other, divided by 𝑘+1. This formula can be applied to create the channel matrix element with a fixed value of correlation estimated by 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) for given multipath conditions and antenna configurations. Furthermore, simulation was done for the correlation and channel matrix eigenvalue indoors, outdoors, and for movement. The simulated and theoretical values of the correlation agree well. The simulated eigenvalue shows that the average of the first eigenvalue 𝜆₁ is hardly dependent on the correlation 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚), but the other ones are dependent on 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) and become close to 𝜆₁ with decreasing 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚). Moreover, the 𝜆₁ state with mobile movement in LOS path is more stable than the 𝜆₁ of NLOS path. The MIMO channel model and derived 𝜌_{𝑖−𝑗,𝑖′−𝑗′}(𝑏𝑚) make it possible to create a MIMO channel matrix with a fixed value correlation and furthermore to study the relation between the correlation and eigenvalue for various cell sites and environments.

Appendices

A. Derivation of (5)

Assuming that 𝑁 is a large number for the delay profile in Section 2.1, then because 𝜏_𝑛 and 𝜙_𝑛 have random values, 𝜃_{𝑖−𝑗,𝑛} obtained by (2) is a random value over the range from 0 to 2π and independent of ℎ_𝑛. So we get ⟨𝑥⟩=⟨∑^𝑁_𝑛=0ℎ_𝑛⟩⟨∑^𝑁_𝑛=0exp(𝑗𝜃_{𝑖−𝑗,𝑛})⟩=0 since the term ⟨∑^𝑁_𝑛=0exp(𝑗𝜃_{𝑖−𝑗,𝑛})⟩ becomes zero, that is, ⟨𝑥⟩ and ⟨𝑦⟩ in (4) are zero. Therefore, the denominator in (4) becomes [⟨𝑥^∗𝑥⟩⟨𝑦^∗𝑦⟩]^1/2, that is, 𝑘+1. On the other hand, ⟨(𝑥−⟨𝑥⟩)^∗(𝑦−⟨𝑦⟩)⟩ in the numerator in (4) can be modified to ⟨𝑥^∗𝑦⟩=⟨𝐸_𝑖−𝑗(𝑡,𝑓_𝑐)^∗𝐸_{𝑖′−𝑗′}(𝑡,𝑓_𝑐)⟩. As a result, the sum of products 𝐸_𝑖−𝑗(𝑡,𝑓_𝑐)^∗ and 𝐸_{𝑖′−𝑗′}(𝑡,𝑓_𝑐) remains the same for the 𝑛th arriving wave on MIMO 𝑐ℎ(𝑖−𝑗) and 𝑐ℎ(𝑖^′−𝑗^′), but vanishes each other differences for the 𝑛th arriving wave on MIMO 𝑐ℎ(𝑖−𝑗) and 𝑐ℎ(𝑖^′−𝑗^′) because the values of 𝜃_{𝑖′−𝑗′,𝑛′}−𝜃_{𝑖−𝑗,𝑛} are random; moreover, ℎ_𝑛ℎ_𝑛′ and 𝜃_{𝑖′−𝑗′,𝑛′}−𝜃_{𝑖−𝑗,𝑛} are independent of each other. Therefore, we get (A.1) while considering ℎ_𝑛 and Δ𝜃_𝑛; moreover, mobile and base site factors are independent of each other, and the directive wave is deterministic and the amplitude ℎ₀ is much larger than the others ℎ_𝑛. (𝑥−⟨𝑥⟩)^∗𝑦=𝐸𝑦−_𝑖−𝑗𝑡,𝑓_𝑐^∗𝐸_{𝑖′−𝑗′}𝑡,𝑓_𝑐=^𝑁_𝑛=0ℎ²_𝑛exp𝑗Δ𝜃_𝑛+^𝑁′_𝑛=0^𝑁_𝑛′^′₌₀ℎ_𝑛ℎ_𝑛′×exp𝑗𝜃_{𝑖′−𝑗′,𝑛′}−𝜃_{𝑖−𝑗,𝑛}=^𝑁_𝑛=0ℎ²_𝑛exp𝑗Δ𝜃_𝑛=ℎ²₀exp𝑗Δ𝜃₀+^𝑁_𝑛=1ℎ²_𝑛^𝑁_𝑛=1exp−𝑗2𝜋𝜆_𝑚𝑖′cos𝜉_𝑛−𝜉_𝑚𝑖′−𝜆_𝑚𝑖cos𝜉_𝑛−𝜉_𝑚𝑖×^𝑁_𝑛=1exp−𝑗2𝜋𝜆_𝑏𝑗′cos𝜁_𝑛−𝜁_𝑏𝑗′−𝜆_𝑏𝑗cos𝜁_𝑛−𝜁_𝑏𝑗, (A.1) where 𝑛≠𝑛^′ in ∑^𝑁′_𝑛=0∑^𝑁_𝑛′^′₌₀. Δ𝜃_𝑛=−2𝜋𝜆_𝑚𝑖′cos𝜉_𝑛−𝜉_𝑚𝑖′+𝜆_𝑏𝑗′cos𝜁_𝑛−𝜁_𝑏𝑗′−𝜆_𝑚𝑖cos𝜉_𝑛−𝜉_𝑚𝑖−𝜆_𝑏𝑗cos𝜁_𝑛−𝜁_𝑏𝑗. (A.2)

The Δ𝜃_𝑛 yielded by (A.2) expresses the phase difference between MIMO 𝑐ℎ(𝑖^′−𝑗^′) and 𝑐ℎ(𝑖−𝑗) for the 𝑛th arriving wave, or Δ𝜃_𝑛=𝜃_{𝑖′−𝑗′,𝑛}−𝜃_{𝑖−𝑗,𝑛}. Equation (A.1) has two terms, that is, directive and no directive wave terms. Moreover, since the second term can be classified into no directive power, mobile-site, and base-site factors, the classified factors are calculated independently of each other. So we used the ensemble average of mobile and base sites as 𝜌_{𝑖−𝑖′}(𝑚) and 𝜌_{𝑗−𝑗′}(𝑏), respectively, where ⟨∑^𝑁_𝑛=1ℎ²_𝑛⟩=1.

Furthermore, by continuously analyzing 𝜌_{𝑖−𝑖′}(𝑚), expanding cos(𝜉_𝑛−𝜉_𝑚𝑖′) and cos(𝜉_𝑛−𝜉_𝑚𝑖) in (A.1), and gathering terms of cos𝜉_𝑛 and sin𝜉_𝑛, and with a little modification we get (A.3) and express that as the third ensemble average in (5). 𝜌_{𝑖−𝑖′}(𝑚)=^𝑁_𝑛=1exp−𝑗2𝜋𝜆_𝑚𝑖′cos𝜉_𝑛−𝜉_𝑚𝑖′−𝜆_𝑚𝑖cos𝜉_𝑛−𝜉_𝑚𝑖=^𝑁_𝑛=1exp−𝑗2𝜋𝜆_𝑚𝑖′cos𝜉_𝑚𝑖′−𝜆_𝑚𝑖cos𝜉_𝑚𝑖cos𝜉_𝑛+𝜆_𝑚𝑖′sin𝜉_𝑚𝑖′−𝜆_𝑚𝑖sin𝜉_𝑚𝑖sin𝜉_𝑛=^𝑁_𝑛=1exp−𝑗2𝜋𝑧_{𝑖−𝑖′}cos𝜉_𝑛−𝜓_{𝑖−𝑖′}. (A.3) Here, 𝑧_{𝑖−𝑖′} and Ψ_{𝑖−𝑖′} are antenna construction parameters obtained from (7) for a mobile site. The 𝜌_{𝑗−𝑗′}(𝑏) for the base site is calculated in a similar manner to 𝜌_{𝑖−𝑖′}(𝑚).

B. Derivation of (9)

We expand the real and imaginary parts for 𝜌_{𝑖−𝑖′}(𝑚) in (5) into a Neumann expansion and get (9). 𝜌_{𝑖−𝑖′}(𝑚)=^𝑁_𝑛=1cos2𝜋𝑧_{𝑖−𝑖′}cos𝜉_𝑛−𝜓_{𝑖−𝑖′}−𝑗sin2𝜋𝑧_{𝑖−𝑖′}cos𝜉_𝑛−𝜓_{𝑖−𝑖′}=^𝑁_𝑛=1^∞_𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑖−𝑖′}cos2𝑙𝜉_𝑛−𝜓_{𝑖−𝑖′}−𝑗^𝑁_𝑛=12^∞_𝑙=0(−1)^𝑙𝐽_2𝑙+12𝜋𝑧_{𝑖−𝑖′}×cos(2𝑙+1)𝜉_𝑛−𝜓_{𝑖−𝑖′}. (B.1)

C. Derivation of (11)

The real and imaginary parts of 𝜌_{𝑖−𝑖′}(𝑚) in (B.1) are denoted 𝐴 and 𝐵. First, we calculate the real part 𝐴. Assuming a large N, when 𝜉_𝑛 has a uniform distribution centered at 𝜉_𝑐 over 𝜉_𝑐−Δ𝜉_𝑛<𝜉_𝑛<𝜉_𝑐+Δ𝜉_𝑛, and the probability density function of 𝜉_𝑛 is 𝑝𝑑𝑓(𝜉_𝑛)=1/2Δ𝜉_𝑛, we can calculate the ensemble average in (9) by integration with respect to 𝜉_𝑛, replacing variable 𝜉_𝑛−Ψ_{𝑖−𝑖′} by u. 𝜌_{𝑖−𝑖′}(𝑚)=𝐴−𝑗𝐵,𝐴=^𝑁_𝑛=1^∞_𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑖−𝑖′}cos2𝑙𝜉_𝑛−𝜓_{𝑖−𝑖′}=^∞_𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑖−𝑖′}12Δ𝜉_𝑛×^{𝜉𝑐+Δ𝜉𝑛}_{𝜉𝑐−Δ𝜉𝑛}cos2𝑙𝜉_𝑛−𝜓_{𝑖−𝑖′}𝑑𝜉_𝑛=^∞_𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑖−𝑖′}12Δ𝜉_𝑛sin(2𝑙𝑢)2𝑙^{𝜉𝑐−𝜓′𝑖−𝑖+Δ𝜉𝑛}_{𝜉𝑐−𝜓′𝑖−𝑖−Δ𝜉𝑛}=^∞_𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑖−𝑖′}12𝑙Δ𝜉_𝑛sin2𝑙Δ𝜉_𝑛×cos2𝑙𝜉_𝑐−𝜓_{𝑖−𝑖′}. (C.1) The imaginary part 𝐵 can be also calculated similarly to A; we get (C.2): 𝐵=2^∞_𝑙=0(−1)^𝑙𝐽_2𝑙+12𝜋𝑧_{𝑖−𝑖′}1(2𝑙+1)Δ𝜉_𝑛sin(2𝑙+1)Δ𝜉_𝑛×cos(2𝑙+1)𝜉_𝑐−𝜓_{𝑖−𝑖′}. (C.2)

D. Derivation of (12)

The real and imaginary parts for 𝜌_{𝑗−𝑗′}(𝑏) in (5) are denoted 𝐴 and 𝐵. We begin by calculating the real part 𝐴. Assuming that 𝜁_𝑛 has a Gaussian distribution centered at 𝜁_𝑐 with standard deviation σ by (D.1), first the real part 𝐴 is expanded into a Neumann expansion and then the ensemble average is calculated by integration with respect to 𝜉_𝑛 as follows. 𝑝𝑑𝑓∶𝑝𝜁_𝑛=1√2𝜋𝜎²𝜁exp^𝑛−𝜁_𝑐²2𝜎²,𝜌_{𝑗−𝑗′}(𝑏)=^𝑁_𝑛=1exp−𝑗2𝜋𝑧_{𝑗−𝑗′}cos𝜁_𝑛−𝜓_{𝑗−𝑗′}=𝐴−𝑗𝐵,𝐴=^𝑁_𝑛=1cos2𝜋𝑧_{𝑗−𝑗′}cos𝜁_𝑛−𝜓_{𝑗−𝑗′}=^𝑁_𝑛=1^∞_𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑗−𝑗′}cos2𝑙𝜁_𝑛−𝜓_{𝑗−𝑗′}=^∞_𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑗−𝑗′}^𝜁𝑐_𝜁^+𝜋_𝑐−𝜋cos2𝑙𝜁_𝑛−𝜓_{𝑗−𝑗′}×𝑝𝜁_𝑛𝑑𝜁_𝑛. (D.1)(D.2)

Putting 𝜁_𝑛−𝜁_𝑐=𝑢, with an odd function of sine and assuming (𝜋/2𝜎)²≫1 and with a little modification, we get 1𝐴=√2𝜋𝜎^2∞𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑗−𝑗′}×^𝜋_−𝜋cos2𝑙𝑢+𝜁_𝑐−𝜓_{𝑗−𝑗′}−𝑢exp²2𝜎²=1𝑑𝑢√2𝜋𝜎^2∞𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑗−𝑗′}cos2𝑙𝜁_𝑐−𝜓_{𝑗−𝑗′}×^𝜋_−𝜋()−𝑢cos2𝑙𝑢exp²2𝜎²𝑑𝑢=^∞_𝑙=0𝜀_𝑙(−1)^𝑙𝐽_2𝑙2𝜋𝑧_{𝑗−𝑗′}cos2𝑙𝜁_𝑐−𝜓_{𝑗−𝑗′}×exp−2𝑙²𝜎². (D.3) The imaginary part 𝐵 can also be calculated in a similar manner to 𝐴. We get (D.4). 𝐵=2^∞_𝑙=0(−1)^𝑙𝐽_2𝑙+12𝜋𝑧_{𝑗−𝑗′}cos(2𝑙+1)𝜁_𝑐−𝜓_{𝑗−𝑗′}−()×exp2𝑙+1²𝜎²2. (D.4)