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

Authors: Yuuki Kanemiyo; Youhei Tsukamoto; Hiroaki Nakabayashi; Shigeru Kozono
Journal: International Journal of Antennas and Propagation
 (2012)

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## 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 formulaρi−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 ρi−i′(m) and base site correlation ρj−j′(b) without LOS path, which are calculated independently of each other. By using ρi−j,i′−j′(bm), it is possible to create the channel matrix element with a fixed correlation value estimated by ρi−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 ρi−j,i′−j′(bm), but the others do depend on ρi−j,i′−j′(bm) and approach λ1 as ρi−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.


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## Body

## 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. Section2 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 Figure1. 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 (ξn,ζn) 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 isN+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 timeτn relative to the shortest path between the two origins and maximum excess delay time τmax. The τn values are random over range 0≤τn≤τmax.(iii)
The amplitude ishn and independent of ξn and ζn, 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 byξn and the base-site angle is denoted by ζn. The ξn and ζn are counterclockwise angles from TN (true north) at the origins on both sites; however, when the mobile station moves, ξn is the angle from the mobile’s movement direction. The arriving wave’s initial phase is ϕn and the values are random over 0≤ϕn<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 Figure1. For the base site, the coordinates are denoted by ANTb(λbj,ζbj), where λbj and ζbj 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 ANTm(λmi,ξmi); however, when the mobile station moves, ξmi 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 ofξn and ζn, but strictly ξn and ζn differ slightly among the elements by value δ. The angular difference δ from ξn or ζn 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 ξn or ζn when the antenna is high and far away and when ξn and ζn have Gaussian distributions, but not so sensitive to ξn or ζn when r/λ is small and ξn and ζn are spread widely, as in the case with mobile stations or indoor cells.Figure 2
Angle differenceδ 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 thejth base- and ith mobile-station antenna elements, is denoted by MIMO ch(i-j), and the complex received signal level Ei-j(t,fc) is given by
(1)Ei-j(t,fc)=∑n=0Nhnejθi-j,n,(2)θi-j,n=2π[(ζn-ζbj)fcτn-fmtcos⁡ξn-λmicos⁡(ξn-ξmi)2ππ-λbjcos⁡(ζn-ζbj)]+ϕn,
where θi-j,n is the path phase of MIMO ch(i-j) for the nth multipath wave, fc is the radio frequency, and fm is the maximum Doppler frequency. The third and fourth terms of θi-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 isM, the MIMO channel matrix E, which describes the connection between both stations, can be expressed as
(3)E=[E1-1E1-2⋯E1-ME2-1E2-2⋯E2-M⋅⋅⋯⋅EM-1EM-2⋯EM-M],
where Ei-j means Ei-j(t,fc) in (1).


### 2.3. Correlation between MIMO Channel Matrix Elements

#### 2.3.1. General Formula

We start by studying the general spatial correlation between MIMOch(i-j) and ch(i′-j′); the complex correlation is denoted by ρi-j,i′-j′(bm). With variables x meaning Ei-j(t,fc) and y meaning Ei′-j′(t,fc) obtained by (1), ρi-j,i′-j′(bm) is expressed by
(4)ρi-j,i′-j′(bm)=〈(x-〈x〉)*(y-〈y〉)〉[〈(x-〈x〉)*(x-〈x〉)〉〈(y-〈y〉)*(y-〈y〉)〉]1/2.
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 hn and θi-j,n (or θi′-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). 										(5)〈(x-〈x〉)*(y-〈y〉)〉=〈h02exp⁡(jΔθ0)〉+〈∑n=1Nhn2〉〈∑n=1N[cos⁡(2πzi-i′cos⁡(ξn-ψi-i′))∑n=1N×-jsin⁡(2πzi-i′cos⁡(ξn-ψi-i′))]∑n=1N〉×〈∑n=1N[cos⁡(2πzj-j′cos⁡(ζn-ψj-j′))∑n=1N×-jsin⁡(2πzj-j′cos⁡(ζn-ψj-j′))]∑n=1N〉(6)=kexp⁡(jΔθ0)+ρi-i′(m)⋅ρj-j′(b)
Here, zi-i′, Ψi-i′, and zj-j′, Ψj-j′ in (5) are antenna construction parameters for the mobile and base sites, respectively. The zi-i′ and Ψi-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. 										(7)zi-i′=[a2+b2]1/2,ψi-i′=tan-1(ba)(8)a=λmi′cos⁡ξmi′-λmicos⁡ξmi,b=λmi′sin⁡ξmi′-λmisin⁡ξmi.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 ρi-i′(m) and ρj-j′(b) in (6). Furthermore, we expand ρi-i′(m) and ρj-j′(b) to a Neumann expansion because the ensemble averages are integrated with respect to ξn and ζn [16], and we get (9) for the mobile site (see Appendix B). 										(9)ρi-i′(m)=〈∑n=1N[∑l=0∞εl(-1)lJ2l(2πzi-i′)cos⁡⁡(2l(ξn-ψi-i′))]〉-j〈∑n=1N[2∑l=0∞(-1)lJ2l+1(2πzi-i′)cos⁡⁡((2l+1)∑l=0∞×(ξn-ψi-i′))∑l=0∞]〉.
Here, εl=1(l=0),εl=2(l≥1),J2l(·) is Bessel function of the first order. We can also get ρj-j′(b) for the base site in a similar manner to that for ρi-i′(m). From the above description, we can get finally the general formula for MIMO channel spatial correlation ρi-j,i′-j′(bm) by rewriting (4) as
(10)ρi-j,i′-j′(bm)=kexp⁡(jΔθ0)+ρi-i′(m)⋅ρj-j′(b)k+1,
where ρi-i′(m) and ρj-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 ρi-i′(m)ρj-j′(b) without LOS, so ρi-j,i′-j′(bm) is the product of ρi-i′(m) and ρj-j′(b).Moreover, (10) shows that ρi-j,i′-j′(bm) is calculated from three items: the mobile- and base-site antenna configurations (zi-i′,zj-j′,Ψi-i′,Ψj-j′), the angle distribution for arriving and incident waves to multiple paths (ξn,ζn), and Rice factor k. By using (10), we can calculate ρi-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 ρi-i′(m)=1 and ρj-j′(b)=1, respectively.


#### 2.3.2. Example of Correlation Coefficientρi-j,i′-j′(bm)

Theρi-j,i′-j′(bm) in (10) contains the product of ρi-i′(m) and ρj-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 ρi-j,i′-j′(bm), it is sufficient to prepare just one side for various situations because we can get ρi-j,i′-j′(bm) by combining them. Therefore, we studied the correlation for two typical distributions with uniform ξn and Gaussian ζn.(i) For Uniform Distribution ofξn
We first calculateρi-i′(m) by (9) when ξn has a uniform distribution centered at ξc over ξc-Δξn<ξn<ξc+Δξn with the probability density function as pdf(ξn)=1/2Δξn. Assuming a large N,we can calculate the ensemble average in (9) by integration with respect to ξn, and get (11) (see Appendix C). 											(11)ρi-i′(m)=∑l=0∞εl(-1)lJ2l(2πzi-i′)12lΔξnsin⁡(2lΔξn)×cos⁡(2l(ξc-ψi-i′))-j2∑l=0∞(-1)lJ2l+1(2πzi-i′)1(2l+1)Δξn×sin⁡((2l+1)Δξn)cos⁡((2l+1)(ξc-ψi-i′)).(ii) For Gaussian Distribution ofζn
Next, we calculateρj-j′(b) when ζn has a Gaussian distribution centered at ζc with deviation σ. Similar to the uniform distribution case, we get (12) (see Appendix D). 											(12)ρj-j′(b)=∑l=0∞εl(-1)lJ2l(2πzj-j′)cos⁡(2l(ζc-ψj-j′))×exp⁡(-2l2σ2)-j2∑l=0∞(-1)lJ2l+1(2πzj-j′)×cos⁡((2l+1)(ζc-ψj-j′))exp⁡(-(2l+1)2σ22).


## 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 Figure1. 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 (ξn,ζn) 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 isN+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 timeτn relative to the shortest path between the two origins and maximum excess delay time τmax. The τn values are random over range 0≤τn≤τmax.(iii)
The amplitude ishn and independent of ξn and ζn, 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 byξn and the base-site angle is denoted by ζn. The ξn and ζn are counterclockwise angles from TN (true north) at the origins on both sites; however, when the mobile station moves, ξn is the angle from the mobile’s movement direction. The arriving wave’s initial phase is ϕn and the values are random over 0≤ϕn<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 Figure1. For the base site, the coordinates are denoted by ANTb(λbj,ζbj), where λbj and ζbj 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 ANTm(λmi,ξmi); however, when the mobile station moves, ξmi 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 ofξn and ζn, but strictly ξn and ζn differ slightly among the elements by value δ. The angular difference δ from ξn or ζn 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 ξn or ζn when the antenna is high and far away and when ξn and ζn have Gaussian distributions, but not so sensitive to ξn or ζn when r/λ is small and ξn and ζn are spread widely, as in the case with mobile stations or indoor cells.Figure 2
Angle differenceδ 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 thejth base- and ith mobile-station antenna elements, is denoted by MIMO ch(i-j), and the complex received signal level Ei-j(t,fc) is given by
(1)Ei-j(t,fc)=∑n=0Nhnejθi-j,n,(2)θi-j,n=2π[(ζn-ζbj)fcτn-fmtcos⁡ξn-λmicos⁡(ξn-ξmi)2ππ-λbjcos⁡(ζn-ζbj)]+ϕn,
where θi-j,n is the path phase of MIMO ch(i-j) for the nth multipath wave, fc is the radio frequency, and fm is the maximum Doppler frequency. The third and fourth terms of θi-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 isM, the MIMO channel matrix E, which describes the connection between both stations, can be expressed as
(3)E=[E1-1E1-2⋯E1-ME2-1E2-2⋯E2-M⋅⋅⋯⋅EM-1EM-2⋯EM-M],
where Ei-j means Ei-j(t,fc) in (1).


## 2.3. Correlation between MIMO Channel Matrix Elements

### 2.3.1. General Formula

We start by studying the general spatial correlation between MIMOch(i-j) and ch(i′-j′); the complex correlation is denoted by ρi-j,i′-j′(bm). With variables x meaning Ei-j(t,fc) and y meaning Ei′-j′(t,fc) obtained by (1), ρi-j,i′-j′(bm) is expressed by
(4)ρi-j,i′-j′(bm)=〈(x-〈x〉)*(y-〈y〉)〉[〈(x-〈x〉)*(x-〈x〉)〉〈(y-〈y〉)*(y-〈y〉)〉]1/2.
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 hn and θi-j,n (or θi′-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). 										(5)〈(x-〈x〉)*(y-〈y〉)〉=〈h02exp⁡(jΔθ0)〉+〈∑n=1Nhn2〉〈∑n=1N[cos⁡(2πzi-i′cos⁡(ξn-ψi-i′))∑n=1N×-jsin⁡(2πzi-i′cos⁡(ξn-ψi-i′))]∑n=1N〉×〈∑n=1N[cos⁡(2πzj-j′cos⁡(ζn-ψj-j′))∑n=1N×-jsin⁡(2πzj-j′cos⁡(ζn-ψj-j′))]∑n=1N〉(6)=kexp⁡(jΔθ0)+ρi-i′(m)⋅ρj-j′(b)
Here, zi-i′, Ψi-i′, and zj-j′, Ψj-j′ in (5) are antenna construction parameters for the mobile and base sites, respectively. The zi-i′ and Ψi-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. 										(7)zi-i′=[a2+b2]1/2,ψi-i′=tan-1(ba)(8)a=λmi′cos⁡ξmi′-λmicos⁡ξmi,b=λmi′sin⁡ξmi′-λmisin⁡ξmi.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 ρi-i′(m) and ρj-j′(b) in (6). Furthermore, we expand ρi-i′(m) and ρj-j′(b) to a Neumann expansion because the ensemble averages are integrated with respect to ξn and ζn [16], and we get (9) for the mobile site (see Appendix B). 										(9)ρi-i′(m)=〈∑n=1N[∑l=0∞εl(-1)lJ2l(2πzi-i′)cos⁡⁡(2l(ξn-ψi-i′))]〉-j〈∑n=1N[2∑l=0∞(-1)lJ2l+1(2πzi-i′)cos⁡⁡((2l+1)∑l=0∞×(ξn-ψi-i′))∑l=0∞]〉.
Here, εl=1(l=0),εl=2(l≥1),J2l(·) is Bessel function of the first order. We can also get ρj-j′(b) for the base site in a similar manner to that for ρi-i′(m). From the above description, we can get finally the general formula for MIMO channel spatial correlation ρi-j,i′-j′(bm) by rewriting (4) as
(10)ρi-j,i′-j′(bm)=kexp⁡(jΔθ0)+ρi-i′(m)⋅ρj-j′(b)k+1,
where ρi-i′(m) and ρj-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 ρi-i′(m)ρj-j′(b) without LOS, so ρi-j,i′-j′(bm) is the product of ρi-i′(m) and ρj-j′(b).Moreover, (10) shows that ρi-j,i′-j′(bm) is calculated from three items: the mobile- and base-site antenna configurations (zi-i′,zj-j′,Ψi-i′,Ψj-j′), the angle distribution for arriving and incident waves to multiple paths (ξn,ζn), and Rice factor k. By using (10), we can calculate ρi-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 ρi-i′(m)=1 and ρj-j′(b)=1, respectively.


### 2.3.2. Example of Correlation Coefficientρi-j,i′-j′(bm)

Theρi-j,i′-j′(bm) in (10) contains the product of ρi-i′(m) and ρj-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 ρi-j,i′-j′(bm), it is sufficient to prepare just one side for various situations because we can get ρi-j,i′-j′(bm) by combining them. Therefore, we studied the correlation for two typical distributions with uniform ξn and Gaussian ζn.(i) For Uniform Distribution ofξn
We first calculateρi-i′(m) by (9) when ξn has a uniform distribution centered at ξc over ξc-Δξn<ξn<ξc+Δξn with the probability density function as pdf(ξn)=1/2Δξn. Assuming a large N,we can calculate the ensemble average in (9) by integration with respect to ξn, and get (11) (see Appendix C). 											(11)ρi-i′(m)=∑l=0∞εl(-1)lJ2l(2πzi-i′)12lΔξnsin⁡(2lΔξn)×cos⁡(2l(ξc-ψi-i′))-j2∑l=0∞(-1)lJ2l+1(2πzi-i′)1(2l+1)Δξn×sin⁡((2l+1)Δξn)cos⁡((2l+1)(ξc-ψi-i′)).(ii) For Gaussian Distribution ofζn
Next, we calculateρj-j′(b) when ζn has a Gaussian distribution centered at ζc with deviation σ. Similar to the uniform distribution case, we get (12) (see Appendix D). 											(12)ρj-j′(b)=∑l=0∞εl(-1)lJ2l(2πzj-j′)cos⁡(2l(ζc-ψj-j′))×exp⁡(-2l2σ2)-j2∑l=0∞(-1)lJ2l+1(2πzj-j′)×cos⁡((2l+1)(ζc-ψj-j′))exp⁡(-(2l+1)2σ22).


## 2.3.1. General Formula

We start by studying the general spatial correlation between MIMOch(i-j) and ch(i′-j′); the complex correlation is denoted by ρi-j,i′-j′(bm). With variables x meaning Ei-j(t,fc) and y meaning Ei′-j′(t,fc) obtained by (1), ρi-j,i′-j′(bm) is expressed by
(4)ρi-j,i′-j′(bm)=〈(x-〈x〉)*(y-〈y〉)〉[〈(x-〈x〉)*(x-〈x〉)〉〈(y-〈y〉)*(y-〈y〉)〉]1/2.
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 hn and θi-j,n (or θi′-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). 										(5)〈(x-〈x〉)*(y-〈y〉)〉=〈h02exp⁡(jΔθ0)〉+〈∑n=1Nhn2〉〈∑n=1N[cos⁡(2πzi-i′cos⁡(ξn-ψi-i′))∑n=1N×-jsin⁡(2πzi-i′cos⁡(ξn-ψi-i′))]∑n=1N〉×〈∑n=1N[cos⁡(2πzj-j′cos⁡(ζn-ψj-j′))∑n=1N×-jsin⁡(2πzj-j′cos⁡(ζn-ψj-j′))]∑n=1N〉(6)=kexp⁡(jΔθ0)+ρi-i′(m)⋅ρj-j′(b)
Here, zi-i′, Ψi-i′, and zj-j′, Ψj-j′ in (5) are antenna construction parameters for the mobile and base sites, respectively. The zi-i′ and Ψi-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. 										(7)zi-i′=[a2+b2]1/2,ψi-i′=tan-1(ba)(8)a=λmi′cos⁡ξmi′-λmicos⁡ξmi,b=λmi′sin⁡ξmi′-λmisin⁡ξmi.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 ρi-i′(m) and ρj-j′(b) in (6). Furthermore, we expand ρi-i′(m) and ρj-j′(b) to a Neumann expansion because the ensemble averages are integrated with respect to ξn and ζn [16], and we get (9) for the mobile site (see Appendix B). 										(9)ρi-i′(m)=〈∑n=1N[∑l=0∞εl(-1)lJ2l(2πzi-i′)cos⁡⁡(2l(ξn-ψi-i′))]〉-j〈∑n=1N[2∑l=0∞(-1)lJ2l+1(2πzi-i′)cos⁡⁡((2l+1)∑l=0∞×(ξn-ψi-i′))∑l=0∞]〉.
Here, εl=1(l=0),εl=2(l≥1),J2l(·) is Bessel function of the first order. We can also get ρj-j′(b) for the base site in a similar manner to that for ρi-i′(m). From the above description, we can get finally the general formula for MIMO channel spatial correlation ρi-j,i′-j′(bm) by rewriting (4) as
(10)ρi-j,i′-j′(bm)=kexp⁡(jΔθ0)+ρi-i′(m)⋅ρj-j′(b)k+1,
where ρi-i′(m) and ρj-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 ρi-i′(m)ρj-j′(b) without LOS, so ρi-j,i′-j′(bm) is the product of ρi-i′(m) and ρj-j′(b).Moreover, (10) shows that ρi-j,i′-j′(bm) is calculated from three items: the mobile- and base-site antenna configurations (zi-i′,zj-j′,Ψi-i′,Ψj-j′), the angle distribution for arriving and incident waves to multiple paths (ξn,ζn), and Rice factor k. By using (10), we can calculate ρi-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 ρi-i′(m)=1 and ρj-j′(b)=1, respectively.


## 2.3.2. Example of Correlation Coefficientρi-j,i′-j′(bm)

Theρi-j,i′-j′(bm) in (10) contains the product of ρi-i′(m) and ρj-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 ρi-j,i′-j′(bm), it is sufficient to prepare just one side for various situations because we can get ρi-j,i′-j′(bm) by combining them. Therefore, we studied the correlation for two typical distributions with uniform ξn and Gaussian ζn.(i) For Uniform Distribution ofξn
We first calculateρi-i′(m) by (9) when ξn has a uniform distribution centered at ξc over ξc-Δξn<ξn<ξc+Δξn with the probability density function as pdf(ξn)=1/2Δξn. Assuming a large N,we can calculate the ensemble average in (9) by integration with respect to ξn, and get (11) (see Appendix C). 											(11)ρi-i′(m)=∑l=0∞εl(-1)lJ2l(2πzi-i′)12lΔξnsin⁡(2lΔξn)×cos⁡(2l(ξc-ψi-i′))-j2∑l=0∞(-1)lJ2l+1(2πzi-i′)1(2l+1)Δξn×sin⁡((2l+1)Δξn)cos⁡((2l+1)(ξc-ψi-i′)).(ii) For Gaussian Distribution ofζn
Next, we calculateρj-j′(b) when ζn has a Gaussian distribution centered at ζc with deviation σ. Similar to the uniform distribution case, we get (12) (see Appendix D). 											(12)ρj-j′(b)=∑l=0∞εl(-1)lJ2l(2πzj-j′)cos⁡(2l(ζc-ψj-j′))×exp⁡(-2l2σ2)-j2∑l=0∞(-1)lJ2l+1(2πzj-j′)×cos⁡((2l+1)(ζc-ψj-j′))exp⁡(-(2l+1)2σ22).


## 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, ρi-i′(m) or ρj-j′(b), and mobile and base stations both-side channel, that is, ρi-j,i′-j′(bm). So we simulated the correlation with uniform-in-ξn and Gaussian-in-ζn 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 fc was 3 GHz, and the channel model with the delay profile in Section 2.1 was used. The nondirective wave amplitude hn exponentially decreases with increasing τn, 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 (ξn, ζn) and antenna parameters (λm, ξm,λb, ζb) 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.
Correlation
Eigenvalue
MIMO channel
One-side channel
Both-side channel
Both-side channel
Environment
Indoors
Outdoors
Indoors
Outdoors
Indoors
Outdoors
Incident/arriving angle
Distribution
Uniform
Gaussian
Uniform-uniform
Uniform-Gaussian
Uniform-uniform
Uniform-Gaussian
ξ
c
(Δξn)
0, 0~π(π,π/2,π/4)
—
0  (π)
0(π)
0(π,π/2,π/4)
0(π)
ζ
c
(σ)
—
π /6, 0~π(π/90,π/36,π/18)
π
/
6
(
π
/
90
,
π
/
36
,
π
/
18
)
0,π/6,π/4,π/2(π/90,π/36,π/18)
Delay profile
N
10 waves
h
n
Exponential distribution (effective amplitude greater than −25 dB)
K
-
∞
,
5
dB
-
∞
dB
-
∞
,
5
dB
-
∞
dB
Antenna
λ
m, (ξm)
0~5,0.5,(0)
—
0~5,0.5,(0)
0~20,0.5,(0)
0.07
,
0
.
2
5
,
(
0
)
0
.
10
,
0.2
1
,
0
.
45
,
(
0
)
λ
b, (ζb)
—
0~20,2,(0)
0~5,(0)
0~20,(0)
0.07
,
0.25
,
(
0
)
2
,
(
0
)
Components
1
×
2
1
×
2
2
×
2
2
×
2
4
×
4


### 3.2. Correlation Coefficient of MIMO Channel

#### 3.2.1. Mobile or Base Station One-Side Channel

Figure3 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=zi-i′(zj-j′) and setting angle Ψi-i′(Ψj-j′)=0. Figures 3(a) and 3(b) have arriving angle ξn with uniform distribution and the parameter Δξn means the nth arriving wave uniformly from the direction ξc-Δξn<ξn<ξc+Δξn. 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 |ρi-i′(m)| calculated by (10) with (11); the simulated and theoretical values agree well. The ρi-i′(m) at Δξn=π in Figure 3(a) becomes Jo(2πzi-i′), as is well known. Figure 3(c) shows the simulated correlation for arriving angle ζn with Gaussian distribution centered at ζc=π/6 with standard deviation σ. The correlation value decreases monotonically with increasing z and the simulated values agree well with the theoretical values |ρj-j′(b)| obtained by (10) using(12)Correlation of one-side channel. (a) NLOS indoors (uniform distribution). (b) LOS indoors (uniform distribution). (c) NLOS outdoors (Gaussian distribution).
(a)
(b)
(c)Figure4 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 ξn with uniform distribution centered on ξc with Δξn, and angle ζn with Gaussian distribution centered on ζc with σ, respectively. The correlation in Figure 4(a) was simulated by changing ξc over the range from 0 to π while keeping zi-i′=0.5 and  Ψi-i′=0. The correlation values for Δξn=π/4 and π/2 have minima at ξc=π/2 and become larger far away from  ξc=π/2, but the value for Δξn=π does not depend on ξc since arriving waves arrive from all directions from 0 to 2π. The theoretical value |ρi-i′(m)| was calculated using (10), that is, (11); the theoretical and simulated values agree well. The correlation |ρj-j′(b)| in Figure 4(b) for the Gaussian distribution was also simulated in a similar way to Figure 4(a), expect for  zj-j′=2. Though |ρj-j′(b)| has a minimum at ξc=π/2 and becomes larger far away from ζc=π/2 like in Figure 4(a), the values at ζc=π/2 depend on standard deviation σ, and the minimum value of |ρj-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.Correlation of dependence on arrival angle for one-side channel. (a) Uniform distribution. (b) Gaussian distribution.
(a)
(b)


#### 3.2.2. Mobile and Base Station Both-Side Channel

Figure5 shows the simulated correlation coefficient |ρi-j,i′-j′(bm)|, that is, i≠i′ at the mobile station and j≠j′ at the base station for incident   ζn and arriving  ξn with the same value of Δξn=Δζn 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=zi-i′=zj-j′ and keeping Ψi-i′=Ψj-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 ρi-j,i′-j′(bm) from (10) is the product of ρi-i′(m) and ρj-j′(b), or here |ρi-j,i′-j′(bm)|=|ρi-i′(m)|2. The simulated and theoretical values agree well. Figure 5(b) is the simulated value when changing z=zj-j′, like Figure 5(a), except keeping zi-i′=0.5. So the value of the correlation at z=0 is less than 1. The theoretical values from (10) at zi-i′=0.5 are also shown in Figure 5(b); the theoretical and simulated values agree well.Correlation of both-side channel (uniform-uniform distribution). (a)Z=Zi-i′=Zj-j′. (b) Zi-i′=0.5.
(a)
(b)Figure6 shows the simulated correlation |ρi-j,i′-j′(bm)| for incident ξn with uniform distribution (ξc=0,Δξn=π) and arriving ζn with Gaussian distribution (ζc=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 ξn with uniform distribution at another site. The theoretical value ρi-j,i′-j′(bm) from (10) was calculated as corresponding to (11) for the mobile station ρi-i′(m) and to (12) for the base station ρj-j′(b), respectively. The theoretical and simulated values agree well. Figure 6(b) is the simulated correlation with changing z=zj-j′ as in Figure 6(a), except keeping zi-i′=0.5. So the simulated value is less than 0.3 whenever z is small since we kept zi-i′=0.5. The theoretical value from (10) was also calculated; the theoretical and simulated values agree well.Correlation of both-side channel (uniform-Gaussian distribution). (a)Z=Zi-i′=Zj-j′. (b) Zi-i′=0.5.
(a)
(b)


### 3.3. Eigenvalue of MIMO Channel Matrix

All of the eigenvalues were simulated by a MIMO antenna with a4×4 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

Figure7 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 ξc=0 and Δξn=π in NLOS; the only difference was the antenna radius λmi=λbj=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 |ρi-i+1(m)|=|ρj-j+1(b)|=0.45 and 0.95, respectively. Here, ρ in Figure 7 means |ρi-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.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).
(a)
(b)
(c)


#### 3.3.2. Dependence of Eigenvalue on Correlation

Figure8 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 ρi-j,i′-j′(bm)=ρi-i+1(m)ρj-j+1(b) in NLOS, where the simulation was done changing antenna space correspondent to |ρi-i+1(m)|=0.2, 0.6, and 0.9 at the mobile station with a uniform distribution keeping a high |ρj-j+1(b)|=0.9 at the base station in the Gaussian distribution with ζc=π/6 and σ=π/36. Figure 8(b) also shows that the 50% cumulative values have little dependence on ρi-j,i′-j′(bm) for λ1, but are dependent on ρi-j,i′-j′(bm) for the other eigenvalues: the smaller ρi-j,i′-j′(bm) is, the closer the values are to λ1. All the eigenvalues tend to have a wider distribution with increasing ρi-j,i′-j′(bm).Dependence of eigenvalue on correlation (4×4 MIMO). (a) Uniform-uniform distribution. (b) Uniform-Gaussian distribution.
(a)
(b)Figure9 shows the dependence of the eigenvalue on correlation ρj-j+1(b) with the properties of standard deviation σ and centering angle ζc in Gaussian distribution, as shown in Figure 4(b). Simulation was done for ρj-j+1(b) with σ and ζc as parameters, while the mobile station condition was set to a constant at |ρi-i+1(m)|=0.9 in a uniform distribution with ξc=0 and Δξn=π. Figure 9(a) shows the influence of σ on the eigenvalue with σ=π/18, π/36, and π/90 as parameters at ζc=π/6 and zj-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, ρj-j+1(b) is small. Figure 9(b) shows the influence of ζc on the eigenvalue, when ζc=π/2, π/4, and 0 at σ=π/18 and zj-j′=2. The dependence of the eigenvalues on ζc is similar to σ  in Figure 9(a). Figure 9 suggests that the σ and ζc are parameters for eigenvalue property.Dependence of eigenvalue on Gaussian distribution (4×4 MIMO). (a) Effect of standard deviation. (b) Effect of centering arriving angle.
(a)
(b)


## 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, ρi-i′(m) or ρj-j′(b), and mobile and base stations both-side channel, that is, ρi-j,i′-j′(bm). So we simulated the correlation with uniform-in-ξn and Gaussian-in-ζn 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 fc was 3 GHz, and the channel model with the delay profile in Section 2.1 was used. The nondirective wave amplitude hn exponentially decreases with increasing τn, 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 (ξn, ζn) and antenna parameters (λm, ξm,λb, ζb) 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.
Correlation
Eigenvalue
MIMO channel
One-side channel
Both-side channel
Both-side channel
Environment
Indoors
Outdoors
Indoors
Outdoors
Indoors
Outdoors
Incident/arriving angle
Distribution
Uniform
Gaussian
Uniform-uniform
Uniform-Gaussian
Uniform-uniform
Uniform-Gaussian
ξ
c
(Δξn)
0, 0~π(π,π/2,π/4)
—
0  (π)
0(π)
0(π,π/2,π/4)
0(π)
ζ
c
(σ)
—
π /6, 0~π(π/90,π/36,π/18)
π
/
6
(
π
/
90
,
π
/
36
,
π
/
18
)
0,π/6,π/4,π/2(π/90,π/36,π/18)
Delay profile
N
10 waves
h
n
Exponential distribution (effective amplitude greater than −25 dB)
K
-
∞
,
5
dB
-
∞
dB
-
∞
,
5
dB
-
∞
dB
Antenna
λ
m, (ξm)
0~5,0.5,(0)
—
0~5,0.5,(0)
0~20,0.5,(0)
0.07
,
0
.
2
5
,
(
0
)
0
.
10
,
0.2
1
,
0
.
45
,
(
0
)
λ
b, (ζb)
—
0~20,2,(0)
0~5,(0)
0~20,(0)
0.07
,
0.25
,
(
0
)
2
,
(
0
)
Components
1
×
2
1
×
2
2
×
2
2
×
2
4
×
4


## 3.2. Correlation Coefficient of MIMO Channel

### 3.2.1. Mobile or Base Station One-Side Channel

Figure3 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=zi-i′(zj-j′) and setting angle Ψi-i′(Ψj-j′)=0. Figures 3(a) and 3(b) have arriving angle ξn with uniform distribution and the parameter Δξn means the nth arriving wave uniformly from the direction ξc-Δξn<ξn<ξc+Δξn. 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 |ρi-i′(m)| calculated by (10) with (11); the simulated and theoretical values agree well. The ρi-i′(m) at Δξn=π in Figure 3(a) becomes Jo(2πzi-i′), as is well known. Figure 3(c) shows the simulated correlation for arriving angle ζn with Gaussian distribution centered at ζc=π/6 with standard deviation σ. The correlation value decreases monotonically with increasing z and the simulated values agree well with the theoretical values |ρj-j′(b)| obtained by (10) using(12)Correlation of one-side channel. (a) NLOS indoors (uniform distribution). (b) LOS indoors (uniform distribution). (c) NLOS outdoors (Gaussian distribution).
(a)
(b)
(c)Figure4 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 ξn with uniform distribution centered on ξc with Δξn, and angle ζn with Gaussian distribution centered on ζc with σ, respectively. The correlation in Figure 4(a) was simulated by changing ξc over the range from 0 to π while keeping zi-i′=0.5 and  Ψi-i′=0. The correlation values for Δξn=π/4 and π/2 have minima at ξc=π/2 and become larger far away from  ξc=π/2, but the value for Δξn=π does not depend on ξc since arriving waves arrive from all directions from 0 to 2π. The theoretical value |ρi-i′(m)| was calculated using (10), that is, (11); the theoretical and simulated values agree well. The correlation |ρj-j′(b)| in Figure 4(b) for the Gaussian distribution was also simulated in a similar way to Figure 4(a), expect for  zj-j′=2. Though |ρj-j′(b)| has a minimum at ξc=π/2 and becomes larger far away from ζc=π/2 like in Figure 4(a), the values at ζc=π/2 depend on standard deviation σ, and the minimum value of |ρj-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.Correlation of dependence on arrival angle for one-side channel. (a) Uniform distribution. (b) Gaussian distribution.
(a)
(b)


### 3.2.2. Mobile and Base Station Both-Side Channel

Figure5 shows the simulated correlation coefficient |ρi-j,i′-j′(bm)|, that is, i≠i′ at the mobile station and j≠j′ at the base station for incident   ζn and arriving  ξn with the same value of Δξn=Δζn 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=zi-i′=zj-j′ and keeping Ψi-i′=Ψj-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 ρi-j,i′-j′(bm) from (10) is the product of ρi-i′(m) and ρj-j′(b), or here |ρi-j,i′-j′(bm)|=|ρi-i′(m)|2. The simulated and theoretical values agree well. Figure 5(b) is the simulated value when changing z=zj-j′, like Figure 5(a), except keeping zi-i′=0.5. So the value of the correlation at z=0 is less than 1. The theoretical values from (10) at zi-i′=0.5 are also shown in Figure 5(b); the theoretical and simulated values agree well.Correlation of both-side channel (uniform-uniform distribution). (a)Z=Zi-i′=Zj-j′. (b) Zi-i′=0.5.
(a)
(b)Figure6 shows the simulated correlation |ρi-j,i′-j′(bm)| for incident ξn with uniform distribution (ξc=0,Δξn=π) and arriving ζn with Gaussian distribution (ζc=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 ξn with uniform distribution at another site. The theoretical value ρi-j,i′-j′(bm) from (10) was calculated as corresponding to (11) for the mobile station ρi-i′(m) and to (12) for the base station ρj-j′(b), respectively. The theoretical and simulated values agree well. Figure 6(b) is the simulated correlation with changing z=zj-j′ as in Figure 6(a), except keeping zi-i′=0.5. So the simulated value is less than 0.3 whenever z is small since we kept zi-i′=0.5. The theoretical value from (10) was also calculated; the theoretical and simulated values agree well.Correlation of both-side channel (uniform-Gaussian distribution). (a)Z=Zi-i′=Zj-j′. (b) Zi-i′=0.5.
(a)
(b)


## 3.2.1. Mobile or Base Station One-Side Channel

Figure3 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=zi-i′(zj-j′) and setting angle Ψi-i′(Ψj-j′)=0. Figures 3(a) and 3(b) have arriving angle ξn with uniform distribution and the parameter Δξn means the nth arriving wave uniformly from the direction ξc-Δξn<ξn<ξc+Δξn. 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 |ρi-i′(m)| calculated by (10) with (11); the simulated and theoretical values agree well. The ρi-i′(m) at Δξn=π in Figure 3(a) becomes Jo(2πzi-i′), as is well known. Figure 3(c) shows the simulated correlation for arriving angle ζn with Gaussian distribution centered at ζc=π/6 with standard deviation σ. The correlation value decreases monotonically with increasing z and the simulated values agree well with the theoretical values |ρj-j′(b)| obtained by (10) using(12)Correlation of one-side channel. (a) NLOS indoors (uniform distribution). (b) LOS indoors (uniform distribution). (c) NLOS outdoors (Gaussian distribution).
(a)
(b)
(c)Figure4 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 ξn with uniform distribution centered on ξc with Δξn, and angle ζn with Gaussian distribution centered on ζc with σ, respectively. The correlation in Figure 4(a) was simulated by changing ξc over the range from 0 to π while keeping zi-i′=0.5 and  Ψi-i′=0. The correlation values for Δξn=π/4 and π/2 have minima at ξc=π/2 and become larger far away from  ξc=π/2, but the value for Δξn=π does not depend on ξc since arriving waves arrive from all directions from 0 to 2π. The theoretical value |ρi-i′(m)| was calculated using (10), that is, (11); the theoretical and simulated values agree well. The correlation |ρj-j′(b)| in Figure 4(b) for the Gaussian distribution was also simulated in a similar way to Figure 4(a), expect for  zj-j′=2. Though |ρj-j′(b)| has a minimum at ξc=π/2 and becomes larger far away from ζc=π/2 like in Figure 4(a), the values at ζc=π/2 depend on standard deviation σ, and the minimum value of |ρj-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.Correlation of dependence on arrival angle for one-side channel. (a) Uniform distribution. (b) Gaussian distribution.
(a)
(b)


## 3.2.2. Mobile and Base Station Both-Side Channel

Figure5 shows the simulated correlation coefficient |ρi-j,i′-j′(bm)|, that is, i≠i′ at the mobile station and j≠j′ at the base station for incident   ζn and arriving  ξn with the same value of Δξn=Δζn 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=zi-i′=zj-j′ and keeping Ψi-i′=Ψj-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 ρi-j,i′-j′(bm) from (10) is the product of ρi-i′(m) and ρj-j′(b), or here |ρi-j,i′-j′(bm)|=|ρi-i′(m)|2. The simulated and theoretical values agree well. Figure 5(b) is the simulated value when changing z=zj-j′, like Figure 5(a), except keeping zi-i′=0.5. So the value of the correlation at z=0 is less than 1. The theoretical values from (10) at zi-i′=0.5 are also shown in Figure 5(b); the theoretical and simulated values agree well.Correlation of both-side channel (uniform-uniform distribution). (a)Z=Zi-i′=Zj-j′. (b) Zi-i′=0.5.
(a)
(b)Figure6 shows the simulated correlation |ρi-j,i′-j′(bm)| for incident ξn with uniform distribution (ξc=0,Δξn=π) and arriving ζn with Gaussian distribution (ζc=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 ξn with uniform distribution at another site. The theoretical value ρi-j,i′-j′(bm) from (10) was calculated as corresponding to (11) for the mobile station ρi-i′(m) and to (12) for the base station ρj-j′(b), respectively. The theoretical and simulated values agree well. Figure 6(b) is the simulated correlation with changing z=zj-j′ as in Figure 6(a), except keeping zi-i′=0.5. So the simulated value is less than 0.3 whenever z is small since we kept zi-i′=0.5. The theoretical value from (10) was also calculated; the theoretical and simulated values agree well.Correlation of both-side channel (uniform-Gaussian distribution). (a)Z=Zi-i′=Zj-j′. (b) Zi-i′=0.5.
(a)
(b)


## 3.3. Eigenvalue of MIMO Channel Matrix

All of the eigenvalues were simulated by a MIMO antenna with a4×4 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

Figure7 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 ξc=0 and Δξn=π in NLOS; the only difference was the antenna radius λmi=λbj=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 |ρi-i+1(m)|=|ρj-j+1(b)|=0.45 and 0.95, respectively. Here, ρ in Figure 7 means |ρi-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.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).
(a)
(b)
(c)


### 3.3.2. Dependence of Eigenvalue on Correlation

Figure8 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 ρi-j,i′-j′(bm)=ρi-i+1(m)ρj-j+1(b) in NLOS, where the simulation was done changing antenna space correspondent to |ρi-i+1(m)|=0.2, 0.6, and 0.9 at the mobile station with a uniform distribution keeping a high |ρj-j+1(b)|=0.9 at the base station in the Gaussian distribution with ζc=π/6 and σ=π/36. Figure 8(b) also shows that the 50% cumulative values have little dependence on ρi-j,i′-j′(bm) for λ1, but are dependent on ρi-j,i′-j′(bm) for the other eigenvalues: the smaller ρi-j,i′-j′(bm) is, the closer the values are to λ1. All the eigenvalues tend to have a wider distribution with increasing ρi-j,i′-j′(bm).Dependence of eigenvalue on correlation (4×4 MIMO). (a) Uniform-uniform distribution. (b) Uniform-Gaussian distribution.
(a)
(b)Figure9 shows the dependence of the eigenvalue on correlation ρj-j+1(b) with the properties of standard deviation σ and centering angle ζc in Gaussian distribution, as shown in Figure 4(b). Simulation was done for ρj-j+1(b) with σ and ζc as parameters, while the mobile station condition was set to a constant at |ρi-i+1(m)|=0.9 in a uniform distribution with ξc=0 and Δξn=π. Figure 9(a) shows the influence of σ on the eigenvalue with σ=π/18, π/36, and π/90 as parameters at ζc=π/6 and zj-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, ρj-j+1(b) is small. Figure 9(b) shows the influence of ζc on the eigenvalue, when ζc=π/2, π/4, and 0 at σ=π/18 and zj-j′=2. The dependence of the eigenvalues on ζc is similar to σ  in Figure 9(a). Figure 9 suggests that the σ and ζc are parameters for eigenvalue property.Dependence of eigenvalue on Gaussian distribution (4×4 MIMO). (a) Effect of standard deviation. (b) Effect of centering arriving angle.
(a)
(b)


## 3.3.1. Eigenvalue Property with Movement

Figure7 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 ξc=0 and Δξn=π in NLOS; the only difference was the antenna radius λmi=λbj=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 |ρi-i+1(m)|=|ρj-j+1(b)|=0.45 and 0.95, respectively. Here, ρ in Figure 7 means |ρi-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.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).
(a)
(b)
(c)


## 3.3.2. Dependence of Eigenvalue on Correlation

Figure8 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 ρi-j,i′-j′(bm)=ρi-i+1(m)ρj-j+1(b) in NLOS, where the simulation was done changing antenna space correspondent to |ρi-i+1(m)|=0.2, 0.6, and 0.9 at the mobile station with a uniform distribution keeping a high |ρj-j+1(b)|=0.9 at the base station in the Gaussian distribution with ζc=π/6 and σ=π/36. Figure 8(b) also shows that the 50% cumulative values have little dependence on ρi-j,i′-j′(bm) for λ1, but are dependent on ρi-j,i′-j′(bm) for the other eigenvalues: the smaller ρi-j,i′-j′(bm) is, the closer the values are to λ1. All the eigenvalues tend to have a wider distribution with increasing ρi-j,i′-j′(bm).Dependence of eigenvalue on correlation (4×4 MIMO). (a) Uniform-uniform distribution. (b) Uniform-Gaussian distribution.
(a)
(b)Figure9 shows the dependence of the eigenvalue on correlation ρj-j+1(b) with the properties of standard deviation σ and centering angle ζc in Gaussian distribution, as shown in Figure 4(b). Simulation was done for ρj-j+1(b) with σ and ζc as parameters, while the mobile station condition was set to a constant at |ρi-i+1(m)|=0.9 in a uniform distribution with ξc=0 and Δξn=π. Figure 9(a) shows the influence of σ on the eigenvalue with σ=π/18, π/36, and π/90 as parameters at ζc=π/6 and zj-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, ρj-j+1(b) is small. Figure 9(b) shows the influence of ζc on the eigenvalue, when ζc=π/2, π/4, and 0 at σ=π/18 and zj-j′=2. The dependence of the eigenvalues on ζc is similar to σ  in Figure 9(a). Figure 9 suggests that the σ and ζc are parameters for eigenvalue property.Dependence of eigenvalue on Gaussian distribution (4×4 MIMO). (a) Effect of standard deviation. (b) Effect of centering arriving angle.
(a)
(b)


## 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 formulaρi-j,i′-j′(bm) between MIMO channel matrix elements was derived: the formula ρi-j,i′-j′(bm) was expressed as a directive wave term added to the product of mobile site correlation ρi-i′(m) and base site correlation ρj-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 ρi-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  ρi-j,i′-j′(bm), but the other ones are dependent on  ρi-j,i′-j′(bm) and become close to λ1 with decreasing  ρi-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 ρi-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.



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*Source: 569864-2012-08-01.xml*