#### 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 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 and base site correlation without LOS path, which are calculated independently of each other. By using , it is possible to create the channel matrix element with a fixed correlation value estimated by 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 *λ*_{1} hardly depends on the correlation , but the others do depend on and approach as decreases. Moreover, as the path moves into LOS, the state with mobile movement becomes more stable than the 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 for mobile site and for base site, respectively. The delay profile has both horizontal azimuth angles () 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 , waves are independent of each other, and the th-path wave is denoted by subscript , where means a directive wave and means no directive waves.(ii)The waves have excess delay time relative to the shortest path between the two origins and maximum excess delay time . The values are random over range *. *(iii)The amplitude is and independent of and , the power ratio of the directive and nondirective waves is denoted by ( in dB: Rice factor ), and ( 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 and the base-site angle is denoted by . The and are counterclockwise angles from TN (true north) at the origins on both sites; however, when the mobile station moves, is the angle from the mobile’s movement direction. The arriving wave’s initial phase is and the values are random over .

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 , where and mean the radius normalized by wavelength *λ* and counterclockwise angle from TN for the th antenna, respectively. For a mobile site, the coordinates are similarly denoted by ; however, when the mobile station moves, 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 and , but strictly and differ slightly among the elements by value * δ*. The angular difference

*from or for the normalized distance between the origin and multipath scattering when the antenna element is set at spacing*

*δ**away from the origin is shown in Figure 2. This*

*λ**is less than 0.01 rad when . As shown later, the spatial correlation is sensitive to or when the antenna is high and far away and when and have Gaussian distributions, but not so sensitive to or when is small and and are spread widely, as in the case with mobile stations or indoor cells.*

*δ*##### 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 th base- and th mobile-station antenna elements, is denoted by MIMO , and the complex received signal level is given by where is the path phase of MIMO for the th multipath wave, is the radio frequency, and is the maximum Doppler frequency. The third and fourth terms of 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
where means 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 and ; the complex correlation is denoted by . With variables meaning and meaning obtained by (1), is expressed by Here, the symbols and * mean ensemble average and conjugate complex, respectively. Under the conditions in Section 2.1, and are zero owing to the independence of and (or ) with random values from 0 to . Therefore, the denominator in (4) is , that is, received power. On the other hand, in the numerator of (4) is expressed by (5), where expresses the directive wave’s path phase difference between MIMO and (see Appendix A). Here, , , and , in (5) are antenna construction parameters for the mobile and base sites, respectively. The and are given by (7) and mean, respectively, the spacing between the th and th antenna elements and the counterclockwise angle from TN or the movement direction to a line with both these elements.

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 and in (6). Furthermore, we expand and to a Neumann expansion because the ensemble averages are integrated with respect to and [16], and we get (9) for the mobile site (see Appendix B).
Here, is Bessel function of the first order. We can also get for the base site in a similar manner to that for . From the above description, we can get finally the general formula for MIMO channel spatial correlation by rewriting (4) as
where and 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, and , it becomes without LOS, so is the product of and *. *Moreover, (10) shows that is calculated from three items: the mobile- and base-site antenna configurations , the angle distribution for arriving and incident waves to multiple paths , and Rice factor . By using (10), we can calculate for any MIMO channel matrix since ,and can be chosen freely within . Here and mean and , respectively.

###### 2.3.2. Example of Correlation Coefficient

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

*(i) For Uniform Distribution of *

We first calculate by (9) when has a uniform distribution centered at over +*Δ * with the probability density function as . Assuming a large ,we can calculate the ensemble average in (9) by integration with respect to , and get (11) (see Appendix C).

* (ii) For Gaussian Distribution of *

Next, we calculate when has a Gaussian distribution centered at with deviation * σ*. Similar to the uniform distribution case, we get (12) (see Appendix D).

#### 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, or , and mobile and base stations both-side channel, that is, . So we simulated the correlation with uniform-in- and Gaussian-in- 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 was 3 GHz, and the channel model with the delay profile in Section 2.1 was used. The nondirective wave amplitude exponentially decreases with increasing , 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 (, ) and antenna parameters (, ,, ) were set as shown in Table 1. Each simulated value was calculated from an ensemble average for more than 10^{6} delay profiles, except for eigenvalue variation with movement in Section 3.3.

##### 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 and setting angle . Figures 3(a) and 3(b) have arriving angle with uniform distribution and the parameter means the th arriving wave uniformly from the direction . Figures 3(a) and 3(b) are for NLOS and LOS with dB and paths, respectively, and the correlation of NLOS fluctuates less with increasing , but that of LOS is higher than that of NLOS and is close to a fixed value fluctuating with increasing . Figures 3(a) and 3(b) also show the theoretical value calculated by (10) with (11); the simulated and theoretical values agree well. The at in Figure 3(a) becomes , as is well known. Figure 3(c) shows the simulated correlation for arriving angle with Gaussian distribution centered at with standard deviation * σ*. The correlation value decreases monotonically with increasing and the simulated values agree well with the theoretical values obtained by (10) using(12)

**(a)**

**(b)**

**(c)**

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 with uniform distribution centered on with , and angle with Gaussian distribution centered on with *σ*, respectively. The correlation in Figure 4(a) was simulated by changing over the range from 0 to while keeping and . The correlation values for and have minima at =*π*/2 and become larger far away from , but the value for does not depend on since arriving waves arrive from all directions from 0 to 2*π*. The theoretical value was calculated using (10), that is, (11); the theoretical and simulated values agree well. The correlation in Figure 4(b) for the Gaussian distribution was also simulated in a similar way to Figure 4(a), expect for . Though has a minimum at and becomes larger far away from like in Figure 4(a), the values at depend on standard deviation * σ*, and the minimum value of is large when

*is small. The theoretical value was calculated using (10), that is, (12); the theoretical and simulated values agree well.*

*σ***(a)**

**(b)**

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

Figure 5 shows the simulated correlation coefficient , that is, at the mobile station and at the base station for incident and arriving with the same value of in the both-side uniform distribution, assuming that the mobile and base stations are indoors. Figure 5(a) shows the simulated correlation when changing and keeping . The correlation decreases faster than in Figure 3(a). The reason for this can be seen from the theory that the theoretical value from (10) is the product of and , or here . The simulated and theoretical values agree well. Figure 5(b) is the simulated value when changing , like Figure 5(a), except keeping . So the value of the correlation at is less than 1. The theoretical values from (10) at are also shown in Figure 5(b); the theoretical and simulated values agree well.

**(a)**

**(b)**

Figure 6 shows the simulated correlation for incident with uniform distribution and arriving with Gaussian distribution , *σ*: parameter), assuming mobile and base sites outdoors. Figure 6(a) was simulated like Figure 5(a), that is, changing and keeping *σ* fixed; compared with those in Figure 3(c), the simulated values become small rapidly with increasing owing to the with uniform distribution at another site. The theoretical value from (10) was calculated as corresponding to (11) for the mobile station and to (12) for the base station , respectively. The theoretical and simulated values agree well. Figure 6(b) is the simulated correlation with changing as in Figure 6(a), except keeping . So the simulated value is less than 0.3 whenever is small since we kept . The theoretical value from (10) was also calculated; the theoretical and simulated values agree well.

**(a)**

**(b)**

##### 3.3. Eigenvalue of MIMO Channel Matrix

All of the eigenvalues were simulated by a MIMO antenna with a 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 and in NLOS; the only difference was the antenna radius and 0.07 to make low and high correlations by (11) or the theoretical correlation between one MIMO antenna element and the next one and 0.95, respectively. Here, * ρ* in Figure 7 means from (10)

*.*Moreover, Figure 7(c) was simulated under the condition in Figure 7(a) with just the addition of a direct wave with [dB] to the delay profile, when the total power of the profile was normalized to 1, and the

*by (10) is 0.7. Comparing Figures 7(a) and 7(b), we see that the first eigenvalue at in Figure 7(a) is almost equal to that at in Figure 7(b) on average but that the other eigenvalues , , and in Figure 7(a) are larger than that in Figure 7(b) ( is less than 10*

*ρ*^{−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 , than that in NLOS.

**(a)**

**(b)**

**(c)**

###### 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 and 0.9 in NLOS and at 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 10^{6} profiles. As assumed in Figure 7, Figure 8(a) seems to suggest the following: though the 50% cumulative values in are weakly dependent on * ρ* and almost equal, the other eigenvalues , , and are dependent on

*and*

*ρ**K*, and the in LOS is the most stable. Figure 8(b) shows the dependence of the eigenvalue on correlation for in NLOS, where the simulation was done changing antenna space correspondent to , 0.6, and 0.9 at the mobile station with a uniform distribution keeping a high at the base station in the Gaussian distribution with and . Figure 8(b) also shows that the 50% cumulative values have little dependence on for , but are dependent on for the other eigenvalues: the smaller is, the closer the values are to . All the eigenvalues tend to have a wider distribution with increasing .

**(a)**

**(b)**

Figure 9 shows the dependence of the eigenvalue on correlation with the properties of standard deviation * σ* and centering angle in Gaussian distribution, as shown in Figure 4(b). Simulation was done for with

*and as parameters, while the mobile station condition was set to a constant at in a uniform distribution with and . Figure 9(a) shows the influence of*

*σ**on the eigenvalue with , , and as parameters at and . These 50% cumulative values are also weakly dependent on*

*σ**for , but do depend on*

*σ**for the other eigenvalues, and they become larger when*

*σ**becomes large, that is, is small. Figure 9(b) shows the influence of on the eigenvalue, when , , and 0 at and . The dependence of the eigenvalues on is similar to*

*σ**in Figure 9(a). Figure 9 suggests that the*

*σ**and are parameters for eigenvalue property.*

*σ***(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 formula between MIMO channel matrix elements was derived: the formula was expressed as a directive wave term added to the product of mobile site correlation and base site correlation , which are calculated independently of each other, divided by . This formula can be applied to create the channel matrix element with a fixed value of correlation estimated by 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 is hardly dependent on the correlation , but the other ones are dependent on and become close to with decreasing . Moreover, the state with mobile movement in LOS path is more stable than the of NLOS path. The MIMO channel model and derived 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 is a large number for the delay profile in Section 2.1, then because and have random values, obtained by (2) is a random value over the range from 0 to 2*π* and independent of . So we get since the term becomes zero, that is, and in (4) are zero. Therefore, the denominator in (4) becomes , that is, . On the other hand, in the numerator in (4) can be modified to . As a result, the sum of products and remains the same for the th arriving wave on MIMO and , but vanishes each other differences for the th arriving wave on MIMO and because the values of are random; moreover, and are independent of each other. Therefore, we get (A.1) while considering and ; moreover, mobile and base site factors are independent of each other, and the directive wave is deterministic and the amplitude is much larger than the others .
where in .

The yielded by (A.2) expresses the phase difference between MIMO and for the th arriving wave, or . 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 and , respectively, where .

Furthermore, by continuously analyzing , expanding and in (A.1), and gathering terms of and , and with a little modification we get (A.3) and express that as the third ensemble average in (5). Here, and are antenna construction parameters obtained from (7) for a mobile site. The for the base site is calculated in a similar manner to .

#### B. Derivation of (9)

We expand the real and imaginary parts for in (5) into a Neumann expansion and get (9).

#### C. Derivation of (11)

The real and imaginary parts of in (B.1) are denoted and . First, we calculate the real part . Assuming a large *N, *when has a uniform distribution centered at over , and the probability density function of is , we can calculate the ensemble average in (9) by integration with respect to , replacing variable by *u*.
The imaginary part can be also calculated similarly to* A*; we get (C.2):

#### D. Derivation of (12)

The real and imaginary parts for in (5) are denoted and . We begin by calculating the real part . Assuming that has a Gaussian distribution centered at with standard deviation * σ* by (D.1), first the real part is expanded into a Neumann expansion and then the ensemble average is calculated by integration with respect to as follows.

Putting , with an odd function of sine and assuming and with a little modification, we get The imaginary part can also be calculated in a similar manner to . We get (D.4).