Radio Science

On the ensemble average capacity of multiple-input multiple-output channels in outdoor line-of-sight multipath urban environments

Authors


Abstract

[1] Many statistical multiple-input multiple-output (MIMO) channel models have been extensively developed over the past few years. However, to better understand and exploit such MIMO channels, there is a need for a MIMO channel model that has a basis in the physical propagation mechanisms. In this paper, a wideband semideterministic MIMO model is derived to quantify the mean capacity in an urban line-of-sight (LOS) microcell. Specifically, the model considers scattering from large-scale discontinuities on building faces randomly distributed along a street canyon. Using geometrical optics and the uniform theory of diffraction, the spatial correlations between any two receiving signals are evaluated and are used to compute the upper bound of the average MIMO capacity.

1. Introduction

[2] In recent years, multiple-input multiple-output (MIMO) systems have emerged as one of the most promising approaches for maximizing capacity in wireless systems [Winters, 1987; Telatar, 1999; Foschini and Gans, 1998]. In principle, MIMO systems are able to provide multiple independent channels, leading to a substantial increase in channel capacity. In reality, this capacity improvement can be severely degraded by signal correlations that may arise at both the transmitter and receiver [Shiu et al., 2000]. There is thus a great interest in characterizing and modeling MIMO channels for different environments, so that the channel capacity can be accurately predicted.

[3] MIMO models are usually built upon fundamental principles of conventional single-input single-output (SISO) models, while incorporating additional spatial information on the multiple antenna elements. One approach is to construct a parametric model and record a large number of typical channel realizations in order to establish the statistical distribution of the relevant parameters [Molisch, 2004]. Such generic models are able to reproduce certain MIMO propagation effects, but rely heavily on channel measurements. Another approach is to construct a geometrically based scattering model and assume a suitable probability density function (PDF) for the location of the scatterers. Such semideterministic models are advantageous as they can relate directly to the actual propagating waves and do not depend on expensive and time-consuming channel sounding measurements. However, many researchers have found that a considerable amount of power is contributed from multiple reflections in microcellular environments [Steinbauer et al., 2001; Gabriela Marques and Correia, 2001; Laurila et al., 2002], suggesting that single-scattering geometrical models are inadequate in terms of modeling angle of arrival and time dispersion in the channel. In this paper, a wideband semideterministic MIMO model considering multiple reflections for a microcell street scenario is presented. A typical urban main street is usually lined with irregular high-rise buildings, occasionally separated by alleys and side streets. Following Blaunstein [1998] and Constantinou and Mughal [2005], the large-scale discontinuities in building faces and/or streets in an urban environment can be modeled statistically to represent a more generic scenario. Additional scattering from such irregularities and multiple reflections along the street canyon are then evaluated analytically to construct our model.

[4] In the current model, diffractions of higher order are not considered. Furthermore, there can be instances when a diffraction is followed by a reflection and vice versa. This kind of ray component is complicated by the fact that the combined reflected and diffracted ray can have higher orders in either the reflection or diffraction part. Modeling these higher-order contributions can improve model predictions at the expense of higher complexity in the analysis as well as computational load. However, such contributions are often insignificant when compared to higher-order reflections, hence providing only a slight improvement in accuracy [Mazar et al., 1998]. In order to keep the model simple and computationally efficient, all such contributions will not be taken into account.

2. Summary of Key Results on MIMO Capacity

[5] The input and output relationship for a MIMO system with nT transmitters and nR receivers can be expressed as

equation image

where s(t) is the transmitted signal vector, r(t) is the received signal vector, n(t) is additive white Gaussian noise vector (AWGN) and ‘⊗’ denotes convolution. H(t) is the nR × nT normalized channel impulse response matrix whose components hij corresponds to the response of the ith receiver to the signal sent by the jth transmitter. Assuming nT independent transmitted signals with equal branch power, the link capacity of the MIMO channel is given [Foschini and Gans, 1998] by

equation image

where equation image is the identity matrix of order nR, ρ is the average (over all nR receiver ports) signal-to-noise ratio, H is the normalized channel impulse response matrix and † denotes a Hermitian adjoint. The capacity equation can be expressed as a function of the normalized joint second-order moment to show the effect of correlation on the channel capacity. For an n × n MIMO system with equal transmit power distribution in each branch, the correlation can be expressed explicitly as [Loyka, 2001]

equation image

where R is defined by Loyka [2001] as the ‘instantaneous’ correlation matrix. Using Jensen's inequality [Loyka and Kouki, 2001], the ergodic capacity for a random propagating environment can be written as

equation image

The reciprocity of (2) means that 〈R〉 can represent correlation at either the transmitter or the receiver. The elements of 〈R〉 can thus be represented by rijT and rijR, respectively:

equation image
equation image

where index i,j for rijR here denotes correlation between the ith and jth receiver (likewise for the transmitters). The symbol ‘*’ denotes complex conjugation and σi is the received power at each receiver branch given by

equation image

By considering both transmitter and receiver correlations, a compound upper bound of capacity can be derived which is tighter than just considering the receiver or transmitter bound alone [Loyka and Kouki, 2002]

equation image

3. Channel Model

[6] The signal correlation between two spatially separated receivers is evaluated for a street scenario with infinitely high buildings as shown in Figure 1. Using reciprocity and superposition, this can be readily extended to a MIMO system by considering the correlation between two transmitting antennas.

Figure 1.

Layout of the main street model, showing the location of the random walls, transmitter, and receiver in the assumed coordinates system.

[7] The row of building faces along one side of the street resembles a broken line along the y axis, where the line represents a wall and the gap may represent a side street, alleyway or even a window. This random distribution of walls and gaps can be best described by a random telegraph function [Blaunstein, 1998]

equation image

The transitions from a wall to a gap and vice versa are assumed to be randomly distributed and can be modeled as a Poisson process, chosen for simplicity [Papoulis, 1984]. Hence the length of wall and gap segments follows an exponential distribution given by

equation image

where λw,g is the mean rate of occurrence of a wall or gap,

equation image

and 〈w〉, 〈g〉 are the average lengths of a wall and a gap respectively. The buildings on the two sides of the street canyon can be modeled as two separate telegraph functions fo,a at x = 0 and x = a respectively, both of which are assumed to be statistically independent, thereby ignoring any correlation that could be introduced by street intersections. For convenience, we hereafter refer to the respective random wall as the upper (at x = 0) and lower wall (at x = a), with the respective averages given by 〈w0,a〉 and 〈g0,a〉.

[8] The street layout is that of a broken waveguide with one open wall. This waveguide can support multiple reflections and diffractions, which in a wideband model contribute considerably to the delay and angular spread of the received signal. In our model up to sixth-order multiple reflections are considered (but this can be trivially increased to any desired order), but multiple diffractions and mixed multiple reflection diffractions are limited at the expense of slight improvements in prediction accuracy [Mazar et al., 1998], for the sake of bounding the model complexity.

[9] For a single realization of the random wall, the received scattered field at a location R can be written as a summation of the directly scattered field and contributions from the corresponding ground reflected field components,

equation image

where

equation image
equation image

and P is the maximum number of multiple reflections considered. Here, ELOS is the field due to a direct path between the transmitter and receiver, Eref is the specularly reflected field and Ediff is the total diffracted field from all edges along the wall. The prime indicates the field from a transmitter ground plane image and RLOS,refg represent the ground reflection coefficients for the LOS and reflected field respectively. The ground reflected fields can be evaluated using the method of images where the transmitter image is at T″(xt, yt, −zt). We consider a Hertzian dipole transmitter whose radiation pattern is symmetric about equation image = equation image. Hence the contribution from the ground reflected field can be simply found by replacing the z coordinate of the directly scattered field and multiplying with the appropriate ground reflection coefficient.

[10] The spatial correlation between two received fields is simply given by

equation image

where equation image1 and equation image2 are the polarization vectors of the two receiving antennas. The use of angled brackets will henceforth denote ensemble averages over all realizations of the random wall, or both random walls in the case of a street canyon. In the current model, we consider the maximum possible correlation that can be achieved by two spatially separated colinear antennas. Hence the spatial correlation is simply a dot product of the two received field vectors,

equation image

where subscripts 1 and 2 are used to represent the field associated with receivers at R1 and R2, respectively.

3.1. Line-of-Sight Fields

[11] For a vertical Hertzian dipole transmitter, the far-field radiated electric field at a distance r is given by

equation image

where Eo is the constant field amplitude at unit distance from the transmitter, sin equation image is the antenna radiation pattern and equation imageθd is the unit vector of the vertically polarized field. Expressing the radiation pattern and unit vector as ratios of distances corresponding to the position of the transmitter T (xt, yt, zt) and receiver R (xr, yr, zr) in the chosen Cartesian coordinate system,

equation image

where X = xrxt, Y = yryt, Z = zrzt and rd = equation image. In the current model, a Hertzian dipole is assumed, but an expression for a real antenna can be derived by substituting the corresponding radiation pattern and recomputing equation (18).

[12] A further complication arises when real antennas rather than Hertzian dipoles are considered. Closely spaced linear antennas are electromagnetically coupled and their mutual impedance needs to be considered explicitly in the calculation of HH which will differ from equation (15). However, the mutual impedance of Hertzian dipoles whose separation is much larger than their length is negligible [Kraus and Marhefka, 2002], which simplifies our analysis considerably. If real antennas are to be considered a straightforward but more complicated-looking correspondence between HH and the field correlation terms needs to be computed.

3.2. Average Reflected Field

[13] The method of images is used to account for the multiple reflections in the street canyon. Figure 2 shows a plan view of the main street, illustrating the first two orders of multiply reflected rays. The transmitter images can be classified into four distinct groups {To+, To, Te+, Te} to simplify the analysis. The superscript +/− denotes whether the image is on the positive or negative x axis and the subscript o/e represents either odd or even numbers of reflection, p. The position of the qth transmitter image source is given by

equation image
equation image
equation image
equation image

where the qth transmitter image corresponds to a ray undergoing p = 2q−1 or 2q numbers of reflections (qN). An analysis of the resultant field polarization shows that the perpendicular component to the wall remains unchanged while the parallel component differs depending on the direction of the incident ray and number of multiple reflections it has undergone. The total ensemble averaged multiply reflected ray can thus be written as a summation of the individual transmitter image contributions, as

equation image

where U is used to represent the scalar field and Q, the number of transmitter images considered. The individual transmitter image terms can be written as a product of the telegraph functions, f0 and fa,

equation image
equation image
equation image
equation image

where ro/e+/− is the path length from the respective transmitter image to the receiver. The superscripts +, 0 in ye,k+,0 are used to show that the transmitter image is on the positive x axis and the specular reflection point is on the random wall at x = 0 respectively. The subscript o/e represents either odd or even number of reflections while the number k represents the kth specular reflection point for the qth transmitter image. This is best shown in Figure 3, where the resultant multiply reflected ray due to the fifth transmitter image on the negative x axis, To,5 is shown.

Figure 2.

Plan view of the multiple reflected ray projections, showing the four classified cases of the transmitter image source.

Figure 3.

Plan view showing the resultant multiply reflected ray and respective specular reflection points due to the fifth transmitter image on the negative x axis, To,5.

[14] The statistical independence of the upper and lower wall telegraph functions implies equation imagef0faequation image = 〈f0〉 〈fa〉. Using the memoryless property of Markovian processes, the qth-order correlation can be written as a product of successive pairwise correlations of the random telegraph function [Papoulis, 1984] as

equation image

where Pw = equation image and S(τ) = 1 + equation image. The pairwise correlation expression can be simplified by noting that the difference ∣yq−1yq∣ is always constant for each of the 4 cases. Using (24) and denoting the constant difference as ▵ye+, the average reflected field for a transmitter image at Te+ becomes

equation image

where the superscripts 0, a in Pw and S are used to associate them with the statistics of the upper and lower wall, respectively. Rs,h is the soft and hard reflection coefficients given by Parsons [1992] as

equation image
equation image

where θ is the incidence angle which varies with the number of reflections,

equation image

[15] Using the same method, the remaining average reflected field terms can be easily derived and are omitted here for brevity. The polarimetrically correct expressions can now be considered by resolving the average reflected field into parallel and perpendicular components to the incidence plane,

equation image

Using (23), the perpendicular component of the received electric field can be found by a direct substitution of Uo with E,

equation image

where e is the polarization vector which includes the antenna radiation pattern,

equation image

The resultant polarization of the parallel component of the received electric field differs for each of the 4 classified cases of the transmitter image. A straightforward geometrical analysis of the resultant polarization from each wall yields

equation image

where

equation image
equation image

and Xo,e+,− are the corresponding separations between the transmitter image and receiver in the x direction.

3.3. Average Diffracted Field

[16] The average edge diffracted field on the statistically independent upper and lower random walls is

equation image

We consider first the average diffracted field from the upper random wall, which can be calculated using the statistics of the random wall distribution, by

equation image

where M represents a finite number of edges within the region of interest, bounded by the lower and upper limits, a and b, respectively. Ed(yj) is the diffracted field due to an edge at yj and pm(y1, y2.ym) is the joint PDF for the location of m edges. Using the fact that the high-frequency diffracted field decreases rapidly away from the reflection shadow boundary, only edges within the first Fresnel zone around the specular reflection point will contribute significantly to the average diffracted field [Bertoni, 2000].

[17] The half width of the first Fresnel zone, measured along the wall, is given by Constantinou and Mughal [2005] as

equation image

where λ is the wavelength and equation image′, sr and sr′ are depicted in Figure 4. Only edges within the range (yQW) ≤ y ≤ (yQ + W) will be considered in the calculation of the average diffracted field. The uniform theory of diffraction (UTD) field at R due to an edge at Qe is given by McNamara et al. [1990] as

equation image

where Ei(Qe) is the incident field at the edge Qe, D is the dyadic UTD diffraction coefficient, s′ and s are the distances from the specular reflection point to the transmitter and receiver respectively. The unit polarization vector of the incident field equation imageθi depends on the location of the diffracting edge and is thus random as well. For edges within the first Fresnel zone, it can be shown that eθd ≡ sin θ.equation imageθi.D does not vary significantly: For a transmitter and receiver located 5 m from the wall, the maximum magnitude discrepancy is less than 3%, while the maximum direction discrepancy is less than 5°. Hence eθd can be approximated as a constant evaluated at the specular reflection point. Using the edge-fixed coordinates system [McNamara et al., 1990], the vector eθd is

equation image

where

equation image

The UTD diffraction coefficient, D, comprises four terms of which only the last term is significant for the chosen orientation of transmitter and receiver. For edges that are near the specular reflection point, the diffraction coefficient for the three-dimensional case is given by Constantinou and Mughal [2005] as

equation image

where β° = cos −1 (equation image) is the skew incidence angle, L = equation imagesin 2β°, and ɛ is the displacement angle shown in Figure 4,

equation image

For all sW, the distance parameters (s and s′) can be expressed in terms of the specular reflection point distance parameters (sr and sr′) and the small displacement angle ɛ. Applying this approximation to the diffracted field expression and using the Taylor expansion for the phase term up to second order in ɛ and the magnitude term up to first order, we have

equation image

where

equation image
equation image
equation image
equation image

The phase error for this approximation has been shown [Constantinou and Mughal, 2005] to remain very low when sr/sr′ ≈ 1 and when the transmitter and receiver are not positioned too close to the wall, which is the case for most instances of practical scenario.

Figure 4.

Diffraction edge from a trailing edge near the specular reflection point [after Constantinou and Mughal, 2005] (© 2005 IEEE).

[18] To calculate the average diffracted field, we employ the joint PDF for the location of m edges within a finite interval (shown in Appendix A). Using the Taylor series expansion of the complementary error function, the average diffracted field expression can be solved analytically for any arbitrary number of edges. It is worth noting the treatment of a π radians phase ambiguity that arises for leading and trailing edges is discussed by Constantinou and Mughal [2005] and is not repeated here for brevity. By evaluating and inspecting the solution for m edges, the average diffracted field for 2N edges within the first Fresnel zone is

equation image

where

equation image
equation image
equation image
equation image

and B = λgλw, C = βequation image and Crn is the binomial coefficient.

[19] The diffracted field from the lower wall at x = a can be found be using simple coordinate transformations and the result is omitted for brevity.

3.4. Correlation Between Reflected Fields

[20] The evaluation of the correlation between two multiply reflected fields in three dimensions can be simplified by first considering the correlation between the multiply reflected scalar fields,

equation image

As before, the reflected field can be written as 4 separate cases corresponding to the transmitter image. The correlation between the qth and rth orders of the multiply reflected rays then becomes

equation image

The Markovian property of the Poisson process allows us to evaluate the qth order correlation using the product of successive pairwise correlations given by (28). For brevity, the full expression will not be presented here as it can be easily derived using (24)–(27). The polarimetrically correct expression for the correlation between reflected fields is

equation image

3.5. Correlation Between Reflected and Diffracted Fields

[21] The correlation between the multiply reflected and diffracted fields from any one wall is given by

equation image

where M is the number of diffracting edges within the first Fresnel zone and Q is the number of transmitter images considered. The reflected field term can be taken out of the integral by imposing a constraint on the integration limits, so that the specular reflection point lies on a wall. If the ‘unfolded’ paths (found by using the method of images and rotating around diffraction cones) corresponding to different field components in an expression such as 〈Eref1*. Ediff2〉 do not lie within the first Fresnel zone of each other, then the double ensemble average over the two statistically uncorrelated random walls on opposite sides of the street simplify to 〈Eref1*〉.〈Ediff2〉. All such expressions considered henceforth that fall in this category will be simplified whereas those that do not, will be computed explicitly.

[22] For p multiple reflections, there will be p specular reflection points. Since we are only considering the first-order diffracted field, only the specular reflection points of the first receiver that fall within the first Fresnel zone centered around the first-order specular reflection point of the second receiver will be considered. In a typical environment, the distance between consecutive specular reflection points, Δy, is likely to be greater than the width of the first Fresnel zone for small orders of multiple reflections. It has been shown by Rustako et al. [1991] that considering multiple reflections up to sixth order is sufficient for a microcell. Hence, for such small number of multiple reflections, it is reasonable to assume that only one specular reflection point will lie within a typical first Fresnel zone: Specifically, consider the case for xr = xt = equation image and yrytd. Simple geometrical considerations show that the minimum possible separation of Δy is given by equation image, where Q, the number of transmitter images considered, is proportional to the maximum number of reflections. The width of the first Fresnel zone, 2W, simplifies to equation image (a2 + d2)3/4. Thus for Q < equation image (a2 + d2)−3/4, the separation Δy will be greater than the width of the first Fresnel zone and our assumption holds. In a typical microcell scenario with a street width of 20 m and a transmitter-receiver separation of 100 m, Q must be less than 11.2 for a system operating at 2.5 GHz. This corresponds to a maximum of about 20 multiple reflections, which is more than sufficient for our considerations.

[23] A simplification can be made when sr/sr′ ≈ 1, as the first-order specular reflection point of the receivers will be located near the midpoint of the transmitter and receiver separation. An analysis of the geometry of the pth-order multiply reflected path shows that only the (equation image)th specular reflection point is likely to be within the first Fresnel zone when p is odd and small. Furthermore, for this specular reflection point to be on the same wall in order to constitute any correlation, only an alternating odd number of transmitter images needs to be considered (e.g., first, fifth, etc.). Specifically, the correlation with the diffracted field from the upper wall 〈 Eref1*.Ediff2,0〉 exists only when q is odd for the transmitter image at To,q and even for To,q+. Conversely, the correlation with the diffracted field from the lower wall 〈Eref1*.Ediff2,a〉 exists only when q is even for To,q and odd for To,q+. Taking into account all the assumptions above, the correlation expressions are now given by

equation image
equation image

where

equation image
equation image

In the above expression, the superscript ‘+’ for the reflected field term E1,e,2q+* denotes that the transmitter image is on the positive x axis while ‘*’ denotes complex conjugation. The comma-separated subscripts ‘1, e, 2q’ represents the receiver number (1 or 2), odd or even number of reflection (o or e) and the number of multiply reflections respectively.

[24] We are only concerned with the correlation terms 〈E1,o,4r−3*.Ediff2,0(M), 〈E1,o,4r−1+*.Ediff2,0(M), 〈E1,o,4r−3+*.Ediff2,a(M) and 〈E1,o,4r−1*.Ediff2,a(M) as the rest of the terms of the form 〈Eref1*〉.〈Ediff2〉 can be found from the preceding sections. To simplify the analysis, we first evaluate the scalar correlation and consider the effects of polarization later. The scalar correlation terms can be expressed in the form

equation image

Consider first 〈E1,o,4r−3*Ediff2,0(M); the magnitude of the correlation depends on the separation of the two specular reflection points with respect to the first and second receiver ∣yQ1,o,r−,0yQ2,o,1−,0∣ and the probability that the rest of the specular reflections points are on a wall. The superscripts ‘−,0’ in yQ1,o,r−,0 are used to denote that location of the transmitter image (+ or − on the x axis) and the wall (x = 0 or a) respectively; while the subscripts ‘Q1, o, r’ are used to represent the receiver number (1 or 2), odd or even number of reflection (o or e) and the rth specular reflection point in a (4r − 3) multiply reflected ray respectively.

[25] The correlation can thus be written in the general form of Er × Prob × Corr, where ‘Er’ represents the multiple specularly reflected field, ‘Prob’ represents the probability of this field being supported as described above, and ‘Corr’ represents the correlation due to the separation ∣yQ1,o,r−,0yQ2,o,1−,0∣. The probability term can be quantified using successive pairwise correlations of the telegraph function as discussed earlier. The product term ‘Er × Prob’ will thus yield an expression that is of a similar form to the average scalar multiply reflected field equation imageEo,4r−3〉 derived earlier, the only difference being the exclusion of the specular reflection point yQ1,o,r−,0. In the analysis that follows, the term ‘Er × Prob’ will be replaced by 〈 Eo,4r−3

equation image

where ‘‡’ here is used to indicate a modified 〈E1,o,4r−3〉 expression excluding the above mentioned specular reflection point. The same analysis can be applied to the rest of the correlation terms to obtain the respective product terms, ‘Er × Prob’. The ‘Corr’ term for each of the four cases can be evaluated by considering first the correlation due to the separation of two specular reflection points |yQ1yQ2| ≡ D. For λgλw considering 2 N edges within the first Fresnel zone, the correlation expression is found to be

equation image

where the constants A, B, C, W are defined previously and

equation image
equation image
equation image

The functions g0+,−(m), gx1+,−(m) and gx2+,−(m) are given by

equation image
equation image
equation image
equation image
equation image
equation image

where the function R(m,n) for m, n > 0 is defined by

equation image

with the initial conditions R(1, n) = 2n−2 for n > 1 and R(1, 1) = 1. Hence the 4 polarimetrically correct correlation terms can be expressed as

equation image
equation image
equation image
equation image

The polarization vector of 〈E1,o follows exactly that of the reflected field and equation image is the corresponding diffracted field polarization from the upper and lower wall given by (4). As discussed earlier, the corresponding expressions for the lower wall can be determined through a trivial coordinate transformation. Hence the resultant correlation between the multiply reflected field at R1 and the diffracted field at R2 is

equation image

Similarly for 〈 Ediff1*.Eref2〉, the correlation expression can be found from 〈Eref1*.Ediff2〉 by simply swapping the receiver positions and conjugating the final result.

3.6. Correlation Between Diffracted Fields

[26] Since no multiple diffractions are considered, the correlation between first-order diffracted fields can be written as

equation image

Only correlation terms between diffracted fields arising on the same random wall have to be evaluated. Consider first the correlation between diffracted field from the upper random wall equation imageEdiff1*.Ediff200. As discussed earlier, the polarization of the diffracted field can be assumed to be approximately independent of the location of the corresponding diffracting edge as long as it is within the first Fresnel zone, giving

equation image
equation image

Ed1(yj) is the diffracted field at R1 due to an edge at yj and (Ed2) (yk) is the diffracted field at R2 due to an edge at yk. For λgλw, the correlation for 2N edges within the two overlapping first Fresnel zones corresponding to the two adjacent receiver locations is

equation image

where H = equation imageK1* K2, and K1,2 were defined in (48). Finally,

equation image

[27] The overlapping of the first Fresnel zone with respect to the first and second receivers will only affect the integral limits and is discussed shortly. equation image(zequation image)2 erfc (Aa (zequation image)), where A = equation image and the subscript a is used to represent the respective receivers. The function h0 (m) is given by

equation image

where

equation image
equation image

The function h (m) depends on the variables x, y (not shown explicitly and so cannot be taken out of the integral) is given by

equation image

where

equation image
equation image
equation image
equation image

When considering the diffracted field at only one receiver, the lower and upper limits of integration are simply the edges of the first Fresnel zone. However, when more than one receiver is involved, their corresponding Fresnel zones overlap. The four possible overlap region bounds are given by yQ2W2yyQ2 + W2, yQ1W1yyQ1 + W1, yQ2W2yyQ1 + W1 and yQ1W1yyQ2 + W2. The different first Fresnel zone overlap regions can be taken into account by considering them as four independent scenarios. From (87), integrals of the following general form have to be evaluated in order to quantify the correlation,

equation image
equation image

The first integral (96) can be evaluated analytically using the same technique as before, by employing the Taylor series expansion of the complementary error function, only this time with two erfc functions. The second integral (97), however, involves a two dimensional integration of the product of any arbitrary order of x and y with I1* (x) I2 (y). The analytic solution can be found using the same Taylor series expansion technique, but when compared to numerical integration it is found to be valid only for very limited transmitter-receiver separations. Beyond such small separations, the analytic solution breaks down and tends to diverge. This problem may be alleviated by adopting a higher-order truncation in the Taylor series expansion of the error function. However, a systematic analysis failed to yield a numerically stable expression and a purely numerical solution was pursued instead. For this, Romberg numerical quadrature [Press et al., 1992] is employed as it converges quickly and is simple to implement. As before, any remaining similar terms can be evaluated using coordinate transformations.

4. Theoretical Validation

[28] Monte Carlo simulations have been carried out to validate the current model. Although we recognize that they are idealizations of real microcellular environments, they are useful to ensure that the mathematical analysis and assumptions used to derive the model are valid and do not result in significant loss of prediction accuracy. The simulation is performed at 2.5 GHz for a 1 × 2 SIMO system for over 4000 realizations of the random wall. The first random wall is located on the y axis, while the second wall is located at x = 20 m, parallel to the first wall. In order to show the accuracy range of the model, the first receiver location is varied along the street up to 150 m away from the transmitter. The second receiver is held fixed at a distance δ = 0.3λ and oriented at ψ = π/2, measured clockwise from the negative y axis, from the first receiver. Hence the receiving antennas are broadside to the transmitter. The rest of the simulation parameters are summarized in Table 1. To reduce the computational time and complexity of the simulation program, we have ignored any ground reflections in both the model and the Monte Carlo simulations.

Table 1. Monte Carlo Simulation Parameters
ParametersValues
Transmitter, T(xt, yt, zt), m(5.0, 10.0, 10.0)
Receiver, R1(xr1, yr1, zr1), m(5.0, y, 2.0)
Upper wall average, 〈w0, m10
Upper gap average, 〈g0, m10
Lower wall average, 〈wa, m10
Lower gap average, 〈ga, m10
Relative dielectric constant, ɛ8.7
Conductivity, σ, S m−1156.9 × 10−3
Maximum number of multiple reflections6

[29] The simulated and modeled average reflected fields and the correlation between the reflected fields are in exact agreement and their discussion is omitted for brevity. Figure 5 compares the diffracted field intensity for 2,4 and 6 edges within the first Fresnel zone. Here, it is observed that all three cases correspond closely to the simulated value until a point (around 55 m for 2 edges, 110 m for 4 edges) when the resultant width of the first Fresnel zone becomes so wide that the probability of only m or fewer edges occurring within the region is very small. Hence our assumptions become invalid after this point and the model breaks down. Similar trends can be observed in both 〈Eref1*Ediff2〉 and 〈Ediff1* Ediff2〉. Assuming that a model validity range of 100–150 m is sufficient for most practical implementations in a microcell, up to four to six edges will have to be considered. Increasing the number of edges to be taken into account will increase the validity range of the model, but at the expense of a longer computation time.

Figure 5.

Comparison of simulated and modeled average diffracted field for varying distance from the transmitter.

[30] Three MIMO systems (2 × 2, 4 × 4 and 6 × 6) are investigated. The first transmitter and receiver antenna element positions are held fixed at (5, 10) m and (5, 110) m, respectively, while the remaining antennas are arranged in a linear configuration oriented either parallel (end fire) or perpendicular (broadside) to the random walls. The average length of walls and gaps are both assumed to be 10 m and up to 4 edges are considered in the evaluation of the fields and their correlations. The effect of spatial correlation is shown by varying the antenna spacing at the transmitter and receiver up to 10λ.

[31] Figure 6 shows the average MIMO link capacity plots for a broadside linear array at both the transmitter and receiver ends when the average signal-to-noise ratio is taken to be 10 dB. For increasing transmitter and receiver separation along the street, the average signal-to-noise ratio variation with distance can be simply computed from the average line-of-sight (LOS) electric field intensity as given by Blaunstein [2000]. As the two receiver antennas are separated by a small distance on the scale of a wavelength in Figure 6, we treat the signal-to-noise ratio as a constant.

Figure 6.

Comparison of modeled and ideal LOS capacity for a 2 × 2, 4 × 4, and 6 × 6 MIMO system arranged in a broadside linear configuration.

[32] For increasing receiver separation, it is observed that the capacity of the MIMO system increases and slowly tends to that of the ideal MIMO capacity. However, in a practical system scenario, antenna separation is at a premium and for small antenna separations, the ensemble averaged capacity upper bound is very much smaller than that of the ideal capacity.

[33] The effect of the antenna element configuration can be investigated by comparing the average capacity plots for a 4 × 4 system with its elements arranged in broadside, end fire and square configuration as shown in Figure 7. From the capacity plots, it can be concluded that if the direction of propagation is roughly known, the capacity can be maximized by placing the linear antenna array perpendicular to the angle of arrival. However, when the angle of arrival is uniform, or unknown, the square array offers the best compromise.

Figure 7.

Comparison of the modeled capacity for a 4 × 4 MIMO system arranged in a broadside linear, inline linear, and square configuration at the receiver.

5. Conclusions

[34] A semideterministic model for a MIMO system has been derived, taking into consideration scattering due to large-scale structures, such as side streets and windows, in a typical urban environment. The mean fields and spatial correlations are ensemble averaged over many realizations of a randomly “broken” wall, which is described using a simple telegraph function. The completed model can be used to predict the (ensemble) average capacity in a LOS urban microcell, taking into consideration the correlation between the antenna elements at both the transmitting and receiving ends. As such a model is fast and cost effective (the computational times are of the order of 1s or less on a desktop PC), such information is particularly useful for system designers to determine an optimum MIMO system setup. Comparing with the Monte Carlo simulations, the time taken to compute each simulation is found to be several orders of magnitude greater (minutes to hours) than the model prediction time, hence dramatically increasing the prediction efficiency when the model is employed. Furthermore, the model allows the MIMO capacity to be directly associated with the environment's physical parameters and propagation mechanisms, and so can be used in MIMO performance evaluative studies. If only a small number of (e.g., geometrical) parameters need to be varied, a fully numerical Monte Carlo approach in evaluating the MIMO link performance would be acceptable. However, if an extensive exploration of the behavior of the channel model across the entire environment parameter space is needed, a Monte Carlo simulation approach will be prohibitively expensive computationally. On the other hand a semianalytical model such as the one presented can provide a better understanding of the system performance at a very small computational cost.

[35] Although our model was only derived for a limited scenario, it can be further extended to include more practical scenarios. For instance, a NLOS scenario can be approximated by removing the LOS component, or performing a more rigorous NLOS channel model construction as in Mughal [2001]. However, there is still a need to perform MIMO measurements, so that such models can be properly validated.

Appendix A:: Construction of the Probability Density Function

[36] We construct the joint PDF, pe(y1, y2.ym; m; Δy), with m being the number of edges within the interval Δy, and yi being the locations of transitions between walls and gaps. For convenience, we simplify the notation to pm (y1, y2.ym). There are 4 possible ways in which the random wall can start and end within the interval; either start with a wall and finish on a wall (wall-wall), start with a gap and finish on a gap (gap-gap), start with a wall and finish on a gap (wall-gap) or start with a gap and finish on a wall (gap-wall). The joint PDF for m edges can then be written as a summation of the PDFs for the above cases,

equation image

where the subscript ‘w’ represents a wall and ‘g’ represents a gap. The joint PDF for m edges can be further grouped under odd or even numbers of edges. When the total number of edges is odd, only wall to gap (pwg) and gap to wall (pgw) cases are possible. Similarly, for m even, only wall to wall (pww) and gap to gap (pgg) cases are possible. Thus

equation image
equation image

Each of the PDFs pw/g,w/g can be represented as follows,

equation image
equation image
equation image
equation image

Pw,g are the probabilities of the occurrence of a wall or a gap at any random position, equation image (x) \are the probabilities that there are no transitions in the interval x and a, b are the lower and upper end points of the interval Δy, respectively:

equation image
equation image
equation image

where equation imagew〉, 〈g〉 are the average length of walls and gaps, respectively:

equation image

To show how the PDFs are constructed, consider (A4) for the wall to gap case, pwg The first multiplier Pw represents the probability that a wall occurs at the starting point a. The second multiplier is the probability that this wall will remain uninterrupted (no transition) until the first edge at y1, after which the transition to a gap is represented by the rate of change of wall, λw. In this manner, the product pair of PDF and transitional probability can be used to represent a continuous random wall for any arbitrary number of edges. The last product pair omits the transition probability to a wall as we expect a gap in this case. Using the same method, the remaining cases can be constructed. In our case, where the wall and gap are assumed to be exponentially distributed, equation image is simply given by

equation image

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