# MIMO Channel Capacity in 2D and 3D Isotropic Environments

Authors: Ryan J. Pirkl; Kate A. Remley
Journal: International Journal of Antennas and Propagation
 (2012)

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## Abstract

We analyze theoretical distributions of MIMO channel capacity for different antennas in 2D and 3D statistically isotropic environments, which may be generated by multiprobe anechoic and reverberation chambers, respectively. We observe that the two environments yield comparable capacity distributions provided that (1) the 2D statistically isotropic environment’s capacity data are taken at many different antenna orientations and (2) the radiation elements have a low directivity. When these conditions are met, we find that the relative error between the 2D statistically isotropic environment’s orientation-combined capacity distribution and the 3D statistically isotropic environment’s capacity distribution is typically less than 10% for signal-to-noise ratios greater than 5 dB.


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## Body

## 1. Introduction

Multiprobe anechoic chambers and reverberation chambers are capable of generating 2D and 3D statistically isotropic environments, respectively, suitable for over-the-air (OTA) testing of multiple-input multiple-output (MIMO) wireless terminals [1–3]. In CTIA, 3GPP’s RAN4, and COST 2100 both multiprobe anechoic and reverberation chambers are being considered for adoption into certification test procedures for MIMO handsets [1, 4–6]. It is possible that test procedures based on both chambers may be adopted, but it is unclear to what extent their test results are comparable. Here, we compare the statistically isotropic environments generated by multiprobe anechoic and reverberation chambers in terms of their capacity distributions for different MIMO antennas-under-test (AUTs).Capacity determines the maximum obtainable throughput for a wireless device. The relationship between capacity and throughput implies that 2D and 3D statistically isotropic environments will only provide comparabledevice performance test results if their capacity statistics are comparable for any given AUT. Prior work has used channel simulations to compare the mean capacity of 2D and 3D statistically isotropic environments [2]. For a single orientation of the AUT, large differences in the channel’s mean capacity were observed for the 2D and 3D statistically isotropic environments. In contrast, by changing the orientation (and position) of the AUT between simulation trials, the mean capacities of the antennas converged to the same value in both environments.We expand on the results presented in [2] by comparing the two environments’ theoretical distributions of MIMO channel capacity for several AUTs. Our procedure extends the analysis presented in [7] to the problem of comparing capacity distributions for MIMO OTA test environments. We observe that for a single orientation of the AUT, the capacity distributions obtained from the two environments are unlikely to be comparable due to the orientation dependence of the 2D isotropic environment’s capacity statistics. This is due to the well-known environment dependence of each AUT’s complex correlation coefficient and mean effective gains (cf., [7–9]). We also observe that differences in the 2D and 3D environments’ capacity distributions may be reduced substantially by comparing the 3D environment’s orientation-independent capacity distribution to the 2D environment’s orientation-combined capacity distribution, which is the distribution formed when capacity realizations are combined from different orientations of the AUT. However, we find that the difference between these two capacity distributions is small (e.g., within 10%) only if the AUTs use low-directivity elements; for high-directivity elements, the difference between the capacity distributions may exceed 20%. This suggests that discrepancies between device performance tests in statistically isotropic environments generated by multiprobe anechoic and reverberation chambers will be small provided that (1) the DUT’s performance in the multiprobe anechoic chamber is evaluated at many different orientations, and (2) the DUT’s dimensions are on the order of a wavelength or less such that the directivities of the antenna elements are low.We begin in Section2 by reviewing the covariance between the signals received by the elements of an AUT when it is placed in a random electromagnetic field. In Section 3, we define the power-angle spectra for 2D and 3D statistically isotropic environments, as well as the set of antenna configurations used in our numerical study of MIMO channel capacity. Capacity distributions for the AUTs in each test environment are presented and analyzed in Section 4. Section 5 explores why certain antenna configurations are more sensitive to orientation in 2D statistically isotropic environments than others. Section 6 summarizes the paper.


## 2. Antennas in Random Electromagnetic Fields

Let us consider a two-element antenna in a time-harmonic electromagnetic field. Assuming the antenna is in the far-field, whereby the electromagnetic field is composed entirely of homogeneous plane waves, we may express the signalxi received by the ith antenna element as [8](1)xi=∫dΩAi(Ω)⋅F(Ω),
where i∈{1,2}, · denotes the vector dot product, Ai(Ω) is the ith element’s complex vector radiation pattern defined as(2)Ai(Ω)=Aih(Ω)ĥ+Aiv(Ω)v̂,
and F(Ω) is the field’s complex vector plane-wave spectrum defined as(3)F(Ω)=Fh(Ω)ĥ+Fv(Ω)v̂.
In (2) and (3), ĥ and v̂ are the two orthogonal unit vectors defining the horizontal and vertical polarizations, respectively, and Ω denotes a point on the unit sphere with ∫dΩ corresponding to an integration over the unit sphere.To constrain the scope of our capacity analysis, we consider power-normalized radiation patterns for which(4)∫dΩ‖Ai(Ω)‖2=4π.
That is, ∥Ai(Ω)∥2 is equal to the directivity of the ith antenna element, where we neglect impedance mismatches and ohmic loss [8]. We recognize that mismatches and loss do affect the capacity of the MIMO wireless channel by way of a reduction in the signal-to-noise ratio (SNR) and, potentially, a gain imbalance at the receiver [10]. However, because impedance mismatches and ohmic losses are solely antenna-related quantities that are independent of the environment’s power-angle spectrum, their effect on capacity does not provide insight into the differences between MIMO device tests in 2D and 3D statistically isotropic environments.To describe a random electromagnetic field, we may specify the components,Fh(Ω) and Fv(Ω), of the plane-wave spectrum to be random variables characterized by the following covariances [11–13]:(5)E{Fh(Ω)Fh*(Ω′)}=Ph(Ω)δ(Ω-Ω′),E{Fv(Ω)Fv*(Ω′)}=Pv(Ω)δ(Ω-Ω′),E{Fh(Ω)Fv*(Ω′)}=E{Fv(Ω)Fh*(Ω′)}=0,
where ℰ{·} denotes the expectation operator, δ(·) is the Dirac delta function, and Ph(Ω) and Pv(Ω) are real quantities that describe the power density versus angle-of-arrival for horizontally and vertically polarized incident plane waves, respectively. The electromagnetic field’s vector power-angle spectrum is given by(6)P(Ω)=Ph(Ω)ĥ+Pv(Ω)v̂.
Denoting P0 as the total power available to an AUT, we define the vector power-angle spectrum such that(7)∫dΩ[P(Ω)⋅ĥ+P(Ω)⋅v̂]=P0.The covariance between the signals received by theith and jth antenna elements is defined as [13](8)E{xixj*}=ρijσiσj,
where σi and σj are the standard deviations of the signals received by ith and jth antenna elements, respectively, and ρij is the complex correlation coefficient of the two signals. Combining (1)–(8), the covariance may be written as(9)ρijσiσj=∫dΩ[Aih(Ω)Ajh*(Ω)Ph(Ω)+Aiv(Ω)Ajv*(Ω)Pv(Ω)].Interpretingσi2 (the variance of xi) as the power received by the ith antenna element, we may define the element’s mean effective gain (MEG) g¯i as the ratio of received power to available power: [8, 14](10)g¯i=σi2P0.
An element’s MEG accounts for the interplay between the element’s radiation pattern and the distribution of incident power versus angle-of-arrival. For an angle-invariant radiation pattern or power-angle spectrum, g¯i=1/2, indicating that one-half of the available power is received by the antenna [8]. By use of (8) and (9) with i=j corresponding to the variance of xi(note that ρii=1), we may express g¯i in terms of the ith antenna’s vector radiation pattern and power-angle spectrum:(11)g¯i=1P0∫dΩ[|Aiv(Ω)|2Pv(Ω)+|Aih(Ω)|2Ph(Ω)].
Finally, by combining (9)–(11), the environment-dependent correlation coefficient ρij may be expressed as(12)ρij=1P0g¯1g¯2×∫dΩ[Aiv(Ω)Ajv*(Ω)Pv(Ω)+Aih(Ω)Ajh*(Ω)Ph(Ω)].


## 3. Simulation Setup

In the following sections, we define the vector power-angle spectra and antenna configurations used in our numerical study of capacity in 2D and 3D statistically isotropic environments.


### 3.1. Power-Angle Spectra

For statistically isotropic environments, we expect that(13)Ph(Ω)=Pv(Ω),
whereby the power density versus angle-of-arrival is independent of the polarization of incident waves. For the 2D isotropic environment, the vector power-angle spectrum is only nonzero along the unit sphere’s equator and is given by(14)P2D(Ω)=P02πδ(θ-π/2)12(ĥ+v̂),
where θ denotes the zenith angle. In contrast, the 3D isotropic environment’s vector power-angle spectrum is constant across the entire unit sphere and is given by(15)P3D(Ω)=P04π12(ĥ+v̂).
Equations (14) and (15) are idealizations of the statistically isotropic environments that may be generated in multiprobe anechoic and reverberation chambers, respectively. Thus, we neglect non-idealities due to, for example, using a finite number of probes in the multiple-probe anechoic chamber [4–6] and unstirred energy in the reverberation chamber [15–17].


### 3.2. Antennas under Test

In the following sections, we will compare the capacity distributions for three classes of two-element antennas.(i)
Two closely-spaced and vertically polarized Hertzian dipoles separated by a distanced [18].(ii)
The recently developed CTIA MIMO reference antennas (so-called “Good’’, “Nominal’’, and “Bad’’) [19].(iii)
Two colocated and cross-polarized Hertzian dipoles aligned parallel to thex-y plane and located (a) in free space and (b) at a height h=λ/4 above a perfectly electrically conducting (PEC) surface with surface normal ẑ [20].The different antenna classes and configurations are intended to span the spectrum of antennas typically used by physically small handheld or mobile devices such as phones or laptops. Where applicable, our antenna descriptions include the effects of mutual coupling between elements.Table1 summarizes the different antenna configurations and compares the antennas’ correlation coefficient magnitudes and MEGs for the 2D and 3D statistically isotropic environments as calculated by use of (11)–(15). Due to (4)’s power normalization, g¯i=0.5 for all antennas in the 3D statistically isotropic environment. For the cross-polarized dipoles above a PEC surface, g¯i=0 for the 2D statistically isotropic environment, because the gain patterns of the antenna’s elements are zero for angles at and below the horizon [20]. The last column in Table 1 presents the maximum directivity of each antenna’s elements. We will use these values later in Section 5.Table 1
Specified antenna configurations.
ClassConfiguration2D3Dmax(∥Ai(Ω)∥2)|ρ12|g¯1g¯2|ρ12|g¯1g¯2d=λ0.020.800.800.120.500.501.85Closely spaced dipoles [18]d=λ/20.170.730.730.010.500.502.20d=λ/40.180.740.740.000.500.503.14“Good’’0.340.420.420.040.500.502.96CTIA reference antennas [19]“Nominal’’0.730.480.480.580.500.501.57“Bad’’0.900.540.540.900.500.501.77Cross-polarized dipoles [20]Free space0.000.380.380.000.500.501.50PEC0.000.000.000.000.500.505.21


## 3.1. Power-Angle Spectra

For statistically isotropic environments, we expect that(13)Ph(Ω)=Pv(Ω),
whereby the power density versus angle-of-arrival is independent of the polarization of incident waves. For the 2D isotropic environment, the vector power-angle spectrum is only nonzero along the unit sphere’s equator and is given by(14)P2D(Ω)=P02πδ(θ-π/2)12(ĥ+v̂),
where θ denotes the zenith angle. In contrast, the 3D isotropic environment’s vector power-angle spectrum is constant across the entire unit sphere and is given by(15)P3D(Ω)=P04π12(ĥ+v̂).
Equations (14) and (15) are idealizations of the statistically isotropic environments that may be generated in multiprobe anechoic and reverberation chambers, respectively. Thus, we neglect non-idealities due to, for example, using a finite number of probes in the multiple-probe anechoic chamber [4–6] and unstirred energy in the reverberation chamber [15–17].


## 3.2. Antennas under Test

In the following sections, we will compare the capacity distributions for three classes of two-element antennas.(i)
Two closely-spaced and vertically polarized Hertzian dipoles separated by a distanced [18].(ii)
The recently developed CTIA MIMO reference antennas (so-called “Good’’, “Nominal’’, and “Bad’’) [19].(iii)
Two colocated and cross-polarized Hertzian dipoles aligned parallel to thex-y plane and located (a) in free space and (b) at a height h=λ/4 above a perfectly electrically conducting (PEC) surface with surface normal ẑ [20].The different antenna classes and configurations are intended to span the spectrum of antennas typically used by physically small handheld or mobile devices such as phones or laptops. Where applicable, our antenna descriptions include the effects of mutual coupling between elements.Table1 summarizes the different antenna configurations and compares the antennas’ correlation coefficient magnitudes and MEGs for the 2D and 3D statistically isotropic environments as calculated by use of (11)–(15). Due to (4)’s power normalization, g¯i=0.5 for all antennas in the 3D statistically isotropic environment. For the cross-polarized dipoles above a PEC surface, g¯i=0 for the 2D statistically isotropic environment, because the gain patterns of the antenna’s elements are zero for angles at and below the horizon [20]. The last column in Table 1 presents the maximum directivity of each antenna’s elements. We will use these values later in Section 5.Table 1
Specified antenna configurations.
ClassConfiguration2D3Dmax(∥Ai(Ω)∥2)|ρ12|g¯1g¯2|ρ12|g¯1g¯2d=λ0.020.800.800.120.500.501.85Closely spaced dipoles [18]d=λ/20.170.730.730.010.500.502.20d=λ/40.180.740.740.000.500.503.14“Good’’0.340.420.420.040.500.502.96CTIA reference antennas [19]“Nominal’’0.730.480.480.580.500.501.57“Bad’’0.900.540.540.900.500.501.77Cross-polarized dipoles [20]Free space0.000.380.380.000.500.501.50PEC0.000.000.000.000.500.505.21


## 4. Capacity

We consider a 2-by-2 MIMO channel with uncorrelated transmit antennas and potentially correlated receive antennas. For the case of an uninformed transmitter and a receiver with perfect channel state information, the cumulative distribution function (CDF) of MIMO capacity is given by [21](16)FC(c)=∑i=12∑j=1,j≠i21γiγj(γi-γj)×{e1/γi+1/γjγj-γiγi(∫12cdvve-v/γi-2c/vγj)+[γi+(γi-γj)(γi+2c)]e(1-2c)/γie1/γi+1/γjγj-γiγi(∫12cdvve-v/γi-2c/vγj)},
where γi are the eigenvalues of the 2-by-2 matrix Γ defined as(17)Γ=1N0nt𝚺r.
In (17), nt=2 is the number of transmit antenna elements, N0 is the noise power density at the receiver, and Σr is the receiver’s covariance matrix given by(18)𝚺r=[E{|x1|2}E{x1x2*}E{x2x1*}E{|x2|2}].
Using (8)–(12), Σr may be expressed as(19)𝚺r=P0[g¯1ρg¯1g¯2ρ*g¯1g¯2g¯2],
where the eigenvalues of Γ are(20)γ21=P0N0nt[g¯1+g¯2±g¯12+g¯22+2g¯1g¯2(2|ρ|2-1)2],
and (P0/2)/N0 defines the SNR of the corresponding single-input single-output link for an isotropic antenna.


### 4.1. Single Orientation

Figures1(a) and 1(b) compare the analytic 2-by-2 MIMO capacity CDFs for the different AUTs in the 2D and 3D statistically isotropic environments, respectively. The CDFs were calculated from (16)–(20) with the correlation coefficients and MEGs reported in Table 1. We observe that the different test environments may yield considerably different capacity CDFs for identical AUTs. The cross-polarized dipoles above a PEC surface are particularly problematic, because the MEGs of the antenna elements are zero (see Table 1) for the 2D environment. This implies an SNR of zero and results in a capacity CDF given by a unit step function with a transition at zero bps/Hz.Comparison of the capacity CDFs for the different AUTs in statistically isotropic environments at a single orientation: (a) capacity CDFs for the 2D environment, (b) capacity CDFs for the 3D environment, (c) relative error in the 2D environment’s capacity CDFs with respect to the 3D environment’s capacity CDFs. The center plot’s legend applies to all three plots.
(a)
2D environment(b)
3D environment(c)
relative errorFigure1(c) illustrates the shift in the capacity CDFs for the 2D statistically isotropic environment with respect to the 3D statistically isotropic environment. We quantify this shift in terms of a relative error of the 2D environment’s capacity curves with respect to the 3D environment’s capacity CDFs. We note that the relative error for the cross-polarized dipoles above a PEC surface is 100% due to its step-function capacity CDF in the 2D isotropic environment; this error is outside of the range of relative errors considered in Figure 1(c). Depending on the antenna configuration, the relative error in the 2D environment’s capacity CDF may exceed 20%. This suggests that capacity statistics for a given AUT in a 2D and 3D statistically isotropic environment may not be comparable for a single orientation of the AUT. This is in agreement with the conclusions presented in [2] for ergodic capacity based on a single orientation of the AUT.


### 4.2. Multiple Orientations

As suggested in [2], a possible solution for obtaining comparable capacity statistics is to combine capacity data from different orientations of the AUT in the 2D statistically isotropic environment. Assuming N orientations of the AUT are sampled equally (i.e., an equal number of independent capacity realizations are taken at N unique orientations), the CDF of the combined set of capacity realizations is equal to the mean of the N orientation-dependent capacity CDFs: [22](21)FĈ(c)=1N∑n=1NFC(n)(c),
where FC(n)(c) denotes the capacity CDF associated with the nth orientation of the AUT, and FĈ(c) describes the orientation-combined CDF of the resulting (mixture) distribution.As Figure2 illustrates, there are three degrees-of-freedom in a 3D rotation of an object to a new orientation [23–25]. In the rotation sequence illustrated in Figure 2, an object is rotated by α∈[0,π) in a direction β∈[0,2π) such that object’s original z-axis is aligned to an axis z′ defined by (α,β). Then, the object is rotated by an angle χ∈[0,2π) about the z′-axis. To obtain a uniform sampling of the AUT’s 3D orientation space, we apply the successive orthogonal images technique [24]. We specify a set of L angle pairs (α,β) corresponding to L points uniformly distributed on the unit sphere. For each angle pair, χ=2π(m/M) for m=0,1,…,M-1, where M is an integer number of χ rotations per (α,β) pair as given by(22)M=round(πL),
and the total number of orientations is N=ML. In (22), round(·) rounds to the nearest integer. Equation (22) provides the appropriate weighting between M and L to ensure that the N orientations are a uniform sampling of the rotation space (cf., [24, Equation (3.1)]).Figure 2
There are three degrees of freedom in the specification of a rotation to an arbitrary orientation. The diagram defines a rotation of an object in 3D via Euler angles. The object is rotated by an angleα in the direction β and subsequently rotated by an angle χ about the axis z′ defined by (α,β).A set ofL=162 uniformly distributed angle pairs (α,β) were generated by iterative triangularization of an icosahedron. By use of (22), we have M=23 whereby a set of N=ML=3726 uniformly distributed orientations were considered. (Similar results were observed for larger choices of N.) The AUT was rotated to each of the N orientations, and (11), (12), and (16) were used to calculate the resulting N capacity CDFs. For each of the AUTs, Figure 3 compares the N orientation-dependent capacity CDFs obtained from the 2D statistically isotropic environment (solid lines), the resulting orientation-combined CDF from the 2D statistically isotropic environment (solid line), and the orientation-independent capacity CDF obtained from the 3D statistically isotropic environment (dashed line). We observe large variations in the 2D environment’s orientation-dependent capacity CDFs for the closely spaced dipoles (Figures 3(a)–3(c)), as well as the cross-polarized dipoles above a PEC surface (Figure 3(h)).Comparison of the capacity CDFs for multiple orientations of the two-element antennas in statistically isotropic environments. Each plot compares the 2D environment’s orientation-dependent capacity CDFs, the 2D environment’s orientation-combined capacity CDFs, and the 3D environment’s orientation-independent capacity CDFs.
(a)
Closely spaced dipoles:d=λ(b)
Closely spaced dipoles:d=λ/2(c)
Closely spaced dipoles:d=λ/4(d)
CTIA reference antenna: “Good”(e)
CTIA reference antenna: “Nominal”(f)
CTIA reference antenna: “Bad”(g)
Cross-polarized dipoles: free space(h)
Cross-polarized dipoles: PECFigures4(a) and 4(b) compare the 2D environment’s orientation-combined capacity CDFs to those from the 3D environment’s orientation-independent capacity CDFs; Figure 4(c) examines the relative error in the 2D statistically isotropic environment’s orientation-combined capacity CDFs with respect to the 3D statistically isotropic environment’s orientation-independent capacity CDFs. We observe that for the CTIA reference antennas and cross-polarized dipoles in free space, this “error’’ is quite small, with typical magnitudes below 2%. In comparison, the relative error of the cross-polarized dipoles above the PEC surface and, to an extent, the closely-spaced dipoles, is much larger, with magnitudes exceeding 5–20%. Note that these are the same antenna configurations which showed the greatest variation in the 2D environment’s orientation-dependent capacity CDFs (see Figure 3). Thus, Figures 3 and 4(c) suggest that for a given AUT, the 2D environment’s orientation-combined capacity statistics will be comparable to the 3D environment’s orientation-independent capacity statistics provided that the variations in the 2D environment’s orientation-dependent capacity CDFs are small. Repeat studies at SNRs of 5 dB to 20 dB yielded similar errors, albeit with a slight increase in relative error for decreasing SNR. The typical range of error magnitudes remained below 10%.Comparison of the capacity CDFs for the different AUTs in statistically isotropic environments at multiple orientations: (a) orientation-combined capacity CDFs for the 2D environment, (b) orientation-independent capacity CDFs for the 3D environment, (c) relative error in the 2D environment’s orientation-combined capacity CDFs with respect to the 3D environment’s orientation-independent capacity CDFs. The center plot’s legend applies to all three plots.
(a)
2D environment(b)
3D environment(c)
Relative error


## 4.1. Single Orientation

Figures1(a) and 1(b) compare the analytic 2-by-2 MIMO capacity CDFs for the different AUTs in the 2D and 3D statistically isotropic environments, respectively. The CDFs were calculated from (16)–(20) with the correlation coefficients and MEGs reported in Table 1. We observe that the different test environments may yield considerably different capacity CDFs for identical AUTs. The cross-polarized dipoles above a PEC surface are particularly problematic, because the MEGs of the antenna elements are zero (see Table 1) for the 2D environment. This implies an SNR of zero and results in a capacity CDF given by a unit step function with a transition at zero bps/Hz.Comparison of the capacity CDFs for the different AUTs in statistically isotropic environments at a single orientation: (a) capacity CDFs for the 2D environment, (b) capacity CDFs for the 3D environment, (c) relative error in the 2D environment’s capacity CDFs with respect to the 3D environment’s capacity CDFs. The center plot’s legend applies to all three plots.
(a)
2D environment(b)
3D environment(c)
relative errorFigure1(c) illustrates the shift in the capacity CDFs for the 2D statistically isotropic environment with respect to the 3D statistically isotropic environment. We quantify this shift in terms of a relative error of the 2D environment’s capacity curves with respect to the 3D environment’s capacity CDFs. We note that the relative error for the cross-polarized dipoles above a PEC surface is 100% due to its step-function capacity CDF in the 2D isotropic environment; this error is outside of the range of relative errors considered in Figure 1(c). Depending on the antenna configuration, the relative error in the 2D environment’s capacity CDF may exceed 20%. This suggests that capacity statistics for a given AUT in a 2D and 3D statistically isotropic environment may not be comparable for a single orientation of the AUT. This is in agreement with the conclusions presented in [2] for ergodic capacity based on a single orientation of the AUT.


## 4.2. Multiple Orientations

As suggested in [2], a possible solution for obtaining comparable capacity statistics is to combine capacity data from different orientations of the AUT in the 2D statistically isotropic environment. Assuming N orientations of the AUT are sampled equally (i.e., an equal number of independent capacity realizations are taken at N unique orientations), the CDF of the combined set of capacity realizations is equal to the mean of the N orientation-dependent capacity CDFs: [22](21)FĈ(c)=1N∑n=1NFC(n)(c),
where FC(n)(c) denotes the capacity CDF associated with the nth orientation of the AUT, and FĈ(c) describes the orientation-combined CDF of the resulting (mixture) distribution.As Figure2 illustrates, there are three degrees-of-freedom in a 3D rotation of an object to a new orientation [23–25]. In the rotation sequence illustrated in Figure 2, an object is rotated by α∈[0,π) in a direction β∈[0,2π) such that object’s original z-axis is aligned to an axis z′ defined by (α,β). Then, the object is rotated by an angle χ∈[0,2π) about the z′-axis. To obtain a uniform sampling of the AUT’s 3D orientation space, we apply the successive orthogonal images technique [24]. We specify a set of L angle pairs (α,β) corresponding to L points uniformly distributed on the unit sphere. For each angle pair, χ=2π(m/M) for m=0,1,…,M-1, where M is an integer number of χ rotations per (α,β) pair as given by(22)M=round(πL),
and the total number of orientations is N=ML. In (22), round(·) rounds to the nearest integer. Equation (22) provides the appropriate weighting between M and L to ensure that the N orientations are a uniform sampling of the rotation space (cf., [24, Equation (3.1)]).Figure 2
There are three degrees of freedom in the specification of a rotation to an arbitrary orientation. The diagram defines a rotation of an object in 3D via Euler angles. The object is rotated by an angleα in the direction β and subsequently rotated by an angle χ about the axis z′ defined by (α,β).A set ofL=162 uniformly distributed angle pairs (α,β) were generated by iterative triangularization of an icosahedron. By use of (22), we have M=23 whereby a set of N=ML=3726 uniformly distributed orientations were considered. (Similar results were observed for larger choices of N.) The AUT was rotated to each of the N orientations, and (11), (12), and (16) were used to calculate the resulting N capacity CDFs. For each of the AUTs, Figure 3 compares the N orientation-dependent capacity CDFs obtained from the 2D statistically isotropic environment (solid lines), the resulting orientation-combined CDF from the 2D statistically isotropic environment (solid line), and the orientation-independent capacity CDF obtained from the 3D statistically isotropic environment (dashed line). We observe large variations in the 2D environment’s orientation-dependent capacity CDFs for the closely spaced dipoles (Figures 3(a)–3(c)), as well as the cross-polarized dipoles above a PEC surface (Figure 3(h)).Comparison of the capacity CDFs for multiple orientations of the two-element antennas in statistically isotropic environments. Each plot compares the 2D environment’s orientation-dependent capacity CDFs, the 2D environment’s orientation-combined capacity CDFs, and the 3D environment’s orientation-independent capacity CDFs.
(a)
Closely spaced dipoles:d=λ(b)
Closely spaced dipoles:d=λ/2(c)
Closely spaced dipoles:d=λ/4(d)
CTIA reference antenna: “Good”(e)
CTIA reference antenna: “Nominal”(f)
CTIA reference antenna: “Bad”(g)
Cross-polarized dipoles: free space(h)
Cross-polarized dipoles: PECFigures4(a) and 4(b) compare the 2D environment’s orientation-combined capacity CDFs to those from the 3D environment’s orientation-independent capacity CDFs; Figure 4(c) examines the relative error in the 2D statistically isotropic environment’s orientation-combined capacity CDFs with respect to the 3D statistically isotropic environment’s orientation-independent capacity CDFs. We observe that for the CTIA reference antennas and cross-polarized dipoles in free space, this “error’’ is quite small, with typical magnitudes below 2%. In comparison, the relative error of the cross-polarized dipoles above the PEC surface and, to an extent, the closely-spaced dipoles, is much larger, with magnitudes exceeding 5–20%. Note that these are the same antenna configurations which showed the greatest variation in the 2D environment’s orientation-dependent capacity CDFs (see Figure 3). Thus, Figures 3 and 4(c) suggest that for a given AUT, the 2D environment’s orientation-combined capacity statistics will be comparable to the 3D environment’s orientation-independent capacity statistics provided that the variations in the 2D environment’s orientation-dependent capacity CDFs are small. Repeat studies at SNRs of 5 dB to 20 dB yielded similar errors, albeit with a slight increase in relative error for decreasing SNR. The typical range of error magnitudes remained below 10%.Comparison of the capacity CDFs for the different AUTs in statistically isotropic environments at multiple orientations: (a) orientation-combined capacity CDFs for the 2D environment, (b) orientation-independent capacity CDFs for the 3D environment, (c) relative error in the 2D environment’s orientation-combined capacity CDFs with respect to the 3D environment’s orientation-independent capacity CDFs. The center plot’s legend applies to all three plots.
(a)
2D environment(b)
3D environment(c)
Relative error


## 5. Sensitivity of Capacity to AUT Orientation in 2D Statistically Isotropic Environments

From Figures3 and 4(c), we observed that for a given AUT, large orientation-dependent variations in the 2D environment’s capacity CDF led to a large relative error between the 2D environment’s orientation-combined capacity CDF and the 3D environment’s orientation-independent capacity CDF. Here, we investigate the source of these large orientation-dependent variations in capacity and determine why some antennas are more sensitive to orientation in the 2D isotropic environment than others. This provides insight into the range of antenna configurations for which we may expect the 2D environment’s orientation-combined capacity CDF to be comparable to the 3D environment’s orientation-independent capacity CDF. We base our analysis on the orientation-dependence of each AUT’s correlation coefficient and MEGs in the 2D statistically isotropic environment. Note that for the 3D environment, these quantities are orientation invariant and are tabulated in Table 1.Figure5 presents scatter plots of each AUT’s MEGs and correlation coefficient for different orientations in the 2D statistically isotropic environment. The location of each circle corresponds to the MEG of the AUT’s elements; the circle’s size/shade corresponds to the magnitude of the correlation coefficient. We observe that the antennas exhibiting the largest variations in the 2D environment’s orientation-dependent capacity CDFs, namely, the cross-polarized dipoles above a PEC surface and the closely-spaced dipoles tend to have similar antenna element MEGs at each orientation (i.e., g¯1≈g¯2). This indicates that the average power received by each of the antenna elements is identical, whereby a low (or high) average received power at one element implies an equally low (or high) average received power at the other element. This allows for large variations in the cumulative power received by the two antenna elements. In contrast, the AUTs exhibiting the smallest variations in the orientation-dependent capacity CDFs tend to have MEGs that follow g¯1+g¯2≈1. This indicates that the cumulative power received by both antenna elements is approximately constant at different orientations, whereby a low average received power at one element is compensated by a high average received power at the other element.Scatter plots of mean effective gain (MEG) and correlation coefficient magnitudes for each AUT in the 2D statistically isotropic environment forN uniformly distributed orientations of the AUT. The position of each circle indicates the MEGs of the two antenna elements; the size/shade of each circle indicates the magnitude of the correlation coefficient.
(a)
closely-spaced dipoles:d=λ(b)
Closely spaced dipoles:d=λ/2(c)
Closely spaced dipoles:d=λ/4(d)
CTIA reference antenna: “Good”(e)
CTIA reference antenna: “Nominal”(f)
CTIA reference antenna: “Bad”(g)
Cross-polarized dipoles: free space(h)
Cross-polarized dipoles: PECBased on Figure5, the 2D environment’s orientation-dependent capacity CDFs appear to be more sensitive to joint orientation-dependent variations in the MEGs of an AUT’s elements than orientation-dependent variations in its correlation coefficient magnitude (e.g., compare Figures 5(g), 5(h), 3(g) and 3(h)). To confirm this, Figure 6 presents environment-independent contour plots of the median capacity for different correlation coefficient magnitudes and MEGs. For the calculations, we specify a nominal single-input single-output SNR of 10dB (i.e., (P0/2)/N0=10). Figure 6(a) corresponds to the case of equal antenna element MEGs; Figure 6(b) corresponds to the case of constant cumulative received power. As indicated by the range of median capacities for the two cases, the equal MEG case (g¯1=g¯2) allows for larger variations in the median capacity than the constant cumulative received power case (g¯1+g¯2=1). We also observe for both cases that the effect on capacity of large variations in the magnitude of the correlation coefficient is considerably weaker than large variations in MEG. This indicated by the smaller gradient for changes in g¯1 as compared to changes in |ρ12|.Median capacity for ranges of correlation coefficient magnitudes|ρ12| and MEGs, g¯1 and g¯2.
(a)
g¯2=g¯1(b)
g¯2=1-g¯1Based on Figures3–6 and the maximum directivities listed in Table 1, we expect that the 2D environment’s orientation-combined capacity statistics should be comparable to the 3D environment’s orientation-independent capacity statistics provided that the maximum directivities of the AUT’s elements are small (e.g., max(∥Ai(Ω)∥2)≤2). This criterion should lead to smaller orientation-dependent fluctuations in the AUT’s MEGs and, thereby, cumulative received power. In [7, 14], we note that similar conclusions were reached concerning the greater variability of MEG for different orientations of antennas with large directivities. Here, we have demonstrated that the variability in capacity is dominated by this variability in MEG. Antennas with markedly different element directivities may also yield comparable capacity CDFs, because it is less likely that g¯1≈g¯2. However, based on the antenna configurations considered here, we expect that low directivity is the more important criteria.


## 6. Conclusions

Channel capacity determines the maximum theoretical throughput for a given wireless link. By designing test procedures such that the capacity distributions obtained for a given AUT in a 2D and 3D statistically isotropic environment are comparable, we expect that throughput statistics for a corresponding DUT in these two environments will likewise be comparable. More so, we expect that discrepancies between observed throughput statistics for different devices in different test environments will be largely due to fundamental differences in the devices (e.g., antenna loss and mismatch, receiver sensitivity, and algorithm implementation) andnot fundamental differences in the capacity statistics of the test environments, assuming a sufficient number of AUT orientations in the 2D environment.Based on the analysis presented here, provided that (1) the DUT is rotated to different orientations in the 2D statistically isotropic environment, and (2) the dimensions of the DUT are on the order of a wavelength or less such that the element directivities will be low, we expect that throughput statistics for a DUT in 2D and 3D statistically isotropic environments will be within 10% of each other, as suggested by Figures3 and 4(c). This suggests that with properly designed test procedures, over-the-air tests of multiantenna wireless terminals in multiprobe anechoic and reverberation chambers should be comparable for a wide range of devices. Finally, it is worth emphasizing that our conclusions are based solely on the distribution of capacity. That is, our conclusions are independent of the actual number of realizations used to calculate a capacity (or throughput) statistic and apply even for the case where a single capacity (or throughput) measurement is made at each orientation.



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*Source: 676405-2012-05-20.xml*