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International Journal of Antennas and Propagation
Volume 2012 (2012), Article ID 676405, 11 pages
MIMO Channel Capacity in 2D and 3D Isotropic Environments
Electromagnetics Division, National Institute of Standards and Technology, RF Fields Group, 325 Broadway Street, MS 818.02, Boulder, CO 80305, USA
Received 5 January 2012; Accepted 19 March 2012
Academic Editor: Markus Landmann
Copyright © 2012 Ryan J. Pirkl and Kate A. Remley. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 comparable device 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 . 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  by comparing the two environments’ theoretical distributions of MIMO channel capacity for several AUTs. Our procedure extends the analysis presented in  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 Section 2 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 signal received by the th antenna element as  where , denotes the vector dot product, is the th element’s complex vector radiation pattern defined as and is the field’s complex vector plane-wave spectrum defined as In (2) and (3), and are the two orthogonal unit vectors defining the horizontal and vertical polarizations, respectively, and denotes a point on the unit sphere with corresponding to an integration over the unit sphere.
To constrain the scope of our capacity analysis, we consider power-normalized radiation patterns for which That is, is equal to the directivity of the th antenna element, where we neglect impedance mismatches and ohmic loss . 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 . 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, and , of the plane-wave spectrum to be random variables characterized by the following covariances [11–13]: where denotes the expectation operator, is the Dirac delta function, and and 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 Denoting as the total power available to an AUT, we define the vector power-angle spectrum such that
The covariance between the signals received by the th and th antenna elements is defined as  where and are the standard deviations of the signals received by th and th antenna elements, respectively, and is the complex correlation coefficient of the two signals. Combining (1)–(8), the covariance may be written as
Interpreting (the variance of ) as the power received by the th antenna element, we may define the element’s mean effective gain (MEG) as the ratio of received power to available power: [8, 14] 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, , indicating that one-half of the available power is received by the antenna . By use of (8) and (9) with corresponding to the variance of (note that ), we may express in terms of the th antenna’s vector radiation pattern and power-angle spectrum: Finally, by combining (9)–(11), the environment-dependent correlation coefficient may be expressed as
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 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 where 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 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 distance .(ii)The recently developed CTIA MIMO reference antennas (so-called “Good’’, “Nominal’’, and “Bad’’) .(iii)Two colocated and cross-polarized Hertzian dipoles aligned parallel to the - plane and located (a) in free space and (b) at a height above a perfectly electrically conducting (PEC) surface with surface normal .
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.
Table 1 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, for all antennas in the 3D statistically isotropic environment. For the cross-polarized dipoles above a PEC surface, for the 2D statistically isotropic environment, because the gain patterns of the antenna’s elements are zero for angles at and below the horizon . The last column in Table 1 presents the maximum directivity of each antenna’s elements. We will use these values later in Section 5.
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  where are the eigenvalues of the 2-by-2 matrix defined as In (17), is the number of transmit antenna elements, is the noise power density at the receiver, and is the receiver’s covariance matrix given by Using (8)–(12), may be expressed as where the eigenvalues of are and defines the SNR of the corresponding single-input single-output link for an isotropic antenna.
4.1. Single Orientation
Figures 1(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.
Figure 1(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  for ergodic capacity based on a single orientation of the AUT.
4.2. Multiple Orientations
As suggested in , 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 orientations of the AUT are sampled equally (i.e., an equal number of independent capacity realizations are taken at unique orientations), the CDF of the combined set of capacity realizations is equal to the mean of the orientation-dependent capacity CDFs:  where denotes the capacity CDF associated with the th orientation of the AUT, and describes the orientation-combined CDF of the resulting (mixture) distribution.
As Figure 2 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 in a direction such that object’s original -axis is aligned to an axis defined by . Then, the object is rotated by an angle about the -axis. To obtain a uniform sampling of the AUT’s 3D orientation space, we apply the successive orthogonal images technique . We specify a set of angle pairs corresponding to points uniformly distributed on the unit sphere. For each angle pair, for , where is an integer number of rotations per pair as given by and the total number of orientations is . In (22), rounds to the nearest integer. Equation (22) provides the appropriate weighting between and to ensure that the orientations are a uniform sampling of the rotation space (cf., [24, Equation (3.1)]).
A set of uniformly distributed angle pairs were generated by iterative triangularization of an icosahedron. By use of (22), we have whereby a set of uniformly distributed orientations were considered. (Similar results were observed for larger choices of .) The AUT was rotated to each of the orientations, and (11), (12), and (16) were used to calculate the resulting capacity CDFs. For each of the AUTs, Figure 3 compares the 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)).
Figures 4(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%.
5. Sensitivity of Capacity to AUT Orientation in 2D Statistically Isotropic Environments
From Figures 3 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.
Figure 5 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., ). 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 . 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.
Based on Figure 5, 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 dB (i.e., ). 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 () allows for larger variations in the median capacity than the constant cumulative received power case (). 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 as compared to changes in .
Based on Figures 3–6 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., ). 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 . However, based on the antenna configurations considered here, we expect that low directivity is the more important criteria.
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) and not 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 Figures 3 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.
The authors acknowledge the cochair of CTIA’s MIMO OTA Subgroup (MOSG), Scott Prather of AT&T Wireless, and the lead designer of the the CTIA MIMO reference antennas, Istvan Szini of Motorola Mobility, for providing the complex radiation pattern data for the CTIA MIMO reference antennas. This work was supported by a National Research Council Postdoctoral Research Associateship.
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