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A reverberation chamber is a convenient tool for over-the-air testing of MIMO devices in isotropic environments. Isotropy is typically achieved in the chamber through the use of a mode stirrer and a turntable on which the device under test (DUT) rides. The quality of the isotropic environment depends on the number of plane waves produced by the chamber and on their spatial distribution. This paper investigates how the required sampling rate for the DUT pattern is related to the plane-wave density threshold in the isotropic environment required to accurately compute antenna correlations. Once the plane-wave density is above the threshold, the antenna correlation obtained through isotropic experiments agrees with the antenna correlation obtained from the classical definition, as has been proven theoretically. This fact is verified for the good, nominal, and bad reference antennas produced by CTIA. MIMO channel capacity simulations are performed with a standard base station model and the DUT placed in a single-tap plane-wave reverberation chamber model. The capacity curves obtained with the good, nominal, and bad reference antennas are clearly distinguishable.

The reverberation chamber [

In the present paper we investigate the use of reverberation chambers for over-the-air testing of MIMO devices. Both antenna correlation (a quantity that is critically important for MIMO system performance) and MIMO capacity will be simulated in a plane-wave reverberation chamber model [

To ensure accurate and reliable over-the-air test results, the test system must produce an accurate test environment in the entire physical region that contains the DUT. For example, in conventional 2D anechoic tests, the number of antennas in the ring surrounding the DUT must be large enough to accurately reproduce the desired channel conditions in the region occupied by the DUT. Similarly, the reverberation chamber must be large enough to supply enough plane waves to achieve an isotropic environment in the region occupied by the DUT.

By expressing the DUT pattern in terms of a spherical expansion (with a recently derived formula for truncation limit), we determine how closely the DUT pattern must be sampled to properly capture its variation. Using this sampling rate, we obtain rules that determine both the required number of antennas in the anechoic test and the required plane-wave density in the reverberation chamber test. As a byproduct of this investigation, we present accurate Fourier expansions of the DUT pattern that have been used in spherical near-field scanning for many years but appear to be relatively unknown in wireless communications.

The paper is organized as follows. Section

In this section we introduce the plane-wave receiving characteristic and far-field pattern of an arbitrary DUT-mounted antenna. (The term “pattern’’ will be used to refer to both the plane-wave receiving characteristic and to the far-field antenna pattern). A spherical expansion determines the spatial sampling rate required to “capture’’ the pattern of the antenna and provides a Fourier series expansion useful for computing any quantity involving the pattern. The standard spherical coordinates

Spherical coordinates.

The DUT with a mounted antenna is shown in Figure

The DUT with mounted antenna contained in the minimum sphere of radius

The plane-wave receiving characteristic is defined as follows. Assume that the incident plane wave

If the DUT-mounted antenna satisfies reciprocity, its plane-wave receiving characteristic can be expressed in terms of its normalized far-field pattern

The electric far field of the DUT, when it is fed by an input voltage-amplitude

These statements fully define the plane-wave receiving characteristic for any propagating plane wave that may illuminate the DUT. If the source of the incident field is close to the DUT, one must also specify the plane-wave receiving characteristic for evanescent plane waves [

Using (

The transverse vector-wave functions can be expressed in terms of the spherical harmonic

The expressions for the transverse vector-wave functions

Since

Assume that

The plane-wave receiving characteristic

The plane-wave receiving characteristic

Integrals of the form

In contrast, brute-force approximations of the form (with the original sampling rate retained)

In this section we describe the concept of antenna correlation in an isotropic environment like the one observed in a reverberation chamber. However, first we state the classical definition of antenna correlation in terms of the plane-wave receiving characteristics introduced in Section

Consider two receiving antennas, possibly mounted on the same DUT, with plane-wave receiving characteristics

A general specification of the isotropic environment can be found in Hill’s book [

A selection of points

Assume that two plane waves are incoming in each of the directions

It was shown by De Doncker and Meys [

Consider two

Two

We compute the correlation from (

Unlike the points

Figure

The error

For both

To accurately reproduce the isotropic field conditions in a reverberation chamber, one must choose a chamber size large enough to ensure enough plane-wave directions of incidence for a given DUT size. We also note that these numerical simulations validate the general theorem by De Doncker and Meys [

Before leaving this section we investigate the sampling required for a 2D configuration where the correlation is based on incident fields from a small region of the unit sphere. This type of model will be used in Section

The two

Two

We also compute an approximate correlation based on a fixed set of equally spaced directions of incidence illustrated by the ring in Figure

Figure

The error

To expedite the baseline between laboratories participants of CTIA LTE round robin, a set of MIMO 2 × 2 reference antennas has been developed [

We compute the isotropic correlation with 180 plane-wave directions of propagation corresponding to an isotropic sampling of

Correlations computed from the classical formula (

Good antenna | Nominal antenna | Bad antenna | |
---|---|---|---|

Classical formula ( | |||

Isotropic simulation formula ( |

These results have also been verified experientially in a reverberation chamber at NIST [

A reverberation chamber provides a rich scattering environment that is ideal for over-the-air testing of wireless devices. The chamber typically contains a number of wall-mounted transmitting antennas, a mechanical mode stirrer, and a turntable on which the DUT is placed; see Figure

Rectangular reverberation chamber with dimensions

Let the rectangular chamber have the dimensions

There is one additional term (called an irrotational mode) that goes with the mode in (

We associate a frequency

The magnitude of the excitation factor

Normalized magnitude of the excitation factor as function of mode frequency for a 750 MHz driving signal and a 30 ns RMS delay. Only modes in a “3 dB’’ region near the peak get effectively excited.

Let us now show the actual plane-wave directions of incidence for two reverberation chambers with 30 ns RMS delay that are driven by a 750 MHz source. One is electrically large (

Plane-wave directions of incidence for modes in a 50 MHz band around 750 MHz in a large chamber with dimensions

As the stirrer in the reverberation chamber rotates, the amplitudes and phases of the plane waves change to produce an isotropic environment as discussed in the previous section. In addition to the stirrer, the chamber contains a turntable on which the DUT rides. As the turntable rotates, the DUT sees the plane waves from different angles, effectively creating additional directions of incidence. Figure

Figure

Plane-wave directions of incidence for modes in a 50 MHz band around 750 MHz in a small chamber with dimensions

Let us now investigate how these reverberation chambers perform when evaluating the correlation between the two

The error

The error

The error

One of the most important statistical properties of the field in the chamber is the degree to which it is isotropic, that is, how evenly distributed are the directions of propagation and polarizations of the incoming plane waves at the location of the DUT. With the isotropy test developed by the international standards committee (ICE) [

Three orthogonal components of the electric field recorded at the location of the DUT are required to compute the field anisotropy coefficients [

When the DUT is placed near the edge of the turntable in Figure

Figures

Histograms of anisotropy coefficients for large chamber with dimensions

Histograms of anisotropy coefficients for small chamber with dimensions

We have now described the plane-wave environment in a reverberation chamber using the mode expansion of the dyadic Green’s function for the rectangular box. We have seen how the plane-wave directions of incidence as seen from the DUT depend on the dimensions of the chamber and on whether or not a turntable is active. Through numerical simulations, we evaluated the accuracy of correlation experiments in the chamber. It would be nice to have a theory that explicitly determined the accuracy of the chamber as a function of chamber dimension and chamber loading (quality factor

In this section we perform MIMO capacity simulations with a two-antenna DUT receiver in an isotropic environment. The transmitter is a standard two-antenna base station. We employ the good, nominal, and bad reference antennas described in Section

A schematic of the channel model is shown in Figure

Channel model employing an isotropic environment. Each point on the base station pattern is randomly paired with a plane-wave propagation direction illuminating the DUT. In addition, each “pairing’’ is supplied by a random phase. The multiple states of the model are obtained by changing the pairing and the random phases.

In practice, the two-antenna 2D Laplacian base station output would be fed to the reverberation chamber through two or more wall antennas. The directions of propagation from the base station are thus distributed randomly into plane waves in the chamber, and the Laplacian distribution is not preserved. In other words, the chamber does not reproduce the Laplacian distribution. However, the correlation imposed on the two information streams by the base station is preserved. Further, one often feeds a reverberation chamber from a channel emulator that is programmed to produce advanced spatial channel models, which can include both Doppler spectra, long time delays (much longer than the one produced by the chamber alone), and specified directions of incidence. When such channel models are fed to the reverberation chamber, the channel model is said to be evaluated isotropically. In such situations, the specified directions of propagation dictated by the channel model are not preserved. However, if the emulator and reverberation chamber are adjusted properly, the time delays and Doppler spectra of the channel model are preserved in the chamber. The use of advanced channel models adds a lot of flexibility to the reverberation chamber as an over-the-air test tool.

Next we select a set of evenly distributed points

We now have

Figure

Capacity curves as functions of SNR for the “good,” “nominal,” and “bad” reference antennas in an isotropic environment. To achieve a capacity of 7 bps/Hz, the three reference antennas require very different SNR values: the bad reference antenna requires an SNR of 19 dB, whereas the good reference antenna requires only an SNR of 13 dB. The difference in SNR between good and bad reference antennas is in this case 6 dB. Similarly, to achieve a capacity of 12.5 bps/Hz, the difference in SNR between good and bad reference antennas is 7 dB.

We investigated the use of reverberation chambers for over-the-air testing of MIMO devices by examining antenna correlation and throughput in isotropic environments. A truncated spherical-wave expansion was used to derive sampling theorems and the Fourier expansions for the pattern of an arbitrary DUT. The required sampling rate of the pattern depends on the frequency, the physical size of the entire DUT (not just its antenna), and the relative location of the DUT to the spherical coordinate system.

Through numerical investigations involving Hertzian dipoles, it was shown how the sampling rate for the pattern determines the plane-wave density required in the isotropic environment to obtain accurate values for the correlation between antennas. It was also demonstrated that antenna correlation in the isotropic environment is equivalent to the classical definition of antenna correlation, as was proven theoretically by De Doncker and Meys [

Using the dyadic Green’s function for the rectangular box, we computed the plane-wave distribution for realistic reverberation chambers, which were in turn used in simulations of antenna correlations and anisotropy coefficients. No general theory that explicitly determined the accuracy of the chamber as a function of chamber dimension and chamber loading was found. Instead we explained how one can determine accuracy estimates through simulations.

We performed MIMO channel capacity simulations using a standard base station model and the DUT (employing the CTIA reference antennas) placed in a single-tap plane-wave reverberation chamber model. The capacity curves obtained with the good, nominal, and bad reference antennas were clearly distinguishable, as would be expected given the vast difference between the correlations of these antennas; see Table

Hence, we conclude that isotropic tests performed in a reverberation chamber can distinguish between DUTs that employ the different CTIA reference antennas. It would be interesting to perform link-level simulations with multitap isotropic channel models to further investigate this over-the-air test method.

The author would like to thank R. J. Pirkl and K. A. Remley of the NIST for numerous helpful discussions. I. Szini of Motorola is thanked for providing the patterns for the reference antennas.