# Reverberation chamber as a multivariate process: FDTD evaluation of correlation matrix and independent positions.

1. INTRODUCTIONReverberation chambers (RC) are widely used in electromagnetic compatibility (EMC) to carry out immunity and emission tests [1]. Beside these traditional EMC applications, these electrically large cavities are also used for antenna testing [2, 3], to replicate multipath propagation in testing wireless devices [4-8], and also for the material scattering cross section determination [9]. The RC random field properties make them appealing also to study human exposure to electromagnetic field [10,11]. In the last years the RC was used to measure the shielding effectiveness (SE) of enclosure [12-15], and the shielding properties of materials [16, 17], using a nested RC [18]. Particular effort was devoted in studying RC field statistical properties in both ideal [19] and non ideal conditions [20,21], also analyzing what happens approaching chamber boundaries [22]. The analysis of RC field can be accurately done by means of powerful numerical tools able to account for complex geometries such as those based on the FDTD technique [23-26], based on the application of the image theory to account for reflecting boundaries [27], based on the application of hybrid techniques [28], based on the plane-wave integral representation [29-32], and comparing more numerical techniques [33]. These analyses highlighted the importance of the mechanical blades in the stirring process, suggesting a deep investigation of the geometrical parameters that affect stirrer performance [34]. The use of multiple blade systems can enhance stirring efficiency but it depends on the way they are moved [35, 36], and new stirrer shapes that enhance its performance are obtained by FDTD simulations [37]. One of the main parameters used to check stirrer efficiency is the number of statistical independent positions assured during blade rotation. The way to compute this quantity is still under investigation especially because its evaluation in a single chamber point does not give sufficient information about spatial correlation [38].

The common definition of stirrer independence is based on the evaluation of the (auto) correlation function. Even though it brings a simple and effective criterion, this perspective is still incomplete in evaluating whether a member of the cavity ensemble is strictly "independent" or not, with respect to the other realizations. Moreover, in using a RC a certain working volume (WV) is generally adopted to locate the device under test. It would be more reliable to evaluate the stirrer efficiency in terms of independent positions but analyzing the field correlation within the entire WV.

In this paper, we support the view of the reverberation chamber (RC) as a multivariate random field generator [39]. The effect of residual correlation has been proven to have an impact in goodness of fit tests [20]. Here, we discuss an alternative way of evaluating the number of independent positions of mechanical mode-stirrers. In particular, the proposed approach is based on the calculation of the correlation matrix for the magnitude of the electric field computed by FDTD code in a grid of [N.sub.w] spatial points, selected among an arbitrary volume of the chamber. Another recent application of the correlation matrix to evaluated stirrer efficiency was applied to RC measurements based on a single point, adopting different frequency points to populate the matrix and investigating different chamber loading conditions [36].

2. REVERBERATION CHAMBER SETUP

Figure 1 shows the RC geometry used for both simulations and measurements: dimensions are 6.00 x 4.00 x 2.50 [m.sup.3]. The fundamental mode resonance frequency is [f.sub.0] = 45.04 MHz.

Two stirrers were placed along the y-axis (paddle shape) and the z-axis (Z-folded shape), and the investigated volume is between the two stirrers 0.5 m far from all metallic objects. The corner C of the investigated volume has the coordinates (2.5, 1.0, 0.5) (m), and its dimension are 1.0 x 2.0 x 1.5 [m.sup.3]. The vertical stirrer has a Z-folded shape, with a width of 1.2 m and a height of 2.4 m. The horizontal stirrer originally consisted of four 1 m x 0.5 m separated panels subsequently joined by means of aluminum sheets to improve its performance. The mechanical engine, associated to each stirrer axis, allows to move them separately and in both stirring and stepped modes with a resolution of 1[degrees]. The transmitting and receiving antennas are log-periodic operating between 300 MHz and 5 GHz (Schwarzbeck model USLP 9143). A VNA (Agilent E5071) was used to acquire the parameter between the two antennas setting the maximum frequency resolution: 1601 points in each investigated sub-range. The VNA was calibrated adopting the standard procedure for transmission measurements, in order to account for all connecting cables: residual uncertainty is less than [+ or -] 0.2dB and [+ or -] 1[degrees]. The data were acquired in tuned mode, moving the stirrers together, adopting synchronous and interleaved mode. When horizontal and vertical stirrers operate synchronously, they move with a step of 1[degrees] (0[degrees] - 0[degrees], 1[degrees] - 1[degrees], 2[degrees] - 2[degrees], ..., 359[degrees] - 359[degrees]) for a total of 360 positions. When they operate in interleaved mode, they move with a step of 19[degrees] each (0[degrees] - 0[degrees], 0[degrees] - 19[degrees], 0[degrees] - 38[degrees], ..., 19[degrees] - 0[degrees], 19[degrees] - 19[degrees], 19[degrees] - 38[degrees], ... 38[degrees] - 0[degrees], 38[degrees] - 19[degrees], 38[degrees] - 38[degrees], ... 342[degrees] - 0[degrees], 342[degrees] - 19[degrees], 342[degrees] - 38[degrees], ... 342[degrees] - 304[degrees], 342[degrees] - 323[degrees], 342[degrees] - 342[degrees]), for a total of 361 positions [35].

3. FDTD FORMULATION

When a pulse is excited inside an RC, it will disappear only after a long time due to the high quality factor. The losses must be accounted in the simulations because they save computer resources and produce accurate results.

Our FDTD code has been extended and optimized to simulate the whole RC. The influence of wall losses on the equilibrium conditions was investigated for a the whole RC [26]. The same reference shows that it is possible to introduce artificial losses in the RC volume by neglecting the losses on the wall slightly affecting the field distributions. This approximation saves computer resources for a single FDTD simulation, in particular when a parallel computer is used. Moreover, when the statistical properties of the RC are simulated, the simulations must be repeated for each position of the stirrer. As consequence, the analysis of an RC when the stirrer is rotating requires multiple runs of FDTD code changing the angle of the stirrer. Each run is independent of the others, making it a "embarrassingly parallel" problem. Each FDTD simulation runs separately in a CPU of a high performance parallel computer, even without any parallelism of the code.

Usually, simulations of RC are helpful to investigate the frequencies up to about five time the lowest usable frequency (LUF). For the other frequencies, the required performance can be obtained with low design efforts. Under this assumption, the single simulation does not require a big computational effort, and can easily run on a single CPU of the parallel computer, because of the FDTD grid cell size depends on the working frequency.

Our RC was sampled in 201 x 134 x 84 cubic cells with side 30 mm, the time step is [DELTA]t = 50 ps, and the number of iteration is 206748. According to [26], we simulated ideal walls and we introduced some losses in the air. Comparing the measured and simulated decay factor of the RC with aluminum walls the air conductivity of [sigma] = [10.sup.-5] S/m gives the same results. Under this choice, the code is a standard FDTD code for any cell except for the two cells where the transmitting and receiving logperiodic antennas are fed. For those two points, an auxiliary line is connected to the antenna as described in [40]. The computer used for the simulations was an IBM P575 Power 6, with 5376 cores and 168 nodes. Each simulation for a stirrer position requires less than 500 MB and less than 20 hours of CPU.

The numerical simulation uncertainty was evaluated using a power balance [26]. In the present case it is less than 3% because the ratio between the lowest wavelength and the cell size is about 14.

4. CORRELATION BY MULTIVARIATE ANALYSIS

We work out a method for evaluating the number of independent positions based on the statistical analysis of multiple fluctuating variables, namely multivariate analysis, collected inside the WV of a reverberation chamber. The simulation of the chamber of Fig. 1 allows for tracing the random field evolution inside the overall WV. In particular, we established a sampling grid of [N.sub.w] points, each adjacent pair being equally-spaced of [DELTA]x = 0.167 m, [DELTA]y = 0.333 m, [DELTA]z = 0.250 m, in the WV of Fig. 1. For any position of the stirrer and in the whole frequency range we have calculated, and stored, the three Cartesian components of the electromagnetic field in the [N.sub.w] = 7 x 7 x 7 points of the WV.

5. MULTIDIMENSIONAL REVERBERATION

The mode-stirred chamber can be now characterized as a multidimensional random variable. Focusing on the total electric field, we define multidimensional reverberation E a collection

{[[E.bar].sup.([tau])] : [tau] [member of] T} (1)

of R-valued multidimensional random variables [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], in the probability space {[OMEGA], [E.bar], P}, where [tau] is the continuous stirring time instant, the local field [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is picked up at the point ([m.sub.i][DELTA]x, [m.sub.j][DELTA]y, [m.sub.k][DELTA]z) of the sampling grid made of [N.sub.w] (spatial) sampling points, for each stir state [[tau].sub.i] = I[DELTA][theta], i = 1, ..., [N.sub.s], with [DELTA][theta] angular stirrer step, [N.sub.s] total number of stirrer positions considered in the analysis, and T total stirring length. The correlation is evaluated using [N.sub.p] field values. In the case of the total electric field it is [N.sub.p] = [N.sub.w], whereas for the three separate Cartesian components it is [N.sub.p] = 3[N.sub.w].

The number of independent positions are evaluated dividing the number of elements of the correlation matrix, that is [N.sup.2.sub.s], by the number of elements greater than the cutoff value of (11).

In this fashion, random variables and observations are organized as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where thus the columns are random vectors made of fields at grid points, and the rows are (chamber) realizations made of stir states.

With reference to Section 6, the correlation can be evaluated in the frequency domain using two sets of the fields simulated for [N.sub.s] = 360 stirrer locations.

By introducing a spatial grid, we implicitly account for the residual correlation between spatial points in calculating the correlation between stir states. As a matter of fact, the coarser the grid, and the higher the chamber operation frequency, the smaller the residual correlation. However, it is to be remarked that, because of the damped behavior of the spatial correlation function, its value is likely small and different from zero for most of the adjacent and nonadjacent points, but for those distances and excitation frequencies for which n[lambda]/2. Therefore, we expect almost all small but not-null correlations for pair of points in the sampling lattice. It is thus quite natural to invoke the concept of a correlation matrix for analyzing the fluctuations in a multivariate stirring. This develops well on the use of the autocorrelation function [1], and of a two-point correlation function [38], widely used in RC analysis so far.

At low frequencies, many elements in such structured correlation matrix will be well above the uncorrelation limit (11), thus reducing its total number of eigenvalues, i.e., of principal components.

5.1. Correlation Matrix

The multidimensional reverberation can be efficiently used to overcome the drawbacks of the local reverberation point of view, such as the physical incompleteness of the single point autocorrelation in detecting unstirred components by goodness-of-fit tests. Here, the reasoning is driven by the multivariate analysis to explore the influence of a residual spatial correlation on the determination of the number of independent positions ([N.sub.ind]).

Among the other possible ways of estimating [N.sub.ind], we propose a procedure based on the correlation matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], calculated for all the possible pair combinations in the sampling grid, within {[[tau].sub.i]} stirrer positions. In particular, for the single pair of spatial points, we use the correlation coefficient between two random vectors [[e.bar].sub.j] and [[e.bar].sub.k], modeling two random (Cartesian, total) fields collected in the FDTD sampling grid at two different stirrer positions j, and k. The Pearson definition of the correlation coefficient is usually employed in multivariate statistics, viz.,

[[rho].sub.jk] = Cov([[e.bar].sup.(j)], [[e.bar].sup.(k)])/[square root of Var ([[e.bar].sup.(j)]) Var ([[e.bar].sup.(k)]] = [[sigma].sub.jk]/[square root of [[sigma].sub.jj][[sigma].sub.kk]], (3)

where Cov(*, *) stands for the covariance, Var(*) stands for the individual field variance, and [[sigma].sub.jk] has the meaning of an entry of the dispersion of the variance/covariance matrix of the field

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

with [(*).sup.T] representing the matrix transpose, [??] defined as in (2), and [N.sub.s] the number of stirrer positions being considered in the evaluation of independence, thus obtained discretizing the time-domain stirring process as depicted in (2). It is worth remarking that it is expected to have many dependent pairs within [N.sub.s], so typically [N.sub.ind] [less than or equal to] [N.sub.s]. For the sake of completeness, we remind that the quantity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called the total variance, and the determinant of [??], denoted by [absolute value of [??]], is referred as the generalized variance. Accordingly, we define the correlation matrix of [??] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

At this point, the multivariate point of view comes into play: having collected a large amount of data points from FDTD simulations, we can estimate the spatial average [[??].sub.e]. This is actually calculated among the chamber volume covered by the FDTD sampling lattice, thus tracing for the field fluctuation through a reverberation sub-volume established by the mode-stirrer effectiveness.

5.2. Correlation Coefficient among Space Lattices

The multivariate perspective we adopted throughout the paper gives a unified picture of the chamber statistics. Focusing on the Pearson correlation coefficient, we remark it is subject to statistical uncertainty and fluctuations driven by statistical moments of underlying data. This effect can be readily understood and quantified thanks to the Fisher theory on correlation distribution [41]. In the late '40, Fisher has been able to derive closed-form solutions for the distribution of r = [[rho].sub.jk] by exploiting a geometrical perspective of the Pearson expression. Its expression, assuming as [[e.bar].sup.(j)] the vector collecting [N.sub.p] realizations of the j-th i.i.d. zero mean Gaussian variable, among the [N.sub.s] ones, reads [41]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where M is the number of uncorrelated samples, and [F.sub.G] is the Gaussian hypergeometric function.

The correlation distribution has been originally used in evaluating the significance of the autocorrelation values in local RC measurements [42]. Actually, once specified the uncorrelation bound [r.bar], e.g., [r.bar] = 0, or [r.bar] = 1/p, the Fisher distribution (7) indicates that even highly deviated value of correlation can occur with finite probability thus calling for a more precise, and yet physical based, definition of independence given a certain amount of uncorrelation.

Beside this, we should also point out that in a multivariate perspective based on lattices, [[rho].sub.jk] depends on the autocorrelation between (spatial) points used to estimate the second-order statistics. The present study constitutes a starting point towards the definition of independency accounting for all the effects we mentioned so far, i.e., statistical inhomogeneity and anisotropy captured by the multivariate (lattice) perception for each stirrer location, spatial autocorrelation introduced by the sampling grid, and fluctuation of the Pearson correlation due to its pure mathematical definition. However, the results we will show in the next section account for all of this through the numerical estimation of the uncorrelated positions.

6. EVALUATION OF STIRRER UNCORRELATED POSITIONS

A result of the measurements and of the simulations is the [S.sub.21] parameter. Over a stirrer rotation, N received sampled data are acquired and stored in an array a. It is periodic with period N ([a.sub.i] = [a.sub.i+kN] with k [member of] Z). Following the standard [1], the i-value of the autocorrelation array is computed as

[[rho].sub.i] = [N-1.summation over (j=0)]([a.sub.j] - <a>)([a.sub.j+i] - <a>)/(N - 1)[[sigma].sub.2a], (8)

where 0 [less than or equal to] i [less than or equal to] N - 1, and

[[sigma].sub.2a] = 1/N - 1 [N-1.summation over (j=0)][([a.sub.j] - <a>).sup.2] (9)

is the variance of the array a.

We introduce a new criterion for estimating the number of independent stirrer configurations. This is useful to evaluate the effectiveness of different stirrer geometries against working frequencies RC is operated at. A rather natural extension of the normative procedure, based on the autocorrelation function [1], might rely on the correlation matrix (6). One is indeed tempted to say that two stirrer configurations are uncorrelated if and only if [[rho].sub.jk] < 1/e. The number of elements satisfying this condition can thus be treated as the number of uncorrelated positions [N.sub.ind] produced by the stirring procedure. Obviously, we should account for the symmetry properties of (6), thus excluding redundant elements yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where # [*] is the counting operator, [??] is the identity matrix of dimension [N.sub.s], and r is the cutoff value set following the new release of standard [1] to

r = [1/e] [1 - 7.22/[([N.sup.2.sub.s]).sup.0.64]]. (11)

When two stirrers are used in an RC, we follow the procedure described in [35].

7. RESULTS

In this section the number of independent positions computed by multivariate analysis correlation (MAC) is compared to that obtained by the classical circular correlation (CC) adopted for both numerical and experimental data [1]. The comparison is done for both stirrer moving mo des. Moreover, the effects on the MAC results varying the total number of the spatial points and their relative distance are also investigated.

The results obtained by the MAC method are shown in Figs. 2 and 3 for synchronous and interleaved mode respectively. Of course the interleaved mode ensures a higher number of independent positions, that reaches the maximum explored value of 360 around 450 MHz. The two figures also reports what obtained applying the CC method, both numerically and experimentally. The traditional CC method overestimates the independent position number and this is the main reason why we introduce the MAC method. This behavior of the CC results was also observed in the past, for the same chamber and the same stirrer configurations, analyzing the ratio between the maximum and the ensemble averaged field inside the chamber whose value is related, in a statistical sense, to the independent position number [35]. Similar results were observed in [36] with a different work methodology and higher frequency range.

Figure 4 shows the effect produced by the number of field points used to build up the correlation matrix. In particular, the same total volume is considered, but the grid varies from a 7 x 7 x 7 = 343 to a 3 x 3 x 3 = 27 points. When the three Cartesian field components are taken for each point, the correlation is computed over a set three times larger.

The figure refers to the interleaved mode case, because the higher independent position number allows us to better highlight the effect of each variation in the spatial point number used for the correlation matrix. The MAC method exhibits an asymptotic behavior when the field point number increases. More precisely, the 343 values obtaining adopting the total electric field values already give good results that can be improved very little if the three separate rectangular field components are employed, so reaching 1029 values. On the contrary, adopting a 4 x 4 x 4 grid, that gives only 64 values for the total electric field, the MAC results is visible bad. If we consider all the field rectangular components for this last grid, reaching 192 values, the computed independent positions are similar to those obtained by the previous densest grid. This direct link to the total field samples used is further confirmed if we consider a less dense grid, 3 x 3 x 3, but with all field components for a total of 81 values: the result is better than the 4 x 4 x 4 case with only the total field. The previous analysis was carried out assuming the same volume for the grid, so increasing the distance between each spatial point when the grid was less dense. Which is the optimal distance to be taken? The issue of the distance capable to assure spatial uncorrelation was intensively addressed in literature [38, 43], leading to suggest a distance of about one half of wavelength. To investigate this aspect for the MAC method let's consider the results reported in Fig. 5. It compares the MAC computed independent positions for the same spatial point number 4 x 4 x 4 previously shown in Fig. 4 (large volume), and for a lower point distance (reduced volume).

It can be noted that the reduced volume gives the same results of the larger one for frequencies above 600 MHz. The explanation can be found in analyzing the point distance in terms of wavelength at the operating frequencies. More precisely, Fig. 6 reports this point distance normalized to half wavelength for both grids. Because the WV is not a cube, the distance between each point is not uniform along the three Cartesian coordinates. Therefore, Fig. 6 considers an equivalent volume cube side (d = [cube root of [DELTA]x[DELTA]y[DELTA]z]), because it represents a sort of averaged distance between two adjacent points of the grid. It can be clearly seen that this averaged distance becomes compared to a half of wavelength around 600 MHz, where the results obtained by the reduced volume overlap to those of the larger one.

8. CONCLUSIONS

The uncorrelated positions of two stirrers have been computed by means of the correlations method of multiple spatial points in an RC. Two kinds of moving mode for the two stirrers were investigated: synchronous and interleaved. In both cases we showed that the traditional way based on the CC overestimates the uncorrelated position number. The larger is the volume adopted to calculate the correlation matrix, the more reliable is the obtained values. Moreover, the importance of the correlation distance between two adjacent points of the spatial grid has been investigated.

The proposed method is based on the knowledge of field values in a grid of points inside the chamber WV. This can be time consuming for investigation of an existing chamber. On the other hand, it can be easily adopted during the design by numerical simulation of a new chamber and new stirrers.

ACKNOWLEDGMENT

The computational resources were granted by CINECA (Italian supercomputing center) under the project ISCRA HP10BYQKHN.

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G. Gradoni (1), *, V. Mariani Primiani (2), and F. Moglie (2)

(1) Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA

(2) DII, Universita Politecnica delle Marche, Via Brecce Bianche, 12, Ancona 60131, Italy

Received 18 September 2012, Accepted 15 October 2012, Scheduled 19 October 2012

* Corresponding author: Gabriele Gradoni (ggradoni@umd.edu).

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Author: | Gradoni, G.; Primiani, V. Mariani; Moglie, F. |
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Publication: | Progress In Electromagnetics Research |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jan 1, 2013 |

Words: | 5290 |

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