IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 50, NO. 2, MAY2008 227 Estimating the Effective Sample Size to Select Independent Measurements in a Reverberation Chamber Christophe Lemoine, Philippe Besnier, Member, IEEE, and M’hamed Drissi, Senior Member, IEEE Abstract—In reverberation chambers (RCs), measurements are usually performed by changing the boundary conditions using a mode stirrer. The major difficulty is to select uncorrelated samples in order to make a statistical analysis of the data. Furthermore, the knowledge of the number of independent samples is of cru- cial importance to assess the measurement accuracy. To evaluate whether measured data are independent, the conventional method compares the autocorrelation function (ACF) with the critical value 0.37. However, this criterion is generally not appropriate because the ACF probability density function (pdf) depends strongly on the sample size. For a measurement series of length N , the ef- fective sample size (ESS) is defined as the number N ′ <N of independent samples, which would provide the same information as the N -size sample. This paper aims to provide a new method based on autoregressive (AR) models and the central limit theorem (CLT) in the case of dependent data, for estimating the ESS. The proposed method is easy to implement since it requires only the knowledge of simple statistical parameters. Moreover, it provides useful guidelines to assess the maximum number of independent samples available with the mode stirrer. Experimental results are in good agreement with the theoretical models, either for the electric field or the received power. Index Terms—Autocorrelation function (ACF), autoregressive models, effective sample size, independent samples, reverberation chamber (RC). I. INTRODUCTION T HE MODE-STIRRED chamber is an alternative tool for various electromagnetic compatibility (EMC) and non- EMC applications. EMC standards [1], [2] are still in evolution for calibration methods, and for immunity and emission test- ing. Also, methods of assessing measurement uncertainty for antenna characterization are of increasing interest [3]. Recent investigations [4], [5] have improved the characterization of dis- tribution functions of measurements in reverberation chambers (RCs). Reverberation chambers are often considered as a random field generator in the test volume. The independence of samples is of a great importance to correctly quantify the uncertainty of a test performed in the cavity. The IEC 61000-4-21 standard [2] emphasizes that the number of independent samples must be known in order to apply statistics to data obtained from a rever- beration chamber. To evaluate that a stirrer provides independent Manuscript received May 22, 2007; revised September 17, 2007. This work was supported by the R´ egion Bretagne. The authors are with the Institute of Electronics and Telecommunica- tions of Rennes (IETR), Institut National des Sciences Appliqu´ ees (INSA) of Rennes, Rennes 35043, France (e-mail: christophe.lemoine@insa-rennes.fr; philippe.besnier@insa-rennes.fr; mhamed.drissi@insa-rennes.fr). Digital Object Identifier 10.1109/TEMC.2008.919037 field conditions, the autocorrelation function (ACF) is usually calculated for the chosen step angle of the stirrer. Statistical tests [6], [7] can also be used to estimate the true value of the correlation coefficient. In both cases, independence is deduced from the comparison of the experimental ACF with a critical value ρ 0 . In particular, the first-order autocorrelation function, denoted r in the paper, corresponds with a shift equal to one step of the stirrer. It is calculated as follows: r = covar(x, y)  var(x)  var(y) (1) where y is the same collection of data as x selected over one stirrer rotation (360 ◦ ) but shifted by one sample. The notations “covar” and “var” are for the covariance and the variance oper- ators, respectively. The normative part of [2] assumes statistically independent boundary conditions between two successive stirrer positions when the first-order ACF is less than 1/e ≈ 0.37. However, the distribution of the first-order ACF is a function of the sample size N . Lund´ en et al. demonstrate [6] that this criterion is ap- propriate only if N = 30 or N = 50 with, respectively, a 5% or 1% level of significance. Moreover, the authors show that if measurements using N> 100 yield r =0.37, the probabil- ity is very high that data are correlated. The informative part of [2] deals with the determination of independent tuner posi- tions and the point (N = 450,ρ 0 =0.37) is taken as a reference. Nonetheless, if a set of N = 450 samples yields r =0.37, the probability is 10 −16 that the true value is ρ =0. In [8] and [9], other methods for estimating the number of independent samples are proposed, based on either the stirrer geometry or a sample difference method. However, both methods refer to the auto- correlation function with the threshold 0.37. In [10], alternative models are proposed based on the Q-factor of the chamber for mechanical stirring and frequency stirring. Experimental results are still compared with theoretical models based on ρ 0 =0.37. Thus, to our knowledge, no conclusion can be made in terms of accuracy, and calculating the number of independent samples remains a great challenge. In the case of an RC, the measurements of field or received power can be viewed as a time-series process. If a time series of length N is autocorrelated, the number of independent ob- servations is fewer than N . Essentially, the series is not random in time, and the information in each observation is not totally separate from the information in other observations. The amount of information which is furnished varies inversely with the expected first-order autocorrelation function ρ. If ρ =0, 0018-9375/$25.00 © 2008 IEEE