International Journal of Antennas and Propagation

Volume 2019, Article ID 8171978, 7 pages

https://doi.org/10.1155/2019/8171978

## Experimental Verification of the Weibull Distribution in a Reverberation Chamber

Correspondence should be addressed to Qian Xu; moc.liamxof@uxme

Received 24 May 2019; Revised 28 June 2019; Accepted 19 July 2019; Published 7 August 2019

Academic Editor: Mauro Parise

Copyright © 2019 Jianhua Xu et al. 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.

#### Abstract

A Weibull distribution has been proposed for the probability density function (PDF) of the magnitude of the E-field in a reverberation chamber (RC). However, the Weibull distribution has two parameters, and if the parameters are position sensitive, the use of the Weibull distribution could be very limited. We investigate the sensitivity of the estimated parameters of the Weibull distribution in this study; the measurement results show that the parameters of the Weibull distribution depend on the positions of the antenna (or device under test), but not sensitive, and the statistical behavior of the parameters can be quantified. This means that the model of the Weibull distribution has a wider usable frequency range than that of the Rayleigh distribution and the statistical variation of the parameters needs to be considered.

#### 1. Introduction

Reverberation chambers (RCs) have been widely used in the electromagnetic compatibility (EMC) test and over-the-air (OTA) measurement [1–6]; the electrical field inside an RC in the working volume is statistically uniform, and this provides the foundation of the applications of the RC [2]. However, the RC has to work in the overmoded region to satisfy the statistical distribution model (Rayleigh distribution). It is well known that when the resonant modes (or incident waves) increase, the E-field distribution approaches the Rayleigh distribution [2, 3]. When the RC works in the undermoded region, the Weibull distribution is more applicable than the Rayleigh distribution [3].

The Weibull distribution has been verified at a specific antenna position [7–11]; the measurement results show that the Weibull distribution parameters for different polarized E-fields seem quite stable in a very wide frequency range [9], but the position sensitivity of the parameters has not been studied. If the parameters of the Weibull distribution are very sensitive to the positions in an RC, they will limit the application of the model and the behavior of the RC could be very difficult to characterize.

To investigate this problem, we measure the *S*-parameters in an RC at different positions and use the maximum likelihood estimate (MLE) method to obtain the parameters of the Weibull distribution. Section 2 provides the theory and the setup of the measurement, Section 3 analyzes the measurement results, and Section 4 summarizes the paper.

#### 2. Theory and Measurement Setup

The Weibull distribution has been introduced to the RC in [3, 7–11], and its probability density function (PDF) iswhere is called the scale parameter and is the shape parameter. Note that when and , the Weibull distribution reduces to the normalized Rayleigh distribution:where is the random variable normalized to the mean value; when and , the Weibull distribution reduces to the normalized exponential distribution [8]:

Thus, the Weibull distribution is more general than the Rayleigh distribution (or exponential distribution). By applying the MLE method, the estimated parameters can be found as the solution of the equation system [3]:where the measured samples are and is the sample number.

The random variable in (1) can be the measured or [8]. In this paper, we use the measured which corresponds to the E-field magnitude distribution in an RC. The measurement setup is illustrated in Figure 1 [12]. Figure 1(a) shows the schematic plot of the measurement; the computer controls the rotation of the stirrer, the *XY* platform, and the frequency sweep of the vector network analyzer (VNA). For each position of antenna 2 (a probe antenna), *S*-parameters (1601 points, 10 MHz–3 GHz) of 360 stirrer positions were recorded [12]. The measurement scenario is illustrated in Figure 1(b), and the measured sample points are plotted in Figure 1(c). The dimensions of the RC are 0.8 m × 1.2 m × 1.2 m, and the lowest usable frequency is about 1 GHz [12]. Sample points at three different heights were measured: 30 cm, 62 cm, and 98 cm; this makes 3 × 36 = 108 positions of antenna 2. For each position, three polarizations were also measured. Typical measured scattering parameters are illustrated in Figure 1(d) with three polarizations. Because the radiation efficiency of transmitting and receiving antennas and the cavity transfer function are all small at low frequencies, show small values below 100 MHz. We have checked the variation of the insertion loss caused by the movement of the cable, which is lower than 0.2 dB at 3 GHz. Thus, the movement of the cable does not affect our results.