Carlo F. M. Carobbi

The Absolute Maximum of the Likelihood Function of the Rice Distribution: Existence and Uniqueness

The Rice probability density function has received considerable attention for its various important technical and scientific applications. One of the more attractive techniques for extracting the distribution parameters, and possibly the most frequently applied, relies on the maximization of the likelihood function for a given set of experimentally determined samples, and many applications are documented in the literature. This paper offers a mathematical analysis which demonstrates that, subject to conditions universally verified in physical systems, an absolute maximum exists, and it is the unique point internal to the domain of existence which zeroes the gradient of the likelihood function. In all previous results, the presence of additional maxima, which are possibly larger than the one that had numerically been found, could not be excluded. We can incidentally state that this paper demonstrates that all previous results based on numerically finding a maximum indeed corresponded to absolute maxima. The mathematical derivations offered here are also suggestive of actions capable of improving the insight into the maximum-likelihood technique and its numerical implementation.

The high-frequency behavior of the shield in the magnetic-field probes

Shielded loops are commonly used for magnetic-field emission and susceptibility experiments. The shield is demonstrated to play a role which changes from low to high frequencies. In the lower frequency range it operates as an electrostatic shield while at higher frequencies it carries the field-generating/sensing currents. The consequences are discussed for the probe equivalent circuit, terminal impedance and calibration, and the relevant formulas are given, along with the expressions defining the frequency limits. Experiments accurately confirm the shields HF behavior.

Analysis of the common-mode rejection in the measurement and generation of magnetic fields using loop probes

Radio frequency magnetic fields are measured using loop probes. Among the aspects of nonideal behavior, a dominant and elusive one is the unwanted response to common-mode excitation. Little attention has been given to this effect in literature and means for predicting its rejection in comparison to the (intentional) differential-mode response are not available. Formulas to describe the common-mode rejection are derived in this paper and the role of the physical quantities (mainly geometry and frequency) influencing this behavior is quantitatively discussed and made evident. Experiments and computations confirming the theoretical predictions are offered.

Design, Preparation, Conduct, and Result of a Proficiency Test of Radiated Emission Measurements

A proficiency test through interlaboratory comparison of radiated emission measurements was carried out in the period of time comprised between May 2012 and May 2013. Nineteen test houses took part in the exercise providing 91 measurement results in total. Measurements were performed in anechoic chambers and in the frequency range comprised between 200 and 3000 MHz. A traveling sample circulated among the laboratories, generating a reference electromagnetic field whose amplitude was a priori known (with uncertainty) but not revealed to the participants until the end of the comparison. Measurement results were provided by the participants in terms of best estimate and uncertainty. The aggregate measurement result is here compared with the a priori known value and its uncertainty. The performance of the laboratories, quantified in terms of two performance statistics selected from ISO 13528, is analyzed and discussed. The measurement uncertainty declared by the laboratories is compared with the dispersion of the measurement results. Aspects concerning the design and conduct of the comparison are also presented and discussed.

Reproducibility of Radiated Emissions Measurements in Compact, Fully Anechoic, Rooms—The Contribution of the Site-to-Site Variations

In this paper, a new procedure is presented for accurately measuring the difference in performance between sites for radiation tests (emission/susceptibility). Differences as low as 0.5 dB can be detected, and this high sensitivity is a consequence of the basic idea of having the entire transmission/reception (T/R) system itinerate through all the sites under investigation. This rules out the otherwise determinant contribution of the system-associated uncertainties (mainly from non-reproducibility of receiving antenna and receiver). The results of the survey, thus, reliably quantify the amount of disagreement that can be accounted for as being due to the site non-ideality alone. Attention was confined to: 1) short-range, fully anechoic rooms (3-m T/R distance) and 2) the lower frequency range (30 to 300 MHz). These assumptions identify a type of site that is in frequent use today and a frequency range where the measurement conditions are usually very critical. Application of this method to different sites or configurations other than those considered here is straightforward. A total of 14 different sites were investigated, and their level of disagreement is collectively described here, in terms of standard deviation of the sample, and individually, in terms of one-to-one deviation, namely, each site against each other (91 pairs). The one-to-one results exclude that the observed collective deviation was due to the presence of a minority of defective sites, and thus, demonstrating that the collective deviation that we derived effectively describes the amount of statistical disagreement in the whole sample. The measured sample standard deviation can be inserted in an overall uncertainty budget together with the independently derived instrumentation uncertainties. All aspects of the physical design of the experiment are analyzed to demonstrate the steps needed to obtain the high sensitivity that is required here.

The Effect of the Imperfect Realization of the Artificial Mains Network Impedance on the Reproducibility of Conducted Emission Measurements

Closed-form formulas are derived permitting us to evaluate the nonreproducibility of conducted emission measurements due to the imperfect realization of the artificial mains network impedance. Since this is the dominant contribution to the combined standard measurement instrumentation uncertainty, the availability of such formulas permits us to tailor the uncertainty value to the actual measurement capability of testing laboratories. Furthermore, analytical formulas provide guidance both to the technical committees involved in the standardization process, when they set tolerances on the complex impedance of coupling/decoupling networks, and to the manufacturers of instrumentation, since they have to optimize the design of the networks in order to comply with the standards specifications. The nonreproducibility error is quantified in terms of its expected value and standard uncertainty. A simple and reasonably accurate approximation of its probability density function is also obtained. A comparison with the material reported in the relevant international standards is offered in order to highlight the common and the innovative aspects of the results derived here.

Generation and measurement of a reference field for round-robin comparison purposes

An accurately known reference field is often the reference quantity in round-robin comparisons of electromagnetic field measurements. The limits of accuracy inherent to the definition and measurement of a reference electromagnetic field are here discussed and quantified in the case of measurements at 3 m distance from the field source and in the 30-300 MHz frequency range. The insertion loss between the transmitting and receiving antenna is considered here as an alternative reference quantity. A method of validation of insertion loss predictions delivering an accuracy of a few tens of dB is described and experimentally demonstrated. Reference to a practical selection of field source and receiving antenna is made for general applicability of the results here obtained.

Proficiency Testing by Using Traveling Samples With Preassigned Reference Values

In this paper, the results of two proficiency tests of radiated emission measurements are reported. In doing this, several considerations are made concerning the design and realization of the traveling samples, the determination of the reference values and the selection of the statistics through which the performance of the participants to the proficiency test is assessed. Emphasis is placed on the advantages stemming from the assignment of a reference value and its uncertainty to the traveling sample before the start of the proficiency test and based on numerical predictions carried out assuming different test environments and distances between the radiator and the receiving antenna.

Validation of far-field numerical predictions through near-field measurements

Reference electromagnetic field radiators for use in interlaboratory comparisons of far-field measurements can be accurately characterized through the comparison between simulations and measurements, provided that measurements are designed to minimize uncertainty and in particular to avoid electromagnetic environment (site) effects. It is well known that near-field measurements can get rid of the imperfections of the site where measurements take place. The experience gained by the Authors in leading a proficiency test through interlaboratory comparison of electromagnetic field measurements in the 200-3000 MHz frequency range is here reported.

Measurement Error of the Standard Unidirectional Impulse Waveforms Due to the Limited Bandwidth of the Measuring System

Analytical formulas derived here are capable of predicting the measurement error of the parameters of the standard unidirectional impulse waveforms due to the distortion induced by the limited bandwidth of the measuring system. The parameters subject to analysis are the rise time and the peak value. The standard waveforms considered are those defined in the IEC 61000-4-2 (electrostatic discharge), -4-4 (electrical fast transient/burst) and -4-5 (surge 1,2/50 μs and 8/20 μs) standards. The results obtained are of importance for the evaluation of the calibration uncertainty of the standard unidirectional impulse generators and associated coupling/decoupling networks as well as for the design of the corresponding measuring systems. Analytical predictions are confirmed by numerical simulations.