Marco Cati

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.

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.

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.

Note on the Expected Value and Standard Deviation of the Mismatch Correction

It is here demonstrated that the expected value of the mismatch correction is zero whatever the magnitude, small or large, of the product between the reflection coefficients of the source and the load. It is also shown that the standard deviation of the mismatch correction can be evaluated, with negligible error, by using the usual formulas derived assuming low mismatch.

Using the log-normal distribution in the statistical treatment of experimental data affected by large dispersion

There are situations in experimental work where large fluctuations of the measurand are experienced, either because of inherent variations of the observed quantity or because of the complexity of the system process leading to the output quantity. A large spread of the measured values around the center value results. It often appears to the experimenter that the spread of values around the mean is not symmetric, rather, values above the mean obtained by multiplying by a certain factor are approximately as likely as those obtained by dividing the mean value by the same factor. A log-normal distribution thus appears to be a candidate for a representation of the distribution of the observed quantity, at least as a simplifying assumption, when such distribution cannot be assumed a priori on the basis of physical reasoning. A system where large variations of the observed quantity result can be exemplified by the radiator-to-receiver transmission in the dominant presence of reflecting surfaces, such as in a screened room. Overall, large variations are often observed in EMC work, where we are usually faced with complex experimental or predictive processes. In the following, we describe the procedure through which the parameters of the log-normal distribution fitting a given set of experimental outcomes are obtained. This description is applied to the measured field distribution in a screened room.

Experience on proficiency testing in Italy with emphasis on radiated emission measurements from 30 MHz to 1 GHz

In recent years (since 2012, through 2014) two proficiency tests on radiated emission measurements were performed in Italy. The first one was devoted to radiated emission measurements in the frequency range between 200 MHz and 3 GHz in fully-anechoic rooms. A rather complete description of the significant technical aspects of this first experience is in [1]. The second one was devoted to radiated emission measurements in the frequency range between 30 MHz and 1 GHz. Both fully anechoic rooms and semi-anechoic rooms were involved in this second experience and measurement distance was 3 m or 10 m. In both cases nineteen participants took part in the proficiency test. The scope of this work is to report the results of this second proficiency test. In doing that several considerations are made concerning the design and realization of the travelling sample, the determination of the reference values of the electromagnetic field, the selection of the statistics through which the performance of the participants to the proficiency test is assessed, the way by which measurement results originated from different test environments (fully- and semi-anechoic rooms, 3 m and 10 m distances) are processed.

A Double Inequality for the Equivalent Impulse Bandwidth

In this paper, we derive a double inequality that permits one to obtain a lower and an upper limit for the value of the impulse bandwidth of a measuring receiver. The limits are the reciprocal of the integral of the relative envelope of the impulse response (lower limit) and the integral of the relative frequency response (upper limit) of the intermediate frequency (IF) filter. Since the limits are relative quantities, their evaluation does not require the use of a calibrated generator, the only significant sources of error being receivers vertical scale nonlinearity and noise proximity. Here, the deviation between the impulse bandwidth and its limits is quantified for practical IF filter configurations. The dominant contributions to measurement uncertainty are identified and suggestions for reducing their magnitude are also offered.