Electromagnetic fields with complex waveform and radiation protection indices

What is meant by complex waveform?

In radiation protection from non-ionizing electromagnetic fields (EMF), it is commonly said that a radiometric or dosimetric quantity has a complex waveform when its instantaneous trend as a function of time is different from a pure sinusoid. It should be remembered that a pure sinusoid is a waveform with amplitude, frequency and phase that are rigorously constant.
Thanks to spectral analysis (Fourier series, Fourier transform, discrete Fourier transform, fast Fourier transform) it is possible to represent any waveform with a sum of a finite or infinite number of pure sinusoidal spectral components. So we can also say that a complex waveform is a waveform that includes at least two pure sinusoidal spectral components at different frequencies.
In EMF radiation protection we are mostly interested in signals with a limited temporal duration (coinciding, for what concerns us here, with the period of observation, measurement or calculation of the evaluated quantity). In this situation, the appropriate tool to perform a spectral analysis is the Fourier series or, in the case of sampled signals, its numeric versions discrete Fourier transform (DFT) and fast Fourier transform (FFT). By means of these tools, it is possible to represent a complex waveform as the sum of a (generally considered) finite number of perfect sinusoidal components, having a frequency equal to an integer multiple of the inverse of the observation period.

Why is a complex waveform a problem?

All national and international safety standards specify maximum limits for radiometric and dosimetric quantities that are generally different as a function of frequency. Therefore, the comparison of an EMF level with the regulatory limits is strictly immediate and direct only if one is dealing with exposure to a perfectly sinusoidal signal. In this case, it is sufficient to compare the effective value of the field (evaluated over a period of the signal) with the regulatory limit. However, perfectly sinusoidal signals are a conceptual abstraction and are never encountered in real situations.
In various applications, one often has to deal with almost perfectly sinusoidal signals, in which one of the parameters of the sinusoid is varied, automatically or manually, according to the needs of the application, always remaining however very close to the central value. If one calculates the spectrum of such a signal, one finds that its energy is distributed over a certain number of spectral components that are very close to each other, in frequency. In this case, it may still be reasonable to directly compare the intensity of the signal with the average limit value of the frequency region occupied by its spectrum. But in many practical cases this is not possible and one must face the fact that the signal contains components at different frequencies, to which different limit values must be applied. For further considerations on this issue, please refer to question B.4 of the document Operational indications for the application of Legislative Decree 81/2008 - Title VIII - Chapter IV released on 20 June 2019 by the Technical Coordination for safety in the workplace of the Regions and autonomous Provinces.

How is exposure assessed when dealing with a radiometric or dosimetric quantity that has a complex waveform?

In the presence of quantities of this type, except for the special cases described in the previous point, it is necessary to resort to the application of assessment methods based on the use of exposure indices. See also question B.5 of the document Operational indications for the application of Legislative Decree 81/2008 - Title VIII - Chapter IV released on 20 June 2019 by the Technical Coordination for workplace safety of the Regions and autonomous Provinces.

What is an exposure index?

The exposure indices are dimensionless quantities defined through a mathematical expression that aims to take into account:

the intensity of the radiometric or dosimetric quantity that the index wants to represent;
how this intensity is distributed among the various spectral components at different frequencies that make up the waveform of the quantity in question;
the variability of the exposure limit to which one wishes to refer, as a function of the frequency.

The purpose of an exposure index is therefore to provide a synthetic and integrated assessment of exposure in the presence of a complex waveform. Exposure indices are formulated in such a way that a value less than 1 (or 100%, in the case of a percentage expression) indicates compliance with the reference regulation, while a value greater than 1 (or 100%) signals a violation of the same.

What are the exposure indices and when are they used?

The exposure indices currently used in radiation protection from non-ionizing electromagnetic fields are (we limit ourselves here to only indices relating to radiometric quantities):

The linear spectral sum index (I_SSL). Introduced since the IRPA-INIRC Guidelines of 1988 as an algorithm to treat exposures from multiple sources, it is suitable for quantifying the level of exposure to an electric or magnetic field with a complex waveform with the aim of preventing the effects related to the stimulation of electrically excitable tissues. These effects can occur in the frequency range from >0 Hz up to 10 MHz. The linear spectral sum index may be adequate, in particular, in the case of non-repetitive or pseudo-chaotic complex waveforms, but in the case of periodic waveforms it leads to excessively cautious assessments, because it presupposes the temporal coincidence of the peak value of all the spectral components, a phenomenon that in reality never occurs with these waveforms in real conditions.
The quadratic spectral sum index (I_SSQ). Also present since the 1988 IRPA-INIRC Guidelines for dealing with exposures from multiple sources, it is suitable for quantifying the level of exposure to an electric, magnetic or electromagnetic field with a complex waveform with the aim of preventing thermal effects. These effects can occur in the frequency range above 100 kHz.
The weighted peak index (I_WP, from the English weighted peak). It was proposed by the ICNIRP in a specific statement of 2003 on complex waveforms (and then explained in detail and incorporated in the 2010 Guidelines) with the aim of remedying the inadequacy shown by the linear spectral sum index in the case of periodic complex waveforms. Currently, being also cited in the European Directive 2013/35/EU, it can be considered the method of choice for the evaluation of exposure to an electric or magnetic field with a complex waveform with the aim of preventing the effects related to the stimulation of electrically excitable tissues, in the frequency range from >0 Hz up to 10 MHz.

On what basis are the radiation protection indices calculated?

The algorithms for calculating all radiation protection indices are based, in principle, on representing - by means of the Fourier series (or, more often, its numerical versions DFT and FFT) - the radiometric quantity G(t) of interest (electric field or magnetic induction) as the sum of a finite number N of sinusoidal spectral components, numbered with i=1,...,N, each characterised by an effective amplitude , a frequency f_i and a phase term θ_i:

The indices are calculated by appropriately relating the effective spectral amplitudes to the corresponding regulatory limit values VA(f_i) and then combining these ratios, possibly also taking into account the phases θ_i.

How is the linear spectral sum index calculated?

The linear spectral sum index I_SSL is calculated by making the direct ratio between the effective spectral amplitudes and the respective limit values (referring to the stimulation effects or more generally to the non-thermal effects) and then adding these ratios:

How is the quadratic spectral sum index calculated?

The quadratic spectral sum index I_SSQ is calculated by squaring the ratio between the effective spectral amplitudes and the respective limit values (referring to thermal effects) and then adding these ratios:

How is the weighted peak index calculated?

The conceptually simplest way to calculate the weighted peak index I_WP is to apply the operational definition with which it was introduced. It is necessary to construct a new waveform in the time domain starting from the spectrum of the original waveform. In the new waveform, the spectral amplitudes are related to the corresponding limit values, while an additional term must be added to the phase values φ_i. The index sought is equal to the maximum absolute value reached in time by the expression thus obtained:

This approach leads to the determination of what we will properly call weighted peak index in the frequency domain (I_WPFD), since for its determination we started from the spectrum of the investigated signal.
The additional phase term was clearly specified in the ICNIRP Guidelines of 2010. Its presence was probably introduced to allow the application of the weighted peak method (for example in a field sensor) also by means of an electronic analog chain of resistance and capacitance (RC) filters, without significantly altering the basic concept of the evaluation, which consists in taking into account the phases of the spectral contributions, so as to mitigate the excess of evaluation inherent in the linear spectral sum index. In this approach that we could call in hardware, however, both the limit values VA(f_i) and the additional phase terms φ_i must be approximated with respect to the values indicated by the ICNIRP Guidelines (which also indicate the permitted tolerances), leading to the determination of index values that are also appreciably different from those provided by the frequency domain approach.
Alternatively, the weighted peak index can be calculated also in the time domain (I_WPTD), that is, without passing through the spectrum calculation. This approach is based on numerical signal processing techniques, which lead to simulating via software the chain of analog RC filters with which the weighting curve adopted by the index can be approximated. Evidently, this approach suffers from the same difficulty described for the hardware approach.

Comparison of the linear spectral sum index and the weighted peak index

Both the linear spectral sum index and the weighted peak index are applied to the evaluation of electric and magnetic fields with complex waveforms in relation to the prevention of stimulation effects of electrically excitable tissues. It may therefore be useful to compare their characteristics.

Main advantages of the linear spectral summation index

It is calculated in a very simple and rapid way.
Since it does not use the phases of the spectral components, it can also be applied to the spectrum provided by a superheterodyne analyzer, or to field measurements performed independently on each component (in the case of overlapping of several independent sinusoidal sources).

Main limitations of the linear spectral summation index

In many cases, especially when dealing with complex periodic waveforms, it leads to excessively cautious evaluations, which can unnecessarily penalize the operation of the source.
It is significantly affected by the presence of noise in the measured signal, since it ignores the intrinsically chaotic distribution of the phases of the spectral components of the noise and sums the latter as if they were all in phase with each other. them.
If the spectrum is determined numerically, the method may be affected by spectral leakage problems.

Main advantages of the weighted peak index

It corrects the limits of the linear spectral sum index, providing correct evaluations even in the case of periodic waveforms, by means of an algorithm that takes into account the phases of the spectral components.
Thanks to the adopted formulation, it is quite well suited for an implementation in hardware in measuring instruments.

Main limitations of the weighted peak index

In some cases, the calculation may be expensive.
The adopted algorithm requires to know the phases of the spectral components and therefore requires, in general, a spectrum calculated with Fourier analysis starting from the field signal sampled in the time domain.
The hardware approach and the time-domain approach lead to evaluations affected by a certain level of error (however tolerated by the ICNIRP Guidelines).
The frequency-domain approach may be affected by spectral leakage problems.
The time-domain approach may be affected by the signal onset transient.

Please note

The WebNir applications compute the weighted peak index, both in the time domain and in the frequency domain, for magnetic induction and for the electric field: the electric-field filters were added in 2026 and are first-order (the discrepancy against higher-order filters stays within 6% in the worst case; the full reasoning, with plots, is in the filter background page).

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Keywords: Complex waveforms, Radiation protection indices