RF modulation, duty cycle and time averaging

A time average answers a question about a specified quantity over a specified interval. Before calculating, identify what is being averaged, whether the “off” state is actually zero and whether the window can move. This lesson builds from a periodic on/off source to a changing exposure where those shortcuts need revision. The windows below are stipulated teaching examples; they do not prescribe a regulatory averaging time.

Next action: If your measurements use selective instruments, cross-check settings against Broadband vs selective meters.

At a glance

• Use this when: the source is time-varying (TDD, bursts, beam scheduling, radar pulses, intermittent uplink, power control) and you need a defensible time-averaged exposure metric.
• Core idea: most heating-relevant metrics are power-like (SAR, absorbed power density, incident power density). Power-like quantities average linearly in time; field amplitudes generally do not.
• Key deliverable: a statement of (a) averaging window and whether it is sliding, (b) time behavior assumptions (duty/occupancy), and (c) how the instrument/model implements averaging.
• Common trap: applying duty cycle to E instead of to E², S, or SAR.

Key equations (time averaging done correctly)

General time average

The time average is the area under a quantity’s time graph divided by the interval’s length. For example, integrating absorbed power density gives energy per area; integrating SAR gives absorbed energy per mass. The duration of a high state matters as well as its height.

For a quantity x(t) over a window starting at t₀ with duration T:

If the framework defines a time average over a window of length T, the compliance-relevant value is often the maximum over all window placements:

Duty cycle model (rectangular on/off)

For a power-like quantity y(t) that is yₒₙ during “on” and ~0 during “off”:

This applies directly to S bar, SAR bar, and P bar when the stated interval contains the corresponding on-time fraction. If the low state is not zero, include it: y bar = D yₒₙ + (1-D)y_low. A window containing exactly whole periods gives the cycle average, regardless of where it starts. A window with only part of a period need not contain the long-term high-state fraction. Place that window on the actual timeline.

If the window is longer than one period, separate the complete periods from the remaining interval. Each complete period contributes the same accumulated quantity. Only the remaining interval changes when the window moves, so place that remainder where its contribution is largest. For a rectangular two-state signal, that means capturing as much high-state time as the remainder permits. This argument uses periodic repetition; a non-repeating or changing source needs its actual time record.

Fields under duty cycling (far-field plane-wave intuition)

In a plane wave, S ∝ Eᵣₘₛ² and S ∝ Hᵣₘₛ². If S has duty cycle D, then RMS field averaged over the entire window scales as:

Duty factor multiplies power-like quantities; it square-roots into the full-interval RMS field for the stated on/off signal. RMS means the square root of the mean of the field squared, not the arithmetic mean of field amplitudes. If the reported RMS already includes the full interval, do not apply the duty factor again.

Average linear quantities before converting to decibels

Convert to linear units, average, then convert back. Never average dB values directly.

Metric selection guidance (what quantity do you average?)

Start from the quantity defined by your framework:

• SAR (whole-body or local),
• incident power density,
• absorbed/epithelial power density,
• induced-field metrics at lower frequencies,
• field-strength reference levels (E/H) in some bands.

Then implement averaging on the right underlying signal:

• If endpoint is power density S, average S(t) (not E(t)).
• If endpoint is SAR, average SAR(t) (or the validated power-like driver that maps to SAR).
• If you only have far-field E, use S ∝ E² and average E² (not E).

If the environment is near field, be cautious: plane-wave shortcuts can fail.

Intuition: why “peak vs limit” arguments often fail

Heating depends on energy deposition over time, and heat transfer smooths isolated spikes. Two waveforms can share the same peak but deposit very different energy.

Averaging is aligning the evaluated quantity with thermal time constants and repeatable measurement practice encoded in standards.

Average pulsed exposure and check field conversions

The pulse-cycle examples use a constant on state and zero between pulses. The moving-window example includes a nonzero low state. Use each prompt’s stated levels and interval rather than carrying the zero-off assumption into every calculation.

Average power density over a pulse cycle

Convert duty-averaged power density to RMS field

Average SAR over a duty cycle

Choose the worst-case averaging window

Convert a peak-to-average ratio from decibels

“Averaging rules” checklist (what to write in a defensible report)

• Averaging window T (and whether sliding-window maximum was taken).
• Time behavior model: duty/occupancy/scheduling assumptions, beam states.
• Quantity averaged: S(t), SAR(t), or E²(t) (if converting far-field E to S).
• Detector mode and bandwidth: peak/RMS/average; measurement bandwidth; time constant.
• Treatment of variability: worst-case vs typical, and why that choice matches the task.

Common errors

• Applying duty cycle to E instead of E²/S/SAR.
• Comparing peak-hold readings to time-averaged reference levels.
• Averaging dB values directly.
• Using “whole day average” when the framework defines a shorter window.
• Ignoring worst-case window placement for non-stationary signals.
• Using plane-wave conversions in near-field conditions without justification.
• Using traffic/occupancy factors without stating whether the goal is typical characterization or worst-case compliance.

Further reading

• ICNIRP RF guidelines (2020) and IEEE C95.1 for their specific quantity, time and spatial averaging conditions. A time-average calculation alone does not check every applicable short-duration criterion.
• IEC 62232 and ITU RF measurement guidance (site-survey methodology for time-varying sources; detector modes and uncertainty).
• Instrument manuals for your meter/analyzer (how RMS/average/peak and time constants are defined).

Knowledge check

Exercises