Trapezoidal Pulse Spectrum Envelope (Clock Harmonics)
Spectrum envelope of a trapezoidal clock pulse from amplitude, frequency, pulse width and rise time: both corner frequencies and the harmonic envelope level at a frequency of interest.
Worked example
Input: Amplitude A (peak) 3.3 V, Repetition frequency 50 MHz, Pulse width τ (at 50 %) 10 ns, Rise / fall time t_r 2 ns, Frequency of interest 500 MHz
Result: First corner 1/(πτ) 31.831 MHz, Second corner 1/(πt_r) 159.155 MHz, Low-frequency harmonic envelope level 130.37 dBµV, Envelope level at that frequency 96.5049 dBµV
Result: First corner 1/(πτ) 31.831 MHz, Second corner 1/(πt_r) 159.155 MHz, Low-frequency harmonic envelope level 130.37 dBµV, Envelope level at that frequency 96.5049 dBµV
Formula
Low-frequency plateau: 2A·τ/T (T = 1/f₀)
−20 dB/dec above f₁ = 1/(π·τ), −40 dB/dec above f₂ = 1/(π·t_r)
L(f) = L₀ − 20·log₁₀(f/f₁) − 20·log₁₀(f/f₂) (each term applies only when f is above its corner)
How it works
The harmonics of a digital clock stay under an envelope. It is flat at 2A·τ/T at low frequency, falls at −20 dB/decade above 1/(πτ) and at −40 dB/decade above 1/(π·t_r). A faster rise time moves the second corner up and puts more energy at high frequency. With the example values (3.3 V, 50 MHz, τ 10 ns, t_r 2 ns) the envelope at 500 MHz is about 96.5 dBµV.
Practical tipThe result is the peak-voltage harmonic envelope of an ideal trapezoidal pulse, not a receiver reading or a radiated field strength. Real emissions depend on traces, antenna effects and resonances. Find the bands where the envelope is high first; slowing the edge with a series resistor or bead is usually the cheapest fix.
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Results and summaries are for reference. For certification and test reports use the latest official standard text and calibrated instrument data. Last updated: 2026-10-10