Learning Library · Frequency domain · 2
From time to frequency: the FFT
Any repeating waveform can be built by adding sine waves at whole-number multiples of its frequency. An oscilloscope shows the sum over time. A fast Fourier transform (FFT) takes the sum apart and shows how much of each sine is in it.
Top: the sum of the harmonics (blue) against the target waveform (grey). Bottom: the spectrum, each harmonic's amplitude in dB relative to the fundamental. Bars beyond the slider are greyed out.
What the harmonics tell you
| Waveform | Harmonics present | Amplitude of harmonic n |
|---|---|---|
| Square | Odd only: 1, 3, 5, 7… | Falls as 1 ÷ n, so the 3rd is −9.5 dB |
| Triangle | Odd only | Falls as 1 ÷ n², so the 3rd is −19.1 dB |
| Sawtooth | All: 1, 2, 3, 4… | Falls as 1 ÷ n, so the 2nd is −6.0 dB |
The sharper the corners, the slower the harmonics fall away, and the more bandwidth the waveform needs. A square wave's vertical edges need harmonics far above its fundamental. A triangle has no vertical edges, so its harmonics fade quickly and a few of them already make a close copy.
Try this
- Start the square wave at one harmonic. It is a pure sine. Add the 3rd and 5th and the corners start to form.
- Push to the 49th harmonic. The edges sharpen, but an overshoot of about 9 % of the step stays next to each corner however many harmonics you add. That is the Gibbs phenomenon, and it is why band-limited square waves always ring a little.
- Switch to triangle. By the 5th harmonic the shape error is under 3 %, and by the 11th it is under 1 %. The square wave at the 49th harmonic is still at 9 %.
FFT on an oscilloscope
A scope's FFT works on the captured record. Three settings decide what you see:
- Frequency span is half the sample rate. Sample faster to see higher frequencies.
- Frequency resolution is the sample rate divided by the number of FFT points, which is one divided by the record time. Capture longer to separate close frequencies.
- The window shapes the ends of the record so a signal that does not fit a whole number of cycles does not smear across the spectrum. Hanning is a good general choice. Flat top gives the most accurate amplitudes. Rectangular gives the sharpest peaks but the most smearing.
An FFT is how you find what a time-domain view hides: switching-supply noise at a specific frequency, mains hum and its harmonics on an audio line, or a clock leaking into a sensitive analog circuit. For low-level signals over a wide frequency range, a spectrum analyzer has far more dynamic range.
See both domains on one screen. Oscilloscopes with built-in FFT analysis.
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