Square Wave Fourier Harmonics Calculator

Split a square wave into its harmonics and see how much of the signal sits in each one.

What it does

A square wave is the simplest signal that cannot be described by one sine wave, and the way it is built out of sines is the most useful result in signal processing. The MIT notes for computational science and engineering give the result straight away: the odd numbered coefficients of a square wave are 4 over pi times k, and the even numbered ones are all zero.

The reason the even harmonics vanish is symmetry. A square wave spends exactly half its period above zero and half below, and it does that about its midpoint, so any sine term that is not odd in the time variable cannot survive the averaging. What is left is an infinite set of sine terms, each a third, a fifth, a seventh of the size of the one before it.

The coefficients fall as one over n, and that slow decay is why the series is awkward to sum. MIT's differential equations notes make the point with the Gibbs phenomenon: for any periodic function with a jump discontinuity, summing any number of terms always overshoots the jump by about 9 % of the size of the jump. For a square wave with a jump of 2, that is an overshoot of about 0.18, and it never gets smaller however many terms you add.

This page takes the peak amplitude, the fundamental frequency and the number of harmonics you want, and returns the harmonic amplitudes, the series written out, the root mean square value of the wave, the total harmonic distortion and the size of the Gibbs overshoot.

The total harmonic distortion figure is the one that matters in practice, because it is the share of the signal power that sits outside the fundamental. For a square wave it is 48.34 %, which is why a square wave sounds so harsh on a loudspeaker compared with a sine, and why a switching power supply running in square wave mode needs a filter to keep the harmonics out of the output.

How to use

1. Enter the peak amplitude of the square wave.
2. Enter the fundamental frequency, which is the frequency of the square wave itself and of its first harmonic.
3. Choose how many harmonics you want listed. The even ones are always zero, so only the odd ones appear.
4. Read the fundamental and third harmonic amplitudes, and note how each one is a third of the one before it.
5. Read the root mean square value, which is the peak divided by the square root of 2 for a square wave.
6. Read the total harmonic distortion and the Gibbs overshoot, which stay the same however many terms you sum.

Formula

A

Peak amplitude of the square wave.

b n

Amplitude of the nth harmonic.

n

Harmonic number, counting the fundamental as 1.

THD

Total harmonic distortion, the share of the power above the fundamental.

w

Angular frequency of the fundamental, 2 pi times the fundamental frequency.

The total harmonic distortion of a square wave does not depend on the amplitude, because every harmonic scales with it. The figure of 48.34 % comes from the closed form, the square root of pi squared over 8 minus 1, and it is a property of the waveform rather than of any particular circuit. The Gibbs overshoot is also fixed at about 9 % of the jump, which is why adding terms makes the ripple narrower but never lower.

Worked example

Reference table

Limits

• The page covers a symmetric square wave with equal positive and negative halves, and not a unipolar pulse train with a duty cycle other than half.
• Only the ideal square wave is modelled, so there is no rounding, no finite rise time and no overshoot in the real signal.
• The harmonic amplitudes are the ideal Fourier coefficients, and a real circuit with a filter in it will not match them.
• No frequency dependence is included, so a square wave through a lossy line is not represented.

Common mistakes

• Expecting the even harmonics to appear. Symmetry kills them, and a series with a second harmonic describes a different waveform.
• Expecting the partial sum to settle on the square wave. The Gibbs overshoot stays at about 9 % of the jump however many terms you add.
• Confusing the amplitude of a harmonic with its power. The amplitude falls as 1 over n but the power falls as 1 over n squared.
• Applying the square wave coefficients to a triangle wave, where the coefficients fall as 1 over n squared instead.
• Using the peak amplitude where the root mean square value is needed, which overstates the power of the signal by the square root of 2.

Try it yourself

Quick checks. Answers sit tucked under each question.

Questions

Sources

• MIT OpenCourseWare 18.085: Fourier series and integrals, the square wave example
• MIT OpenCourseWare ES.1803: Fourier series introduction, the square wave and the Gibbs phenomenon
• MIT OpenCourseWare 6.003: signals and systems, Fourier representations of a square wave

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