Square wave (waveform)

Definitions

The square wave in mathematics has many definitions, which are equivalent except at the discontinuities:

It can be defined as simply the sign function of a sinusoid: $${\displaystyle \begin{array}{cc}x(t)& =\mathrm{sgn}\left(\sin \frac{2\pi t}{T}\right)=\mathrm{sgn}(\sin 2\pi ft)\\ v(t)& =\mathrm{sgn}\left(\cos \frac{2\pi t}{T}\right)=\mathrm{sgn}(\cos 2\pi ft),\end{array}}$$ which will be 1 when the sinusoid is positive, −1 when the sinusoid is negative, and 0 at the discontinuities. Here, T is the period of the square wave and f is its frequency, which are related by the equation f = 1/T.

A square wave can also be defined with respect to the Heaviside step function u(t) or the rectangular function Π(t): $${\displaystyle \begin{array}{cc}x(t)& =2\left[\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{\Pi }\left(\frac{2(t-nT)}{T}-\frac{1}{2}\right)\right]-1\\ & =2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left[u\left(\frac{t}{T}-n\right)-u\left(\frac{t}{T}-n-\frac{1}{2}\right)\right]-1.\end{array}}$$

A square wave can also be generated using the floor function directly: $${\displaystyle x(t)=2\left(2\lfloor ft\rfloor -\lfloor 2ft\rfloor \right)+1}$$ and indirectly: $${\displaystyle x(t)={\left(-1\right)}^{\lfloor 2ft\rfloor }.}$$

Using the Fourier series (below) one can show that the floor function may be written in trigonometric form $${\displaystyle \frac{2}{\pi }\arctan \left(\tan \left(\frac{\pi ft}{2}\right)\right)+\frac{2}{\pi }\arctan \left(\cot \left(\frac{\pi ft}{2}\right)\right)}$$

Fourier analysis

Using Fourier expansion with cycle frequency f over time t, an ideal square wave with an amplitude of 1 can be represented as an infinite sum of sinusoidal waves: $${\displaystyle \begin{array}{cc}x(t)& =\frac{4}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{\sin \left(2\pi (2k-1)ft\right)}{2k-1}\\ & =\frac{4}{\pi }\left(\sin (\omega t)+\frac{1}{3}\sin (3\omega t)+\frac{1}{5}\sin (5\omega t)+\dots \right),& \text{where }\omega =2\pi f.\end{array}}$$

The ideal square wave contains only components of odd-integer harmonic frequencies (of the form 2π(2k − 1)f).

A curiosity of the convergence of the Fourier series representation of the square wave is the Gibbs phenomenon. Ringing artifacts in non-ideal square waves can be shown to be related to this phenomenon. The Gibbs phenomenon can be prevented by the use of σ-approximation, which uses the Lanczos sigma factors to help the sequence converge more smoothly.

An ideal mathematical square wave changes between the high and the low state instantaneously, and without under- or over-shooting. This is impossible to achieve in physical systems, as it would require infinite bandwidth.

Square waves in physical systems have only finite bandwidth and often exhibit ringing effects similar to those of the Gibbs phenomenon or ripple effects similar to those of the σ-approximation.

For a reasonable approximation to the square-wave shape, at least the fundamental and third harmonic need to be present, with the fifth harmonic being desirable. These bandwidth requirements are important in digital electronics, where finite-bandwidth analog approximations to square-wave-like waveforms are used. (The ringing transients are an important electronic consideration here, as they may go beyond the electrical rating limits of a circuit or cause a badly positioned threshold to be crossed multiple times.)

Characteristics of imperfect square waves

As already mentioned, an ideal square wave has instantaneous transitions between the high and low levels. In practice, this is never achieved because of physical limitations of the system that generates the waveform. The times taken for the signal to rise from the low level to the high level and back again are called the rise time and the fall time respectively.

If the system is overdamped, then the waveform may never actually reach the theoretical high and low levels, and if the system is underdamped, it will oscillate about the high and low levels before settling down. In these cases, the rise and fall times are measured between specified intermediate levels, such as 5% and 95%, or 10% and 90%. The bandwidth of a system is related to the transition times of the waveform; there are formulas allowing one to be determined approximately from the other.