Graph of the Square Wave $y = \operatorname{sgn}(\sin x)$

The function $y = \operatorname{sgn}(\sin x)$ extracts only the sign of the sine, giving a square wave whose value alternates between $+1$ and $-1$.

Definition

The sign function $\operatorname{sgn}$ is defined by

and this function applies it to $t = \sin x$.

Values, domain, and range

Its domain is all real numbers. Where $\sin x > 0$ it equals $+1$, where $\sin x < 0$ it equals $-1$, and where $\sin x = 0$, at $x = n\pi$, it is exactly $0$. Its range is therefore the three values $\{-1,\ 0,\ +1\}$.

Periodicity and symmetry

Its period is $2\pi$, the same as sine: it equals $+1$ on $0 < x < \pi$ and $-1$ on $\pi < x < 2\pi$. Since $\operatorname{sgn}(\sin(-x)) = \operatorname{sgn}(-\sin x) = -\operatorname{sgn}(\sin x)$, it has the odd symmetry of point symmetry about the origin (away from its discontinuities).

Discontinuities

At each $x = n\pi$ the value jumps discontinuously between $+1$ and $-1$. The jump has size $2$, and the value at the instant of the jump is $0$. In contrast to the smooth sine wave, the graph consists only of horizontal segments and vertical jumps — a staircase, or square, shape.

Relation to other functions

The square wave has the Fourier series

a sum of odd-harmonic sines. Approximating it with finitely many terms leaves an overshoot near the jumps known as the Gibbs phenomenon.

Applications

Square waves are fundamental in engineering as clock signals in digital circuits, in on-off control, in pulse-width modulation (PWM), and in electronic sound synthesis.