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

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

Definition

The sign function is defined as follows.

Applying it to $t = \sin x$ gives the function considered here.

Domain and range

The domain is all real numbers, and the range consists of just three values: $-1$, $0$ and $+1$.

Periodicity and symmetry

The period is $2\pi$, the same as that of $\sin x$. Since $\operatorname{sgn}(\sin(-x)) = -\operatorname{sgn}(\sin x)$, the function is odd, with point symmetry about the origin.

Points of discontinuity

At each $x = n\pi$ the value jumps discontinuously between $+1$ and $-1$. The jump has size $2$, and the value at that instant is $0$. Against the smooth wave of $\sin x$, this graph is made only of horizontal segments and vertical jumps.

There are two jumps in each period, and everywhere else the derivative is $0$. The value switches without ever having a slope, a structure typical of the step functions.

Fourier series

A square wave can be written as a sum of odd-numbered sines.

The coefficients fall off only as $\dfrac{1}{2k+1}$, so the convergence is slow and a good approximation needs many terms. Truncating the series leaves an overshoot near each jump, known as the Gibbs phenomenon². Adding terms does not reduce the height of that overshoot, which stays at about $9\%$; only its width narrows.

The general rule shows through: a waveform with jumps has slowly decaying coefficients, while one with mere corners has fast decaying ones.

Applications

• The clock signal of a digital circuit
• On-off control
• Pulse-width modulation
• The synthesis of electronic sound

1. Square wave, Wikipedia
2. Gibbs phenomenon, Wikipedia