Signum function

The signum function returns the sign of its argument:

$\mathrm{sgn}(t)=\{\begin{array}{ll}+1,& t>0\\ 0,& t=0\\ -1,& t<0.\end{array}$

Image: Signum function, CC BY-SA 3.0

A step that jumps from $-1$ to $+1$ at the origin. The value at $t=0$ is a convention; since we only ever integrate it, the point value doesn’t matter, and many texts drop the $\mathrm{sgn}(0)=0$ clause and let the function jump straight from $-1$ to $+1$.

Relation to the unit step

The signum and the unit step are linearly related:

$u(t)=\frac{1}{2}\mathrm{sgn}(t)+\frac{1}{2},\mathrm{sgn}(t)=2u(t)-1.$

Either one builds the other. In practice the unit step is the workhorse; the signum shows up occasionally as a building block, mainly when the symmetric $\pm 1$ form is more convenient than the asymmetric $\{0,1\}$ form.

Fourier transform

$\mathrm{sgn}(t)\stackrel{\mathcal{F}}{\leftrightarrow }\frac{1}{j\pi f}.$

The signum has only an “AC” part (no DC component), so its spectrum is purely $1/(j\pi f)$, blowing up at $f=0$. By comparison, the unit step has the same $1/(j2\pi f)$ AC tail plus a half-impulse at the origin for the DC piece: $u(t)\leftrightarrow \frac{1}{2}\delta (f)+1/(j2\pi f)$.