Heaviside function

The Heaviside step function, using the half-maximum convention

The Heaviside function, often written as H(x), is a non-continuous function whose value is zero for a negative input and one for a positive input.

The function is used in the mathematics of control theory to represent a signal that switches on at a specified time, and which stays switched on indefinitely. It was named after the Englishman Oliver Heaviside.

The Heaviside function is the integral of the Dirac delta function: H′(x) = δ(x). This is sometimes written as

$H(x)=\underset{-\mathrm{\infty }}{\overset{x}{\int }}\delta (t)\mathrm{d}t$

Discrete form

We can also define an alternative form of the Heaviside step function as a function of a discrete variable n:

$H[n]=\{\begin{array}{cc}0,& n<0\\ 1,& n\ge 0\end{array}$

where n is an integer.

Or

$H(x)=\underset{z\to x^{-}}{\lim }((|z|/z+1)/2)$

The discrete-time unit impulse is the first difference of the discrete-time step

$\delta \left[n\right]=H[n]-H[n-1].$

This function is the cumulative summation of the Kronecker delta:

$H[n]=\sum _{k=-\mathrm{\infty }}^n\delta [k]$

where

$\delta [k]={\delta }_{k,0}$

is the discrete unit impulse function.

Representations

Often an integral representation of the Heaviside step function is useful:

$H(x)=\underset{ϵ\to 0^{+}}{\lim }-\frac{1}{2\pi \mathrm{i}}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{\tau +\mathrm{i}ϵ}{\mathrm{e}}^{-\mathrm{i}x\tau }\mathrm{d}\tau =\underset{ϵ\to 0^{+}}{\lim }\frac{1}{2\pi \mathrm{i}}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{\tau -\mathrm{i}ϵ}{\mathrm{e}}^{\mathrm{i}x\tau }\mathrm{d}\tau .$

H(0)

The value of the function at 0 can be defined as H(0) = 0, H(0) = ½ or H(0) = 1. In particular:

$H(x)=\frac{1+\mathrm{sgn}(x)}{2}=\{\begin{array}{cc}0,& x<0\\ \frac{1}{2},& x=0\\ 1,& x>0.\end{array}$

Related pages

• Laplace transform
• Step function