Heaviside Function

The Heaviside step function, using the half-maximum convention

The Heaviside function, H is a non-continuous function whose value is zero for negative argument and one for positive argument.

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

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

$H(x) = \int_{-\infty}^x { \delta(t)} \mathrm{d}t$

Discrete form

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

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

where n is an integer.

Or

$H(x) = \lim_{z \rightarrow x^-} ((|z| / z + 1) / 2)$

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:

$H[n] = \sum_{k=-\infty}^{n} \delta[k] \,$

where

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

is the discrete unit impulse function.

Representations

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

$H(x)=\lim_{ \epsilon \to 0^+} -{1\over 2\pi \mathrm{i}}\int_{-\infty}^\infty {1 \over \tau+\mathrm{i}\epsilon} \mathrm{e}^{-\mathrm{i} x \tau} \mathrm{d}\tau =\lim_{ \epsilon \to 0^+} {1\over 2\pi \mathrm{i}}\int_{-\infty}^\infty {1 \over \tau-\mathrm{i}\epsilon} \mathrm{e}^{\mathrm{i} x \tau} \mathrm{d}\tau.$

H(0)

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

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

Other pages

• Laplace transform