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

[math]\displaystyle{ H(x) = \int_{-\infty}^x { \delta(t)} \mathrm{d}t }[/math]

Discrete form

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

[math]\displaystyle{ H[n]=\begin{cases} 0, & n \lt 0 \\ 1, & n \ge 0 \end{cases} }[/math]

where n is an integer.

Or

[math]\displaystyle{ H(x) = \lim_{z \rightarrow x^-} ((|z| / z + 1) / 2) }[/math]

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

[math]\displaystyle{ \delta\left[ n \right] = H[n] - H[n-1]. }[/math]

This function is the cumulative summation of the Kronecker delta:

[math]\displaystyle{ H[n] = \sum_{k=-\infty}^{n} \delta[k] \, }[/math]

where

[math]\displaystyle{ \delta[k] = \delta_{k,0} \, }[/math]

is the discrete unit impulse function.

Representations

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

[math]\displaystyle{ 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. }[/math]

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:

[math]\displaystyle{ H(x) = \frac{1+\sgn(x)}{2} = \begin{cases} 0, & x \lt 0 \\ \frac{1}{2}, & x = 0 \\ 1, & x \gt 0. \end{cases} }[/math]

Heaviside Function Media

• 

  \tfrac{1}{2} + \tfrac{1}{2} \tanh(kx) = \frac{1}{1+e^{-2kx}}approaches the step function as k → ∞.

Related pages

• Laplace transform
• Step function