Heaviside function

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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[1]

Discrete form[change | change source]

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

where n is an integer.


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

This function is the cumulative summation of the Kronecker delta:


is the discrete unit impulse function.

Representations[change | change source]

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

H(0)[change | change source]

The value of the function at 0 can be defined as H(0) = 0, H(0) = ½ or H(0) = 1. In particular:[2]

Related pages[change | change source]

References[change | change source]

  1. "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-10-06.
  2. Weisstein, Eric W. "Heaviside Step Function". mathworld.wolfram.com. Retrieved 2020-10-06.