Heaviside Function

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The Heaviside step function, using the half-maximum convention

The Heaviside function, H 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 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

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.

Related pages[change | change source]