Heaviside step function

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The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument:

The function is used in the mathematics of control theory and signal processing to represent a signal that switches on at a specified time and stays switched on indefinitely.

It is the cumulative distribution function of a random variable which is almost surely 0. (See constant random variable.)

The Heaviside function is the integral of the Dirac delta function.

The value of u(0) is occasionally of disputed value. Some writers give u(0) = 0, some u(0) = 1. u(0) = 1/2 is the most consistent choice used, since it maximizes the symmetry of the function and becomes completely consistent with the signum function. This makes for a more general definition:

To remove the ambiguity of which value to use for u(0), a subscript specifying which value may be used:

Often an integral representation of the step function is useful:

Discrete form

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

where n is an integer.

This function is the cumulative summation of the Kronecker delta:

where

is the discrete unit impulse function.

See also

• Rectangular function
• Step response
• Dirac delta
• Signum function
• Negative and non-negative numbersca:Funció esglaó

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