Heaviside step function

The Heaviside step function, sometimes called the unit step function Step function

In mathematics [i], a function [i] on the real number [i]s is called a step function if it can ...

and named in honor of Oliver Heaviside Oliver Heaviside

Oliver Heaviside was a self-taught English [i] electrical engineer [i], ...

, is a discontinuous function whose value is zero for negative argument and one for positive argument: It seldom matters what value is used for H, since H is mostly used as a distribution. Some common choices can be seen below Heaviside step function

The Heaviside step function, sometimes called the unit step function [i] and named in honor of Oliver Heaviside [i] ...

. The function is used in the mathematics of control theory Control theory

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Encyclopedia

The Heaviside step function, sometimes called the unit step function Step function

In mathematics [i], a function [i] on the real number [i]s is called a step function if it can ...

and named in honor of Oliver Heaviside Oliver Heaviside

Oliver Heaviside was a self-taught English [i] electrical engineer [i], ...

, is a discontinuous function whose value is zero for negative argument and one for positive argument:
It seldom matters what value is used for H, since H is mostly used as a distribution. Some common choices can be seen below Heaviside step function

The Heaviside step function, sometimes called the unit step function [i] and named in honor of Oliver Heaviside [i] ...

.

The function is used in the mathematics of control theory Control theory

In engineering [i] and mathematics [i], control theory deals with the behavior of dynamical system [i]s. ...

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.

The Heaviside function is an antiderivative Antiderivative

In calculus [i], an antiderivative, primitive or indefinite integral of a function [i] ...

of the Dirac delta function Dirac delta function

The Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by...

, . This is sometimes written as
although this expression isn't mathematically correct.

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.

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

This function is the cumulative summation of the Kronecker delta Kronecker delta

In mathematics [i], the Kronecker delta or Kronecker's delta, named after Leopold Kronecker [i], i ...

:

where

is the discrete unit impulse function Degenerate distribution

In mathematics [i], a degenerate distribution is the probability distribution [i] of a discrete random v ...

.

Analytic approximations

For a smooth approximation to the step function, one can use the logistic function Logistic function

A logistic function or logistic curve models
...

,
where larger k corresponds to a sharper transition at x=0. If we take H = 1/2, equality holds in the limit:

There are many other smooth, analytic approximations to the step function. Some might be:

Representations

Often an integral representation of the step function is useful:

H

The value of H can be defined differently. It can be given as H = 0, H = 1/2 or H = 1. H = 1/2 is the most consistent choice used, since it maximizes the symmetry Symmetry

Symmetry is a characteristic feature of geometrical [i] shapes, system [i]s, equation [i]s, and ...

of the function and becomes completely consistent with the signum function Sign function

In mathematics [i] and especially in computer science [i], the sign function is a logical function [i] ...

. This makes for a more general definition:

To remove the ambiguity of which value to use for H, a subscript specifying which value may be used: