- Heaviside step function
The

**Heaviside step function**, "H", also called the**unit step function**, 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"(0), since "$H$" is mostly used as a distribution. Some common choices can be seen below.The function is used in the mathematics of

control theory andsignal processing to represent a signal that switches on at a specified time and stays switched on indefinitely. It was named in honor of the Englishpolymath Oliver Heaviside .It is the

cumulative distribution function of arandom variable which isalmost surely 0. (Seeconstant random variable .)The Heaviside function is an

antiderivative of theDirac delta function : "H"′ = "δ". This is sometimes written as:$H(x)\; =\; int\_\{-infty\}^x\; \{\; delta(t)\}\; mathrm\{d\}t$although this expansion may not hold (or even make sense) for "x" = 0, depending on which formalism one uses to give meaning to integrals involving "δ".**Discrete form**We can also define an alternative form of the unit step as a function of a discrete variable "n":

:$H\; [n]\; =egin\{cases\}\; 0,\; n\; 0\; \backslash \; 1,\; n\; ge\; 0\; end\{cases\}$

where "n" is an

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

:$delta\; [n]\; =\; H\; [n]\; -\; H\; [n-1]\; .$

This function is the cumulative summation of the

Kronecker delta ::$H\; [n]\; =\; sum\_\{k=-infty\}^\{n\}\; delta\; [k]\; ,$

where

:$delta\; [k]\; =\; delta\_\{k,0\}\; ,$

is the discrete unit impulse function.

**Analytic approximations**For a smooth approximation to the step function, one can use the

logistic function :$H(x)\; approx\; frac\{1\}\{2\}\; +\; frac\{1\}\{2\}\; anh(kx)\; =\; frac\{1\}\{1+mathrm\{e\}^\{-2kx$,where a larger "k" corresponds to a sharper transition at "x" = 0. If we take "H"(0) = ½, equality holds in the limit::$H(x)=lim\_\{k\; ightarrow\; infty\}frac\{1\}\{2\}(1+\; anh\; kx)=lim\_\{k\; ightarrow\; infty\}frac\{1\}\{1+mathrm\{e\}^\{-2kx$There are many other smooth, analytic approximations to the step function [

*MathWorld | urlname=HeavisideStepFunction | title=Heaviside Step Function*] . They include::$H(x)\; =\; lim\_\{k\; ightarrow\; infty\}\; frac\{1\}\{2\}\; +\; frac\{1\}\{pi\}arctan(kx)$

:$H(x)\; =\; lim\_\{k\; ightarrow\; infty\}\; frac\{1\}\{2\}\; +\; frac\{1\}\{2\}operatorname\{erf\}(kx)$

Beware that while these approximations converge pointwise towards the step function, the implied "distributions" do not strictly converge towards the delta distribution. In particular, the

measurable set:$igcup\_\{n=0\}^\{infty\}\; [2^\{-2n\};2^\{-2n+1\}]$has measure zero in the delta distribution, but its measure under each smooth approximation family becomes "larger" with increasing "k".**Representations**Often an integral representation of the Heaviside step function is useful::$H(x)=lim\_\{\; epsilon\; o\; 0^+\}\; -\{1over\; 2pi\; mathrm\{iint\_\{-infty\}^infty\; \{1\; over\; au+mathrm\{i\}epsilon\}\; mathrm\{e\}^\{-mathrm\{i\}\; x\; au\}\; mathrm\{d\}\; au\; =lim\_\{\; epsilon\; o\; 0^+\}\; \{1over\; 2pi\; mathrm\{iint\_\{-infty\}^infty\; \{1\; over\; au-mathrm\{i\}epsilon\}\; mathrm\{e\}^\{mathrm\{i\}\; x\; au\}\; mathrm\{d\}\; au$

**"H"(0)**The value of the function at 0 can be defined as "H"(0) = 0, "H"(0) = ½ or "H"(0) = 1. "H"(0) = ½ is the most consistent choice used, since it maximizes the

symmetry of the function and becomes completely consistent with thesign function . This makes for a more general definition::$H(x)\; =\; frac\{1+sgn(x)\}\{2\}\; =\; egin\{cases\}\; 0,\; x\; 0\; \backslash \; frac\{1\}\{2\},\; x\; =\; 0\; \backslash \; 1,\; x\; 0\; end\{cases\}$

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

:$H\_a(x)\; =\; egin\{cases\}\; 0,\; x\; 0\; \backslash \; a,\; x\; =\; 0\; \backslash \; 1,\; x\; 0\; end\{cases\}$

**Antiderivative and derivative**The

ramp function is theantiderivative of the Heaviside step function: $R(x)\; :=\; int\_\{-infty\}^\{x\}\; H(xi)mathrm\{d\}xi$The

derivative of the Heaviside step function is theDirac delta function :$dH(x)/dx\; =\; delta(x)$**Fourier transform**The

Fourier transform of the Heaviside step function is a distribution. Using one choice of constants for the definition of the Fourier transform we have:$hat\{H\}(s)\; =\; intlimits^\{infty\}\_\{-infty\}\; mathrm\{e\}^\{-2pimathrm\{i\}\; x\; s\}\; H(x),\; dx\; =\; frac\{1\}\{2\}\; left(\; delta(s)\; -\; frac\{\; mathrm\{i\{pi\; s\}\; ight)$Here the $frac\{1\}\{s\}$ term must be interpreted as a distribution that takes a test function $phi$ to the

Cauchy principal value of $intlimits^\{infty\}\_\{-infty\}\; phi(x)/x,\; dx$.**ee also***

Rectangular function

*Step response

*Dirac delta

*Sign function

*Negative and non-negative numbers

*Laplace transform **References**

*Wikimedia Foundation.
2010.*

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