Definitions

# 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 and signal processing to represent a signal that switches on at a specified time and stays switched on indefinitely. It was named in honor of the English polymath Oliver Heaviside.

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

The Heaviside function is an antiderivative of the Dirac delta function: H′ = δ. This is sometimes written as

$H\left(x\right) = int_\left\{-infty\right\}^x \left\{ delta\left(t\right)\right\} mathrm\left\{d\right\}t$
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:

$H\left[n\right]=begin\left\{cases\right\} 0, & n < 0 1, & n ge 0 end\left\{cases\right\}$

where n is an integer.

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

$delta\left[n\right] = H\left[n\right] - H\left[n-1\right].$

This function is the cumulative summation of the Kronecker delta:

$H\left[n\right] = sum_\left\{k=-infty\right\}^\left\{n\right\} delta\left[k\right] ,$

where

$delta\left[k\right] = delta_\left\{k,0\right\} ,$

## Analytic approximations

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

$H\left(x\right) approx frac\left\{1\right\}\left\{2\right\} + frac\left\{1\right\}\left\{2\right\}tanh\left(kx\right) = frac\left\{1\right\}\left\{1+mathrm\left\{e\right\}^\left\{-2kx\right\}\right\}$,
where a larger k corresponds to a sharper transition at x = 0. If we take H(0) = ½, equality holds in the limit:
$H\left(x\right)=lim_\left\{k rightarrow infty\right\}frac\left\{1\right\}\left\{2\right\}\left(1+tanh kx\right)=lim_\left\{k rightarrow infty\right\}frac\left\{1\right\}\left\{1+mathrm\left\{e\right\}^\left\{-2kx\right\}\right\}$

There are many other smooth, analytic approximations to the step function. They include:

$H\left(x\right) = lim_\left\{k rightarrow infty\right\} frac\left\{1\right\}\left\{2\right\} + frac\left\{1\right\}\left\{pi\right\}arctan\left(kx\right)$

$H\left(x\right) = lim_\left\{k rightarrow infty\right\} frac\left\{1\right\}\left\{2\right\} + frac\left\{1\right\}\left\{2\right\}operatorname\left\{erf\right\}\left(kx\right)$

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

$bigcup_\left\{n=0\right\}^\left\{infty\right\}\left[2^\left\{-2n\right\};2^\left\{-2n+1\right\}\right]$
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:

$H\left(x\right)=lim_\left\{ epsilon to 0^+\right\} -\left\{1over 2pi mathrm\left\{i\right\}\right\}int_\left\{-infty\right\}^infty \left\{1 over tau+mathrm\left\{i\right\}epsilon\right\} mathrm\left\{e\right\}^\left\{-mathrm\left\{i\right\} x tau\right\} mathrm\left\{d\right\}tau =lim_\left\{ epsilon to 0^+\right\} \left\{1over 2pi mathrm\left\{i\right\}\right\}int_\left\{-infty\right\}^infty \left\{1 over tau-mathrm\left\{i\right\}epsilon\right\} mathrm\left\{e\right\}^\left\{mathrm\left\{i\right\} x tau\right\} mathrm\left\{d\right\}tau$

## 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 the sign function. This makes for a more general definition:

$H\left(x\right) = frac\left\{1+sgn\left(x\right)\right\}\left\{2\right\} =$
begin{cases} 0, & x < 0 frac{1}{2}, & x = 0
`             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:

$H_a\left(x\right) =$
begin{cases} 0, & x < 0
`             a, & x = 0`
`             1, & x > 0`
end{cases}

## Antiderivative and derivative

The ramp function is the antiderivative of the Heaviside step function: $R\left(x\right) := int_\left\{-infty\right\}^\left\{x\right\} H\left(xi\right)mathrm\left\{d\right\}xi$

The derivative of the Heaviside step function is the Dirac delta function: $dH\left(x\right)/dx = delta\left(x\right)$

## 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} right)

Here the $frac\left\{1\right\}\left\{s\right\}$ term must be interpreted as a distribution that takes a test function $phi$ to the Cauchy principal value of $intlimits^\left\{infty\right\}_\left\{-infty\right\} phi\left(x\right)/x, dx$.