Heaviside step function

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:

H(x)=\begin{cases} 0, & x < 0 \\ 1, & x > 0 \end{cases}

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 an integral of the Dirac delta function.

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

Discrete form

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

H

^*=\begin{cases} 0, & n < 0 \\ 1, & n \ge 0 \end{cases}

where n is an integer.

This function is the cumulative summation of the Kronecker delta:

H = \sum_{k=-\infty}^{n} \delta[k \,

where

\delta

^* = \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}\tanh(kx) = \frac{1}{1+\mathrm{e}^{-2kx}}

where larger k corresponds to a sharper transition at x=0. Equality holds in the limit:

H(x)=\lim_{k \rightarrow \infty}\frac{1}{2}(1+\tanh kx)=\lim_{k \rightarrow \infty}\frac{1}{1+\mathrm{e}^{-2kx}}

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

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

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

Representations

Often an integral representation of the step function is useful:

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

H(0)

The value of H(0) can be defined differently. It can be given as H(0) = 0, H(0) = 1/2 or H(0) = 1. H(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:

H(x) =

\begin{cases} 0, & x < 0 \\ \frac{1}{2}, & x = 0 \\ 1, & x > 0 \end{cases}

H(x) = \frac{1}{2} \left ( 1 + \sgn(x) \right )

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

H_n(x) =

\begin{cases} 0, & x < 0 \\ n, & x = 0 \\ 1, & x > 0 \end{cases}

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Discrete form

Analytic approximations

Representations

H(0)

See also