Heaviside step function

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The Heaviside step function, using the half-maximum convention

The Heaviside step function, also called unit step function, is a discontinuous function whose value is zero for negative argument and one for positive argument:

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

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 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

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

although this expression isn't mathematically correct.

1 Discrete form
2 Analytic approximations
3 Representations
4 H(0)
5 See also

[edit] Discrete form

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

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

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.

[edit] 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. If we take H(0) = 1/2, 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) \$

[edit] Representations

Often an integral representation of the step function is useful:

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

[edit] H(0)

The value of the function at 0 can be defined 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:

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

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