Heaviside step function

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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 after 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 the integral of the Dirac delta function: H′ = δ. This is sometimes written as

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.

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Discrete form

An alternative form of the unit step, as a function of a discrete variable n:

where n is an integer. Unlike the usual (not discrete) case, the definition of H[0] is significant.

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

This function is the cumulative summation of the Kronecker delta:

where

Analytic approximations

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

where a larger k corresponds to a sharper transition at x = 0. If we take H(0) = ½, equality holds in the limit:

There are many other smooth, analytic approximations to the step function.[1] Among the possibilities are: