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 discretetime unit impulse is the first difference of the discretetime step
This function is the cumulative summation of the Kronecker delta:
where
is the discrete unit impulse function.
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:
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