Heaviside step function

related topics
{math, number, function}
{system, computer, user}
{rate, high, increase}

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.

Contents

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

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:

Full article ▸

related documents
Polynomial time
Bézout's identity
Logarithmic integral function
Magma (algebra)
Separated sets
CLU (programming language)
Intersection (set theory)
NP-hard
Twin prime conjecture
Topological ring
Goldbach's weak conjecture
Euler's criterion
BPP
Kernel (category theory)
Decision problem
Partial fractions in integration
Amicable number
Alternative algebra
Whittaker–Shannon interpolation formula
Regular space
Real line
Ordered field
Metrization theorem
Regular language
Syntactic sugar
Graded algebra
C*-algebra
Lipschitz continuity
T1 space
Algebraic number