- Heaviside step function
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 "" is mostly used as a distribution. Some common choices can be seen below.
The function is used in the mathematics of
control theoryand signal processingto represent a signal that switches on at a specified time and stays switched on indefinitely. It was named in honor of the English polymath Oliver Heaviside.
It is the
cumulative distribution functionof a random variablewhich is almost surely0. (See constant random variable.)
The Heaviside function is an
antiderivativeof 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".
We can also define an alternative form of the unit step as a function of a discrete variable "n":
where "n" is an
The discrete-time unit impulse is the first difference of the discrete-time step
This function is the cumulative summation of the
is the discrete unit impulse function.
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 [MathWorld | urlname=HeavisideStepFunction | title=Heaviside Step Function] . They include:
Beware that while these approximations converge pointwise towards the step function, the implied "distributions" do not strictly converge towards the delta distribution. In particular, the
measurableset:has measure zero in the delta distribution, but its measure under each smooth approximation family becomes "larger" with increasing "k".
Often an integral representation of the Heaviside step function is useful::
The value of the function at 0 can be defined as "H"(0) = 0, "H"(0) = ½ or "H"(0) = 1. "H"(0) = ½ is the most consistent choice used, since it maximizes the
symmetryof the function and becomes completely consistent with the sign function. This makes for a more general definition:
To remove the ambiguity of which value to use for "H"(0), a subscript specifying the value may be used:
Antiderivative and derivative
ramp functionis the antiderivativeof the Heaviside step function:
derivativeof the Heaviside step function is the Dirac delta function:
Fourier transformof the Heaviside step function is a distribution. Using one choice of constants for the definition of the Fourier transform we have:
Here the term must be interpreted as a distribution that takes a test function to the
Cauchy principal valueof .
Negative and non-negative numbers
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