Heaviside step function


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 "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 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 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 expansion may not hold (or even make sense) for "x" = 0, depending on which formalism one uses to give meaning to integrals involving "δ".

Discrete form

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

:H [n] =egin{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.

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} anh(kx) = frac{1}{1+mathrm{e}^{-2kx,where a larger "k" corresponds to a sharper transition at "x" = 0. If we take "H"(0) = ½, equality holds in the limit::H(x)=lim_{k ightarrow infty}frac{1}{2}(1+ anh kx)=lim_{k ightarrow infty}frac{1}{1+mathrm{e}^{-2kx

There are many other smooth, analytic approximations to the step function [MathWorld | urlname=HeavisideStepFunction | title=Heaviside Step Function] . They include:

:H(x) = lim_{k ightarrow infty} frac{1}{2} + frac{1}{pi}arctan(kx)

:H(x) = lim_{k ightarrow infty} frac{1}{2} + frac{1}{2}operatorname{erf}(kx)

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 measurable set:igcup_{n=0}^{infty} [2^{-2n};2^{-2n+1}] has measure zero in the delta distribution, but its measure under each smooth approximation family becomes "larger" with increasing "k".

Representations

Often an integral representation of the Heaviside step function is useful::H(x)=lim_{ epsilon o 0^+} -{1over 2pi mathrm{iint_{-infty}^infty {1 over au+mathrm{i}epsilon} mathrm{e}^{-mathrm{i} x au} mathrm{d} au =lim_{ epsilon o 0^+} {1over 2pi mathrm{iint_{-infty}^infty {1 over au-mathrm{i}epsilon} mathrm{e}^{mathrm{i} x au} mathrm{d} au

"H"(0)

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 symmetry of the function and becomes completely consistent with the sign function. This makes for a more general definition:

: H(x) = frac{1+sgn(x)}{2} = egin{cases} 0, & x < 0 \ frac{1}{2}, & x = 0 \ 1, & x > 0 end{cases}

To remove the ambiguity of which value to use for "H"(0), a subscript specifying the value may be used:

: H_a(x) = egin{cases} 0, & x < 0 \ a, & x = 0 \ 1, & x > 0 end{cases}

Antiderivative and derivative

The ramp function is the antiderivative of the Heaviside step function: R(x) := int_{-infty}^{x} H(xi)mathrm{d}xi

The derivative of the Heaviside step function is the Dirac delta function: dH(x)/dx = delta(x)

Fourier transform

The Fourier transform of the Heaviside step function is a distribution. Using one choice of constants for the definition of the Fourier transform we have:hat{H}(s) = intlimits^{infty}_{-infty} mathrm{e}^{-2pimathrm{i} x s} H(x), dx = frac{1}{2} left( delta(s) - frac{ mathrm{i{pi s} ight)

Here the frac{1}{s} term must be interpreted as a distribution that takes a test function phi to the Cauchy principal value of intlimits^{infty}_{-infty} phi(x)/x, dx.

ee also

* Rectangular function
* Step response
* Dirac delta
* Sign function
* Negative and non-negative numbers
* Laplace transform

References


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Step function — In mathematics, a function on the real numbers is called a step function (or staircase function) if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant …   Wikipedia

  • Heaviside (disambiguation) — Heaviside usually refers to the mathematician Oliver Heaviside, but it can refer to:* Heaviside step function, named in honor of him. * Heaviside condition, named in honor of him. * Heaviside layer, or Kennelly Heaviside layer, named in honor of… …   Wikipedia

  • Step potential — The Schrödinger equation for a one dimensional step potential is a model system in quantum mechanics and scattering theory. The problem consists of solving the time independent Schrödinger equation for a particle with a step like potential in one …   Wikipedia

  • Step response — The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the… …   Wikipedia

  • Heaviside-Funktion — Die Heaviside Funktion, auch Theta , Treppen , Schwellenwert , Stufen , Sprung oder Einheitssprungfunktion genannt, ist eine in der Mathematik und Physik oft verwendete Funktion. Sie ist nach dem britischen Mathematiker und Physiker Oliver… …   Deutsch Wikipedia

  • Oliver Heaviside — Heaviside redirects here. For other uses, see Heaviside (disambiguation). Oliver Heaviside Portrait by Francis Edwin Hodge …   Wikipedia

  • Dirac delta function — Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention… …   Wikipedia

  • Ramp function — The ramp function is an elementary unary real function, easily computable as the mean of its independent variable and its absolute value.This function is applied in engineering (e.g., in the theory of DSP). The name ramp function can be derived… …   Wikipedia

  • Sign function — In mathematics, the sign function is a mathematical function that extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function (after the Latin form of sign ).In mathematical… …   Wikipedia

  • Green's function — In mathematics, Green s function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. The term is used in physics, specifically in quantum field theory and statistical field theory, to refer to… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.