Heaviside step function

	Email		Print
	Bookmark		Link

Heaviside step function

The Heaviside step function, H, also called the unit step function, is a discontinuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
function

Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
whose value is zero

0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
for negative argument and one

1 (number)

1 is a number, number names, and the name of the glyph representing that number.It represents a single entity, the unit of counting or measurement....
for positive argument. It seldom matters what value is used for H(0), since ' is mostly used as a distribution

Distribution (mathematics)

In mathematical analysis, distributions are objects which generalize function s. They extend the concept of derivative to all locally integrable functions and beyond, and are used to formulate generalized solutions of partial differential equations....
.

Discussion

Ask a question about 'Heaviside step function'

Start a new discussion about 'Heaviside step function'

Answer questions from other users

Full Discussion Forum

Encyclopedia

The Heaviside step function, H, also called the unit step function, is a discontinuous

Continuous function

Function (mathematics)

0 (number)

1 (number)

Distribution (mathematics)

Heaviside step function

The Heaviside step function, H, also called the unit step function, is a continuous function Function whose value is 0 for negative argument and 1 for positive argument....
.

The function is used in the mathematics of control theory

Control theory

Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The desired output of a system is called the reference....
and signal processing

Signal processing

Signal processing is the analysis, interpretation, and manipulation of signal . Signals of interest include: audio signal processing, , time-varying measurement values and sensor data, for example biological data such as electrocardiograms, control system signals, telecommunication transmission signals such as radio signals, and many others....
to represent a signal that switches on at a specified time and stays switched on indefinitely. It was named after the English

England

Polymath

A polymath is a person whose knowledge is not restricted to one subject area. In less formal terms, a polymath may simply refer to someone who is very knowledgeable....
Oliver Heaviside

Oliver Heaviside

Oliver Heaviside was a autodidact English electrical engineering, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's equations in terms of electric and magnetic forces and flux, and independently co-f...
.

It is the cumulative distribution function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
of a random variable

Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
which is almost surely

Almost surely

In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory....
0. (See constant random variable.)

The Heaviside function is the integral

Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
of the Dirac delta function

Dirac delta function

The Dirac delta or Dirac's delta is a mathematics construct introduced by theoretical physicist Paul Dirac. Informally, it is a function representing an infinitely sharp peak bounding unit area: a function d that has the value 0 everywhere except at x = 0 where its value is infinity in such a way that its total integral is 1....
: H′ = d. 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 d.

Discrete form

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

where n is an integer

Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
.

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

This function is the cumulative summation of the Kronecker delta

Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a Function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise....
:

where

is the discrete unit impulse function

Degenerate distribution

In mathematics, a degenerate distribution is the probability distribution of a discrete random variable whose support consists of only one value....
.

Analytic approximations

For a smooth

Smooth function

In mathematical analysis, a differentiability class is a classification of function according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives....
approximation to the step function, one can use the logistic function

Logistic function

A logistic function or logistic curve is the most common sigmoid curve. It modelsthe S-curve of growth of some set P, where P might...
, 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. They include:

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 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(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

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
of the function and becomes completely consistent with the sign function

Sign function

In mathematics, the sign function is an Even and odd functions function that extracts the negative and non-negative numbers of a real number....
. 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

The ramp function

Ramp function

The ramp function is an elementary function unary function real function, easily computable as the arithmetic mean of its independent variable and its absolute value....
is the antiderivative

Antiderivative

In calculus, an antiderivative, primitive or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f....
of the Heaviside step function:

The derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
of the Heaviside step function is the Dirac delta function

Dirac delta function

Fourier transform

The Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions....
of 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

Generalized function

In mathematics, generalized functions are objects generalizing the notion of function s. There is more than one recognised theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges....
that takes a test function to the Cauchy principal value

Cauchy principal value

In mathematics, the Cauchy principal value of certain improper integrals, named after Augustin Louis Cauchy, is defined as one of the following:...
of

Heaviside step function

Discrete form

Analytic approximations

Representations

H(0)

Antiderivative and derivative

Fourier transform

See also