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Information about Heaviside Step Function

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The Heaviside step function, using the half-maximum convention


The Heaviside step function, H, also called 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 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
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:



where n is an integer.

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



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

Antiderivative and Derivative

The ramp function is the antiderivative of the Heaviside step function:

The derivative of the Heaviside step function is the Dirac delta function:

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


Here the term must be interpreted as a distribution that takes a test function the Cauchy principal value of .

See also

References

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function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.
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0 1 2 3 4 5 6 7 8 9 →

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Dieu et mon droit   (French)
"God and my right"
Anthem
No official anthem specific to England — the anthem of the United Kingdom is "God Save the Queen".
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polymath (Greek polymathēs, πολυμαθής, "having learned much")[1][2] is a person with encyclopedic, broad, or varied knowledge or learning.
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Oliver Heaviside

Portrait of Oliver Heaviside (1850-1925) by Frances Hodge
Born May 18 1850(1850--)
Camden Town, London, England
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antiderivative, primitive or indefinite integral[1] of a function f is a function F whose derivative is equal to f, i.e., F ′ = f.
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Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x
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In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. So, for example, , but .
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degenerate distribution is the probability distribution of a discrete random variable whose support consists of only one value. Examples include a two-headed coin and rolling a die whose sides all show the same number.
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logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.
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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).
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antiderivative, primitive or indefinite integral[1] of a function f is a function F whose derivative is equal to f, i.e., F ′ = f.
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derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
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Fourier transform, named in honor of French mathematician Joseph Fourier, is a certain linear operator that maps functions to other functions. Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components
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