Heaviside step function: Information from Answers.com

The Heaviside step function, using the half-maximum convention

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

$H(x) = \int_{-\infty}^x { \delta(t)} \mathrm{d}t$

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.

1 Discrete form
2 Analytic approximations
3 Representations
4 H(0)
5 Antiderivative and derivative
6 Fourier transform
7 See also
8 References

Discrete form

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

$H[n]=\begin{cases} 0, & n < 0 \\ 1, & n \ge 0 \end{cases}$

where n is an integer.

$H(x) = \lim_{z \rightarrow x^-} ((|z| / z + 1) / 2)$

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

$\delta\left[ n \right] = H[n] - H[n-1].$

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.

The discrete form has the following algebraic representation:

$H[n] = \frac{1}{2} \Bigg( 1+ \frac{|p \cdot n+q|}{p \cdot n+q} \Bigg),$

where $p$ and $q$ are arbitrary integers that satisfy $p > q > 0$ (e.g. $p = 2$ , $q = 1$ : this is the simplest choice). This formula can be very useful for the implementation of the discrete Heaviside step function in spreadsheet softwares.

Analytic approximations

For a smooth approximation to the step function, one can use the logistic function

$H(x) \approx \frac{1}{2} + \frac{1}{2}\tanh(kx) = \frac{1}{1+\mathrm{e}^{-2kx}}$ ,

where a larger k corresponds to a sharper transition at x = 0. If we take H(0) = ½, equality holds in the limit:

$H(x)=\lim_{k \rightarrow \infty}\frac{1}{2}(1+\tanh kx)=\lim_{k \rightarrow \infty}\frac{1}{1+\mathrm{e}^{-2kx}}.$

There are many other smooth, analytic approximations to the step function.^[1] They include:

$H(x) = \lim_{k \rightarrow \infty} \left(\frac{1}{2} + \frac{1}{\pi}\arctan(kx)\right)$

$H(x) = \lim_{k \rightarrow \infty}\left(\frac{1}{2} + \frac{1}{2}\operatorname{erf}(kx)\right).$

These limits hold pointwise and in the sense of distributions. In general, however, pointwise convergence need not imply distributional convergence, and vice-versa distributional convergence need not imply pointwise convergence.

Representations

Often an integral representation of the Heaviside step function is useful:

$H(x)=\lim_{ \epsilon \to 0^+} -{1\over 2\pi i}\int_{-\infty}^\infty {1 \over \tau+i\epsilon} \mathrm{e}^{-i x \tau} \mathrm{d}\tau =\lim_{ \epsilon \to 0^+} {1\over 2\pi i}\int_{-\infty}^\infty {1 \over \tau-i\epsilon} \mathrm{e}^{i x \tau} \mathrm{d}\tau.$

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:

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

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

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

where 0 ≤ a ≤ 1.

Antiderivative and derivative

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

The distributional derivative of the Heaviside step function is the Dirac delta function: $d H (x) / d x = δ(x).$

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

$\hat{H}(s) = \lim_{N\to\infty}\int^N_{-N} \mathrm{e}^{-2\pi i x s} H(x)\, dx = \frac{1}{2} \left( \delta(s) - \frac{i}{\pi}p.v.\frac{1}{s} \right).$

Here $p.v.\frac{1}{s}$ is the distribution that takes a test function $φ$ to the Cauchy principal value of $\int^{\infty}_{-\infty} \varphi(s)/s\, ds.$ The limit appearing in the integral is also taken in the sense of (tempered) distributions.

References

^ Weisstein, Eric W., "Heaviside Step Function" from MathWorld.

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Heaviside step function

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