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The Heaviside step function, also called unit step function, is a discontinuous function whose value is zero for negative argument and one for positive argument:

<math> H(x) =

 \begin{cases} 0,           & x < 0
            \\ \frac{1}{2}, & x = 0
            \\ 1,           & x > 0
 \end{cases}

</math> 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 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, <math>H' = \delta</math>. This is sometimes written as

<math> H(x) = \int_{-\infty}^x { \delta(t)} \mathrm{d}t </math>

although this expansion may not hold (or even make sense) for <math>x=0</math>, depending on which formalism one uses to give meaning to integrals involving <math>\delta</math>.

Contents 1 Discrete form 2 Analytic approximations 3 Representations 4 H(0) 5 See also

[edit] Discrete form

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

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

where n is an integer.

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

<math> \delta[n] = H[n] - H[n-1]\,</math>

This function is the cumulative summation of the Kronecker delta:

<math> H[n] = \sum_{k=-\infty}^{n} \delta[k] \,</math>

where

<math> \delta[k] = \delta_{k,0} \,</math>

is the discrete unit impulse function.

[edit] Analytic approximations

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

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

where larger k corresponds to a sharper transition at x=0. If we take H(0) = 1/2, equality holds in the limit:

<math>H(x)=\lim_{k \rightarrow \infty}\frac{1}{2}(1+\tanh kx)=\lim_{k \rightarrow \infty}\frac{1}{1+\mathrm{e}^{-2kx}}</math>

There are many other smooth, analytic approximations to the step function. Some might be:

<math>H(x) = \lim_{k \rightarrow \infty} \frac{1}{2} + \frac{1}{\pi}\arctan(kx) \ </math>

<math>H(x) = \lim_{k \rightarrow \infty} \frac{1}{2} + \frac{1}{2}\operatorname{erf}(kx) \ </math>

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

<math>\bigcup_{n=0}^{\infty}[2^{-2n};2^{-2n+1}]</math>

has measure zero in the delta distribution, but its measure under each smooth approximation family becomes larger with increasing <math>k</math>.

[edit] Representations

Often an integral representation of the step function is useful:

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

[edit] H(0)

The value of the function at 0 can be defined as H(0) = 0, H(0) = 1/2 or H(0) = 1. H(0) = 1/2 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:

<math> H(x) =

 \begin{cases} 0,           & x < 0
            \\ \frac{1}{2}, & x = 0
            \\ 1,           & x > 0
 \end{cases}

</math>

<math> H(x) = \frac{1}{2} \left ( 1 + \sgn(x) \right ) </math>

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

<math> H_n(x) =

 \begin{cases} 0, & x < 0
            \\ n, & x = 0
            \\ 1, & x > 0
 \end{cases}

</math>

[edit] See also

• Rectangular function
• Step response
• Dirac delta
• Signum function
• Negative and non-negative numbers
• Laplace transformca:Funció esglaó

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