Heaviside step function

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Image:Dirac distribution CDF.png

The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument:

<math>u(x)=\begin{cases} 0, & x < 0 \\ 1, & x > 0 \end{cases}</math>

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 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.

<math> u(x) = \int_{-\infty}^x { \delta(t)} dt </math>

The value of u(0) is occasionally of disputed value. Some writers give u(0) = 0, some u(0) = 1. u(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> u(x) =

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

</math>

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

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

<math> u_n(x) =

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

</math>

Often an integral representation of the step function is useful:

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

Discrete form

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

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

where n is an integer.

This function is the cumulative summation of the Kronecker delta:

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

where

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

is the discrete unit impulse function.

Analytic approximations

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

<math>u(x)={1\over 2}(1+\tanh kx)={1\over 1+e^{-2kx}}</math>,

where larger k corresponds to a sharper transition at x=0.

See also

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

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