Heaviside Step Function

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:
:

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

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 .

:

u(x) = \int_{-\infty}^x { \delta(t)}  dt

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:

:

u(x) =
  \begin{cases} 0,           & x < 0
             \ rac{1}{2}, & x = 0
             \ 1,           & x > 0
  \end{cases}

u(x) = rac{1}{2} \left ( 1 + \sgn(x) 
ight )

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

:

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

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

u(x)=\lim_{ \epsilon 	o 0} -{1\over 2\pi i}\int_{-\infty}^\infty {1 \over 	au+i\epsilon} e^{-i x 	au} d	au

DISCRETE FORM

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

:

u

{Link without Title} =\begin{cases} 0, & n < 0 \ 1, & n \ge 0 \end{cases}

where ''n'' is an integer.

This function is the cumulative summation of the Kronecker Delta :

:

u = \sum_{k=-\infty}^{n} \delta[k \,

where

:

\delta

{Link without Title} = \delta_{k,0} \,

is the Discrete Unit Impulse Function .

ANALYTIC APPROXIMATIONS

For a Smooth approximation to the step function, one can use the Logistic Function
:

u(x)={1\over 2}(1+	anh kx)={1\over 1+e^{-2kx}}

,
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 Numbers

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Heaviside Step Function