Heaviside step function

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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:

[H(x)=\begin 0, & x < 0 \\ 1, & x > 0 \end]

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 an integral of the Dirac delta function.

[ H(x) = \int_^x \mathrmt ]

Contents

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

Discrete form

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

[H[n]=\begin 0, & n < 0 \\ 1, & n \ge 0 \end]

where n is an integer.

This function is the cumulative summation of the Kronecker delta:

[ H[n] = \sum_^ \delta[k] \,]

where

[ \delta[k] = \delta_ \,]

is the discrete unit impulse function.

Analytic approximations

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

[H(x) \approx \frac + \frac\tanh(kx) = \frac^}],

where larger k corresponds to a sharper transition at x=0. Equality holds in the limit:

[H(x)=\lim_\frac(1+\tanh kx)=\lim_\frac^}]

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

[H(x) = \lim_ \frac + \frac\arctan(kx) \ ]

[H(x) = \lim_ \frac + \frac\operatorname(kx) \ ]

Representations

Often an integral representation of the step function is useful:

[H(x)=\lim_ -}\int_^\infty \epsilon} \mathrm^ x \tau} \mathrm\tau ]

H(0)

The value of H(0) can be defined differently. It can be given 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:

[ H(x) = \begin 0, & x < 0 \\ \frac, & x = 0 \\ 1, & x > 0 \end]

[ H(x) = \frac \left ( 1 + \sgn(x) \right ) ]

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

[ H_n(x) = \begin 0, & x < 0 \\ n, & x = 0 \\ 1, & x > 0 \end]

Heaviside step function

Discrete form

Analytic approximations

Representations

H(0)

See also