heaviside step function

The 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)=\left\{\begin{matrix} 0 & : & x < 0 \\ 1 & : & x > 0 \end{matrix}\right.

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. The value of u(0) is occasionally of disputed value. Some writers give u(0) = 0, some u(0) = 1. u(0) = 0.5 is the most consistent choice used, since it maximizes the symmetry of the function and becomes completely consistent with the sgn() function. This makes for a more general definition:

u(x) = \left\{ \begin{matrix} 0 & : & x < 0 \\ \frac{1}{2} & : & x = 0 \\ 1 & : & x > 0 \end{matrix} \right.

u(x) = \frac{1}{2} \left ( 1 + \sgn(x) \right )

Often an integral representation of the step function is useful:

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

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