Heaviside step function
The Heaviside step function, named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative inputs and one elsewhere:
The function is used in the mathematics of signal processing
to represent a signal that switches on at a specified time and stays switched on indefinitely.
The Heaviside function is the integral of the Dirac delta function. The value of H(0) is of very little importance, since the function is often used within an integral. Some writers give H(0) = 0, some H(0) = 1. H(0) = 0.5 is often used, since it maximizes the symmetry of the function. This makes the definition:
The question of the Fourier transform
of H is an interesting example for the theory of distributions. It is often stated that it is 1/x, up to a normalizing constant
. But near x=0 that cannot be justified: the definition must be given in terms of principal value
limit, and the transform isn't to be treated simply as a function. The corresponding convolution operator is the Hilbert transform
Often an integral representation of the step function is useful,
in the limit .