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:
It seldom matters what value is used for
H, since
H is mostly used as a distribution. Some common choices can be seen
below.
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.
Encyclopedia
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:
It seldom matters what value is used for
H, since
H is mostly used as a distribution. Some common choices can be seen
below.
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.
The Heaviside function is an
antiderivative of the
Dirac delta function, . This is sometimes written as
although this expression isn't mathematically correct.
Discrete form
We can also define an alternative form of the unit step as a function of a discrete variable
n:
where
n is an integer.
The discrete-time unit impulse is the first difference of the discrete-time step
This function is the cumulative summation of the
Kronecker delta:
where
is the
discrete unit impulse function.
Analytic approximations
For a smooth approximation to the step function, one can use the
logistic function,
where larger
k corresponds to a sharper transition at
x=0. If we take
H = 1/2, equality holds in the limit:
There are many other smooth, analytic approximations to the step function. Some might be:
Representations
Often an integral representation of the step function is useful:
H
The value of
H can be defined differently. It can be given as
H = 0,
H = 1/2 or
H = 1.
H = 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:
To remove the ambiguity of which value to use for
H, a subscript specifying which value may be used:
See also
...