Step function
From Wikipedia, the free encyclopedia
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of half-open intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.
Let the following quantities be given:
- a sequence of coefficients
- a sequence of interval margins
- a sequence of intervals
(for 
)
Definition: Given the notations above, a function 
is a step function if and only if it can be written as
for all 
where 
is the indicator function of 
:
Note: for all 
and 
it holds: 
.
[edit] Special step functions
A particular step function, the unit step function or Heaviside step function H(x), is obtained by setting n=1, α0=0, α1=1, and x1=0 in the general expression above. It is the mathematical concept behind some test signals, as those used to determine the step response of a dynamical system.
