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Concept
Step function
Summary
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 intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.
A function is called a step function if it can be written as
for all real numbers
where , are real numbers, are intervals, and is the indicator function of :
In this definition, the intervals can be assumed to have the following two properties:
The intervals are pairwise disjoint: for
The union of the intervals is the entire real line:
Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, the step function
can be written as
Sometimes, the intervals are required to be right-open or allowed to be singleton. The condition that the collection of intervals must be finite is often dropped, especially in school mathematics, though it must still be locally finite, resulting in the definition of piecewise constant functions.
A constant function is a trivial example of a step function. Then there is only one interval,
The sign function sgn(x), which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.
The Heaviside function H(x), which is 0 for negative numbers and 1 for positive numbers, is equivalent to the sign function, up to a shift and scale of range (). It is the mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system.
The rectangular function, the normalized boxcar function, is used to model a unit pulse.
The integer part function is not a step function according to the definition of this article, since it has an infinite number of intervals. However, some authors also define step functions with an infinite number of intervals.
The sum and product of two step functions is again a step function. The product of a step function with a number is also a step function.
Official source
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