**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

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

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (1)

Related concepts (11)

Related courses (12)

Step function

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.

Sign function

In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that returns the sign of a real number. In mathematical notation the sign function is often represented as . The signum function of a real number is a piecewise function which is defined as follows: Any real number can be expressed as the product of its absolute value and its sign function: It follows that whenever is not equal to 0 we have Similarly, for any real number , We can also ascertain that: The signum function is the derivative of the absolute value function, up to (but not including) the indeterminacy at zero.

Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one. The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely.

EE-205: Signals and systems (for EL&IC)

This class teaches the theory of linear time-invariant (LTI) systems. These systems serve both as models of physical reality (such as the wireless channel) and as engineered systems (such as electrica

PHYS-441: Statistical physics of biomacromolecules

Introduction to the application of the notions and methods of theoretical physics to problems in biology.

DH-406: Machine learning for DH

This course aims to introduce the basic principles of machine learning in the context of the digital humanities. We will cover both supervised and unsupervised learning techniques, and study and imple

Related lectures (123)

Vibratory Mechanics: QCMsME-332: Mechanical vibrations

Covers multiple-choice questions on vibratory mechanics and interactive polling sessions.

Integration of Piecewise Continuous FunctionsMOOC: Analyse I

Covers the definition and integration of piecewise continuous functions on intervals.

Polymer Behavior: Force-Extension CurvePHYS-441: Statistical physics of biomacromolecules

Delves into the entropic behavior of polymers through force-extension curves.

Let X = {X(t); t ∈ RN} be a (N,d) fractional Brownian motion in Rd of index H ∈ (0,1). We study the local time of X for all temporal dimensions N and spatial dimensions d for which local time exist. W