Step function: Difference between revisions

Revision as of 08:24, 25 November 2006

Revision as of 09:17, 25 November 2006

Line 11:

<li> a sequence of intervals

:<math>A_0 := (-\infty, x_1)</math>

:<math>A_i := [x_i, x_{i+1})\,</math> (for <math>i=1,\cdots,n-2</math>)

:<math>A_n := [x_{n-1},\infty)</math>

(Although the intervals are shown as being closed below and open above, this is not necessary to the definition; all that is required is that the intervals A_n do not intersect, and that their union is the set of real numbers.)

Line 21:

:<math>

f(x) = \sum\limits_{i=0}^n \alpha_i \cdot 1_{A_i}(x)

</math> for all <math>x \in \mathbb{R}</math> where <math>1_A\,</math> is the [[indicator function]] of <math>A\,</math>:

:<math>1_A(x) =

\left\{

Line 32:

'''Note:''' for all <math>i=0,\cdots,n</math> and <math>x \in A_i</math> it holds:

<math>f(x)=\alpha_i.\,</math>.

== Special step functions ==

A versionparticular ofstep function, the ''unit step function'' or [[Heaviside step function]], ''H''₁(''x''), is theobtained specialby casesetting ''n''=1, &alpha;₀=0, &alpha;₁=1, and ''x''₁=0 in the general expression above. It is the mathematical concept behind some test [[signal]]s, as those used to determine the [[step response]] of a [[dynamical system (definition)|dynamical system]].<!-- <math>n=1</math>, <math>\alpha_0=0</math>, <math>\alpha_1=1</math>, and <math>x_1=0</math>.-->

==See also==