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The '''ramp function''' is a [[unary function|unary]] [[real function]], easily computable as the [[arithmetic mean|mean]] of the [[independent variable]] and its [[absolute value]]. | |||
This function is applied in engineering (e.g., in the theory of [[Digital signal processing|DSP]]). The name ''ramp function'' can be derived by the look of its graph. | |||
== Definitions == | |||
[[Image:Ramp_function.svg|[[Graph of a function|Graph]] of the ramp function|thumb|260px|right]] | |||
The ramp function (<math> R(x): \mathbb{R} \rightarrow \mathbb{R}</math>) may be defined analytically in several ways. Possible definitions are: | |||
:<math>R(x) := \begin{cases} x, & x \ge 0; \\ 0, & x<0 \end{cases} </math> | |||
* The mean of a straight line with unity gradient and its modulus: | |||
:<math>R(x) := \frac{x+|x|}{2} </math> | |||
this can be derived by noting the following definition of <math> \operatorname{max}(a,b) </math>, | |||
: <math> \operatorname{max}(a,b) = \frac{a+b+|a-b|}{2} </math> | |||
for which <math>a = x</math> and <math>b = 0</math> | |||
* The [[Heaviside step function]] multiplied by a straight line with unity gradient: | |||
: <math>R\left( x \right) := xH\left( x \right)</math> | |||
* The [[convolution]] of the Heaviside step function with itself: | |||
: <math>R\left( x \right) := H\left( x \right) * H\left( x \right)</math> | |||
* The [[integral]] of the Heaviside step function: | |||
: <math>R(x) := \int_{-\infty}^{x} H(\xi)\,\mathrm{d}\xi</math> | |||
* [[Macaulay brackets]]: | |||
: <math>R(x) := \langle x\rangle</math> | |||
== Analytic properties == | |||
=== Non-negativity === | |||
In the whole [[domain of a function|domain]] the function is non-negative, so its [[absolute value]] is itself, i.e. | |||
<math>\forall x \in \mathbb{R}: R(x) \geqslant 0 </math> | |||
and | |||
<math>\left| R \left( x \right) \right| = R\left( x \right)</math> | |||
* Proof: by the mean of definition [2] it is non-negative in the I. quarter, and zero in the II.; so everywhere it is non-negative. | |||
=== Derivative === | |||
Its derivative is the [[Heaviside function]]: | |||
<math>R'(x) = H(x)\ \mathrm{if}\ x \ne 0</math> | |||
<!--Ugyanis | |||
* ha x<0, akkor R(x)=0 konstans, tehát ezen a tartományon (ℝ<sup>-</sup>) R'(x)=0 (konstans deriváltja 0); ami megegyezik a Heaviside-függvénnyel. | |||
* ha x>0, akkor R(x)=x, tehát ezen a tartományon (ℝ<sup>+</sup>) R'(x)=1 (a valós számokon értelmezett [[identitás]] deriváltja 1); ami megegyezik a Heaviside-függvénnyel. | |||
* 0-ban a függvénynek [[töréspont]]ja van, tehát nem deriválható (jobbról deriválva 0-t, balról deriválva 1-et kapunk, holott a deriválhatóság feltétele, hogy a jobb és bal oldali derivált megegyezzen).--> | |||
From this property definition [5]. goes. | |||
=== [[Fourier transform]] === | |||
<center> <math> \mathcal{F}\left\{ R(x) \right\}(f) </math> <math> = </math> <math> \int_{-\infty}^{\infty}R(x) e^{-2\pi ifx}dx </math> <math> = </math> <math> \frac{i\delta '(f)}{4\pi}-\frac{1}{4\pi^{2}f^{2}} </math> </center> | |||
Where <code>δ(x)</code> is the [[Dirac delta]] (in this formula, its [[derivative]] appears). | |||
=== [[Laplace transform]] === | |||
The single-sided [[Laplace transform]] of <math>R(x)</math> is given as follows, | |||
<center> <math> \mathcal{L}\left\{ R\left( x \right)\right\} (s) = \int_{0}^{\infty} e^{-sx}R(x)dx = \frac{1}{s^2}. </math> </center> | |||
== Algebraic properties == | |||
=== Iteration invariance === | |||
Every [[iterated function]] of the ramp mapping is itself, as<br> | |||
<center><math> R \left( R \left( x \right) \right) = R \left( x \right) </math>. </center><br> | |||
* Proof: <math> R(R(x)):= \frac{R(x)+|R(x)|}{2} = \frac{R(x)+R(x)}{2} </math> <math>=</math> <br> <math>=</math> <math> \frac{2R(x)}{2} = R(x) </math>. | |||
We applied the [[#Non-negativity|non-negative property]]. | |||
== References == | |||
* [http://mathworld.wolfram.com/RampFunction.html Mathworld] | |||
[[Category:Real analysis]] | |||
[[Category:Special functions]] |
Revision as of 14:40, 16 December 2013
The ramp function is a unary real function, easily computable as the mean of the independent variable and its absolute value.
This function is applied in engineering (e.g., in the theory of DSP). The name ramp function can be derived by the look of its graph.
Definitions
The ramp function () may be defined analytically in several ways. Possible definitions are:
- The mean of a straight line with unity gradient and its modulus:
this can be derived by noting the following definition of ,
- The Heaviside step function multiplied by a straight line with unity gradient:
- The convolution of the Heaviside step function with itself:
- The integral of the Heaviside step function:
Analytic properties
Non-negativity
In the whole domain the function is non-negative, so its absolute value is itself, i.e.
and
- Proof: by the mean of definition [2] it is non-negative in the I. quarter, and zero in the II.; so everywhere it is non-negative.
Derivative
Its derivative is the Heaviside function:
From this property definition [5]. goes.
Fourier transform
Where δ(x)
is the Dirac delta (in this formula, its derivative appears).
Laplace transform
The single-sided Laplace transform of is given as follows,
Algebraic properties
Iteration invariance
Every iterated function of the ramp mapping is itself, as
We applied the non-negative property.