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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]].
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| 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.
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| == Definitions ==
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| [[Image:Ramp_function.svg|[[Graph of a function|Graph]] of the ramp function|thumb|260px|right]]
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| The ramp function (<math> R(x): \mathbb{R} \rightarrow \mathbb{R}</math>) may be defined analytically in several ways. Possible definitions are:
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| :<math>R(x) := \begin{cases} x, & x \ge 0; \\ 0, & x<0 \end{cases} </math>
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| * The mean of a straight line with unity gradient and its modulus:
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| :<math>R(x) := \frac{x+|x|}{2} </math>
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| this can be derived by noting the following definition of <math> \operatorname{max}(a,b) </math>,
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| : <math> \operatorname{max}(a,b) = \frac{a+b+|a-b|}{2} </math>
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| for which <math>a = x</math> and <math>b = 0</math>
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| * The [[Heaviside step function]] multiplied by a straight line with unity gradient:
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| : <math>R\left( x \right) := xH\left( x \right)</math>
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| * The [[convolution]] of the Heaviside step function with itself:
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| : <math>R\left( x \right) := H\left( x \right) * H\left( x \right)</math>
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| * The [[integral]] of the Heaviside step function:
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| : <math>R(x) := \int_{-\infty}^{x} H(\xi)\,\mathrm{d}\xi</math>
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| * [[Macaulay brackets]]:
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| : <math>R(x) := \langle x\rangle</math>
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| == Analytic properties ==
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| === Non-negativity ===
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| In the whole [[domain of a function|domain]] the function is non-negative, so its [[absolute value]] is itself, i.e.
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| <math>\forall x \in \mathbb{R}: R(x) \geqslant 0 </math>
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| and
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| <math>\left| R \left( x \right) \right| = R\left( x \right)</math>
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| * 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.
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| === Derivative ===
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| Its derivative is the [[Heaviside function]]:
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| <math>R'(x) = H(x)\ \mathrm{if}\ x \ne 0</math>
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| <!--Ugyanis
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| * 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.
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| * 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.
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| * 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).-->
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| From this property definition [5]. goes.
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| === [[Fourier transform]] ===
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| <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>
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| Where <code>δ(x)</code> is the [[Dirac delta]] (in this formula, its [[derivative]] appears).
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| === [[Laplace transform]] ===
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| The single-sided [[Laplace transform]] of <math>R(x)</math> is given as follows,
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| <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>
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| == Algebraic properties ==
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| === Iteration invariance ===
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| Every [[iterated function]] of the ramp mapping is itself, as<br>
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| <center><math> R \left( R \left( x \right) \right) = R \left( x \right) </math>. </center><br>
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| * 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>.
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| We applied the [[#Non-negativity|non-negative property]].
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| == References ==
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| * [http://mathworld.wolfram.com/RampFunction.html Mathworld]
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| [[Category:Real analysis]]
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| [[Category:Special functions]]
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