11/15/11 Ramp function - Wikipedia, the free encyclopedia

1/3en.wikipedia.org/wiki/Ramp_function

Graph of the ramp function

Ramp functionFrom Wikipedia, the free encyclopedia

The ramp function is an elementary unary real function, easily computable as the mean of its independent variableand its absolute value.

This function is applied in engineering (e.g., in the theory of DSP). The name ramp function can be derived by thelook of its graph.

Contents

1 Definitions

2 Analytic properties

2.1 Non-negativity

2.2 Derivative

2.3 Fourier transform

2.4 Laplace transform

3 Algebraic properties3.1 Iteration invariance

4 References

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 ,

for which a = x and b = 0

11/15/11 Ramp function - Wikipedia, the free encyclopedia

2/3en.wikipedia.org/wiki/Ramp_function

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 R(x) is given as follows,

11/15/11 Ramp function - Wikipedia, the free encyclopedia

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Algebraic properties

Iteration invariance

Every iterated function of the ramp mapping is itself, as

.

Proof: =

= .

We applied the non-negative property.

References

Mathworld (http://mathworld.wolfram.com/RampFunction.html)

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Categories: Mathematical analysis Elementary special functions

This page was last modified on 21 October 2011 at 20:40.

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