Ramp function

Graph of the ramp function

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 is derived from the appearance of its graph.

Definitions

The ramp function (R(x) : ℝ → ℝ) may be defined analytically in several ways. Possible definitions are:

this can be derived by noting the following definition of max(a,b),
for which a = x and b = 0

Analytic properties

Non-negativity

In the whole domain the function is non-negative, so its absolute value is itself, i.e.

and

Derivative

Its derivative is the Heaviside function:

Second derivative

The ramp function satisfies the differential equation:

where δ(x) is the Dirac delta. This means that R(x) is a Green's function for the second derivative operator. Thus, any function, f(x), with an integrable second derivative, f″(x), will satisfy the equation:

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,

Algebraic properties

Iteration invariance

Every iterated function of the ramp mapping is itself, as

This applies the non-negative property.

References

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