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.

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

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

and

Its derivative is the Heaviside function:

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:

where δ(x) is the Dirac delta (in this formula, its derivative appears).

The single-sided Laplace transform of R(x) is given as follows,

Every iterated function of the ramp mapping is itself, as

This applies the non-negative property.

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