Ramp function
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The ramp function is an elementary unary real function, easily computable as the mean of its 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 derivated by the look of its graph.
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[edit] 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:
- A straight line with unity gradient for non-negative x, and zero otherwise:
- 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:
[edit] Analytic properties
[edit] Non-negativity
In the whole domain the function is non-negative, so its absolute value is itself, i.e.
∀x∈ℝ: R(x)≥0
and
|R(x)| = R(x)
.- 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.
[edit] Derivative
Its derivative is the H(x) Heaviside function restricted to ℝ\{0}.
R
'(x) = H(x)
if x≠0 .
From this property definition [5]. goes.
[edit] Fourier transform
Where δ(x)
is the Dirac delta (in this formula, its derivative appears).
[edit] Algebraic properties
[edit] Iteration invariancy
Every iterated function of the ramp mapping is itself, as
R(R(x)) = R(x)
.- Proof: =
= .
We applied the non-negative property.