Ramp function
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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.
Contents
Definitions
The ramp function () may be defined analytically in several ways. Possible definitions are:
- A system of equations:
- The max function:
- The mean of a straight line with unity gradient and its modulus:
- this can be derived by noting the following definition of ,
- for which and
- 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:
- Macaulay brackets:
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:
Fourier transform
Where δ(x)
is the Dirac delta (in this formula, its derivative appears).
Laplace transform
The single-sided Laplace transform of is given as follows,
Algebraic properties
Iteration invariance
Every iterated function of the ramp mapping is itself, as
- .
- Proof:
.
This applies the non-negative property.
References
External links
- Ramp Function at Mathworld