Ramp function

Graph of the ramp function

The ramp function is a unary real function, whose graph is shaped like a ramp. It can be expressed by numerous definitions, for example "0 for negative inputs, output equals input for non-negative inputs".

This function has numerous applications in mathematics and engineering, and goes by various names, depending on the context.

Definitions

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

• A system of equations:

  $R(x):={\begin{cases}x,&x\geq 0;\\0,&x<0\end{cases}}$

• The max function:

  $R(x):=\max(x,0)$

• The mean of an independent variable and its absolute value (a straight line with unity gradient and its modulus):

  $R(x):={\frac {x+|x|}{2}}$

this can be derived by noting the following definition of max(a,b),

$\max(a,b)={\frac {a+b+|a-b|}{2}}$

for which a = x and b = 0

• The Heaviside step function multiplied by a straight line with unity gradient:

  $R\left(x\right):=xH(x)$

• The convolution of the Heaviside step function with itself:

  $R\left(x\right):=H(x)*H(x)$

• The integral of the Heaviside step function:

  $R(x):=\int _{-\infty }^{x}H(\xi )\,d\xi$

• Macaulay brackets:

  $R(x):=\langle x\rangle$

Applications

The ramp function has numerous applications in engineering, e.g., in the theory of DSP.

Analytic properties

Non-negativity

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

$\forall x\in \mathbb {R} :R(x)\geq 0$

and

$\left|R(x)\right|=R(x)$

• Proof: by the mean of definition 2, it is non-negative in the first quarter, and zero in the second; so everywhere it is non-negative.

Derivative

Its derivative is the Heaviside function:

$R'(x)=H(x)\quad {\mbox{for }}x\neq 0.$

Second derivative

The ramp function satisfies the differential equation:

${\frac {d^{2}}{dx^{2}}}R(x-x_{0})=\delta (x-x_{0}),$

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:

$f(x)=f(a)+(x-a)f'(a)+\int _{a}^{b}R(x-s)f''(s)\,ds\quad {\mbox{for }}a<x<b.$

Fourier transform

${\mathcal {F}}{\big \{}R(x){\big \}}(f)=\int _{-\infty }^{\infty }R(x)e^{-2\pi ifx}\,dx={\frac {i\delta '(f)}{4\pi }}-{\frac {1}{4\pi ^{2}f^{2}}},$

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,

${\mathcal {L}}{\big \{}R(x){\big \}}(s)=\int _{0}^{\infty }e^{-sx}R(x)dx={\frac {1}{s^{2}}}.$

Algebraic properties

Iteration invariance

Every iterated function of the ramp mapping is itself, as

$R{\big (}R(x){\big )}=R(x).$

• Proof:

$R{\big (}R(x){\big )}:={\frac {R(x)+|R(x)|}{2}}={\frac {R(x)+R(x)}{2}}=R(x).$

This applies the non-negative property.