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 can be derived by the look of its graph.
Definitions Graph of the ramp functionThe ramp function (
R
(
x
)
:
R
→
R
{\displaystyle R(x):\mathbb {R} \rightarrow \mathbb {R} }
R
(
x
)
:=
{
x
,
x
≥
0
;
0
,
x
<
0
{\displaystyle R(x):={\begin{cases}x,&x\geq 0;\\0,&x<0\end{cases}}}
The mean of a straight line with unity gradient and its modulus:
R
(
x
)
:=
x
+
|
x
|
2
{\displaystyle R(x):={\frac {x+|x|}{2}}}
this can be derived by noting the following definition of
max
(
a
,
b
)
{\displaystyle \operatorname {max} (a,b)}
max
(
a
,
b
)
=
a
+
b
+
|
a
−
b
|
2
{\displaystyle \operatorname {max} (a,b)={\frac {a+b+|a-b|}{2}}}
for which
a
=
x
{\displaystyle a=x}
b
=
0
{\displaystyle b=0}
R
(
x
)
:=
x
H
(
x
)
{\displaystyle R\left(x\right):=xH\left(x\right)}
The convolution of the Heaviside step function with itself:
R
(
x
)
:=
H
(
x
)
∗
H
(
x
)
{\displaystyle R\left(x\right):=H\left(x\right)*H\left(x\right)}
The integral of the Heaviside step function:
R
(
x
)
:=
∫
−
∞
x
H
(
ξ
)
d
ξ
{\displaystyle R(x):=\int _{-\infty }^{x}H(\xi )\,\mathrm {d} \xi }
R
(
x
)
:=
⟨
x
⟩
{\displaystyle R(x):=\langle x\rangle }
Analytic properties Non-negativity In the whole domain the function is non-negative, so its absolute value is itself, i.e.
∀
x
∈
R
:
R
(
x
)
⩾
0
{\displaystyle \forall x\in \mathbb {R} :R(x)\geqslant 0}
and
|
R
(
x
)
|
=
R
(
x
)
{\displaystyle \left|R\left(x\right)\right|=R\left(x\right)}
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 :
R
′
(
x
)
=
H
(
x
)
i
f
x
≠
0
{\displaystyle R'(x)=H(x)\ \mathrm {if} \ x\neq 0}
F
{
R
(
x
)
}
(
f
)
{\displaystyle {\mathcal {F}}\left\{R(x)\right\}(f)}
=
{\displaystyle =}
∫
−
∞
∞
R
(
x
)
e
−
2
π
i
f
x
d
x
{\displaystyle \int _{-\infty }^{\infty }R(x)e^{-2\pi ifx}dx}
=
{\displaystyle =}
i
δ
′
(
f
)
4
π
−
1
4
π
2
f
2
{\displaystyle {\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).
The single-sided Laplace transform of
R
(
x
)
{\displaystyle R(x)}
L
{
R
(
x
)
}
(
s
)
=
∫
0
∞
e
−
s
x
R
(
x
)
d
x
=
1
s
2
.
{\displaystyle {\mathcal {L}}\left\{R\left(x\right)\right\}(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
(
R
(
x
)
)
=
R
(
x
)
{\displaystyle R\left(R\left(x\right)\right)=R\left(x\right)}
We applied the non-negative property .
References 
