From Wikipedia, the free encyclopedia
Graph of the ramp functionThe 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.
Definitions
The ramp function (
R
(
x
)
:
R
→
R
{\displaystyle R(x):\mathbb {R} \rightarrow \mathbb {R} }
A system of equations :
R
(
x
)
:=
{
x
,
x
≥
0
;
0
,
x
<
0
{\displaystyle R(x):={\begin{cases}x,&x\geq 0;\\0,&x<0\end{cases}}}
The max function :
R
(
x
)
:=
max
(
x
,
0
)
{\displaystyle R(x):=\operatorname {max} (x,0)}
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}
The Heaviside step function multiplied by a straight line with unity gradient:
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:[ 1]
R
(
x
)
:=
∫
−
∞
x
H
(
ξ
)
d
ξ
{\displaystyle R(x):=\int _{-\infty }^{x}H(\xi )\,\mathrm {d} \xi }
Macaulay brackets :
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}
Second Derivative
The ramp function satisfies the differential equation:
d
2
d
x
2
R
(
x
−
x
0
)
=
δ
(
x
−
x
0
)
,
{\displaystyle {\frac {\operatorname {d} ^{2}}{\operatorname {d} x^{2}}}R(x-x_{0})=\delta (x-x_{0}),}
where
δ
(
x
)
{\displaystyle \delta (x)}
Dirac delta . This means that
R
(
x
)
{\displaystyle R(x)}
Green's function for the second derivative operator. Thus, any function,
f
(
x
)
{\displaystyle f(x)}
f
″
(
x
)
{\displaystyle f''(x)}
f
(
x
)
=
f
(
a
)
+
(
x
−
a
)
f
′
(
a
)
+
∫
a
b
R
(
x
−
s
)
f
″
(
s
)
d
s
,
{\displaystyle f(x)=f(a)+(x-a)f'(a)+\int _{a}^{b}R(x-s)f''(s)\operatorname {d} s,}
for
a
<
x
<
b
{\displaystyle a<x<b}
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
)
{\displaystyle \delta (x)}
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)}
Proof:
R
(
R
(
x
)
)
:=
R
(
x
)
+
|
R
(
x
)
|
2
=
R
(
x
)
+
R
(
x
)
2
{\displaystyle R(R(x)):={\frac {R(x)+|R(x)|}{2}}={\frac {R(x)+R(x)}{2}}}
=
{\displaystyle =}
=
{\displaystyle =}
2
R
(
x
)
2
=
R
(
x
)
{\displaystyle {\frac {2R(x)}{2}}=R(x)}
This applies the non-negative property .
References
External links
