The Heaviside step function, using the half-maximum convention The Heaviside function , often written as H (x), is a non-continuous function whose value is zero for a negative input and one for a positive input.
The function is used in the mathematics of control theory to represent a signal that switches on at a specified time, and which stays switched on indefinitely. It was named after the Englishman Oliver Heaviside .
The Heaviside function is the integral of the Dirac delta function : H ′(x) = δ (x). This is sometimes written as[ 1]
H
(
x
)
=
∫ ∫ -->
− − -->
∞ ∞ -->
x
δ δ -->
(
t
)
d
t
{\displaystyle H(x)=\int _{-\infty }^{x}{\delta (t)}\mathrm {d} t}
We can also define an alternative form of the Heaviside step function as a function of a discrete variable n :
H
[
n
]
=
{
0
,
n
<
0
1
,
n
≥ ≥ -->
0
{\displaystyle H[n]={\begin{cases}0,&n<0\\1,&n\geq 0\end{cases}}}
where n is an integer .
Or
H
(
x
)
=
lim
z
→ → -->
x
− − -->
(
(
|
z
|
/
z
+
1
)
/
2
)
{\displaystyle H(x)=\lim _{z\rightarrow x^{-}}((|z|/z+1)/2)}
The discrete-time unit impulse is the first difference of the discrete-time step
δ δ -->
[
n
]
=
H
[
n
]
− − -->
H
[
n
− − -->
1
]
.
{\displaystyle \delta \left[n\right]=H[n]-H[n-1].}
This function is the cumulative summation of the Kronecker delta :
H
[
n
]
=
∑ ∑ -->
k
=
− − -->
∞ ∞ -->
n
δ δ -->
[
k
]
{\displaystyle H[n]=\sum _{k=-\infty }^{n}\delta [k]\,}
where
δ δ -->
[
k
]
=
δ δ -->
k
,
0
{\displaystyle \delta [k]=\delta _{k,0}\,}
is the discrete unit impulse function .
Representations
Often an integral representation of the Heaviside step function is useful:
H
(
x
)
=
lim
ϵ ϵ -->
→ → -->
0
+
− − -->
1
2
π π -->
i
∫ ∫ -->
− − -->
∞ ∞ -->
∞ ∞ -->
1
τ τ -->
+
i
ϵ ϵ -->
e
− − -->
i
x
τ τ -->
d
τ τ -->
=
lim
ϵ ϵ -->
→ → -->
0
+
1
2
π π -->
i
∫ ∫ -->
− − -->
∞ ∞ -->
∞ ∞ -->
1
τ τ -->
− − -->
i
ϵ ϵ -->
e
i
x
τ τ -->
d
τ τ -->
.
{\displaystyle H(x)=\lim _{\epsilon \to 0^{+}}-{1 \over 2\pi \mathrm {i} }\int _{-\infty }^{\infty }{1 \over \tau +\mathrm {i} \epsilon }\mathrm {e} ^{-\mathrm {i} x\tau }\mathrm {d} \tau =\lim _{\epsilon \to 0^{+}}{1 \over 2\pi \mathrm {i} }\int _{-\infty }^{\infty }{1 \over \tau -\mathrm {i} \epsilon }\mathrm {e} ^{\mathrm {i} x\tau }\mathrm {d} \tau .}
H (0)The value of the function at 0 can be defined as H (0) = 0, H (0) = ½ or H (0) = 1. In particular:[ 2]
H
(
x
)
=
1
+
sgn
-->
(
x
)
2
=
{
0
,
x
<
0
1
2
,
x
=
0
1
,
x
>
0.
{\displaystyle H(x)={\frac {1+\operatorname {sgn}(x)}{2}}={\begin{cases}0,&x<0\\{\frac {1}{2}},&x=0\\1,&x>0.\end{cases}}}
Related pages
References
↑ "List of Calculus and Analysis Symbols" . Math Vault . 2020-05-11. Retrieved 2020-10-06 .↑ Weisstein, Eric W. "Heaviside Step Function" . mathworld.wolfram.com . Retrieved 2020-10-06 .
![{\displaystyle H[n]={\begin{cases}0,&n<0\\1,&n\geq 0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efb65c686bdec46712eae1997c363f7ef8709b79)
![{\displaystyle \delta \left[n\right]=H[n]-H[n-1].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c417f161a7987a4db18818743eeb23494b0feb)
![{\displaystyle H[n]=\sum _{k=-\infty }^{n}\delta [k]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5db28db3163e2940aaf93d0fc180edb19d865d32)
![{\displaystyle \delta [k]=\delta _{k,0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97a99ab7d257c74f38fbb2d42c8e42e1cb16d0e9)
