The Dirac function
δ
(
t
)
{\displaystyle \delta (t)}
is a "signal" with unit energy that is concentrated around
t
=
0
{\displaystyle t=0}
δ
(
x
)
=
{
∞
,
x
=
0
0
,
x
≠
0
{\displaystyle \delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}}
δ
(
t
)
=
lim
σ
→
0
1
σ
(
2
π
)
exp
(
−
t
2
2
σ
2
)
{\displaystyle \delta (t)=\lim _{\sigma \to 0}{\frac {1}{\sigma {\sqrt {(}}2\pi )}}\exp(-{\frac {t^{2}}{2\sigma ^{2}}})}
This is a gaussian distribution with spread 0.
E
=
∫
−
∞
∞
δ
(
t
)
2
d
t
=
∞
{\displaystyle E=\int _{-\infty }^{\infty }\delta (t)^{2}dt=\infty }
NB:
δ
(
t
)
2
{\displaystyle \delta (t)^{2}}
has no mathematical meaning, as
δ
(
t
)
{\displaystyle \delta (t)}
isn't an ordinary function but a distribution.
The special nature of
δ
(
t
)
{\displaystyle \delta (t)}
appears clearly e.g. when you try to square the same Gaussian distribution above and try to compute the same limit of the integral in
−
∞
,
∞
{\displaystyle -\infty ,\infty }
.
The result will be quite surprising: it is
∞
{\displaystyle \infty }
!
y
(
t
)
∗
δ
(
t
)
=
∫
−
∞
∞
y
(
τ
)
δ
(
t
−
τ
)
d
τ
=
y
(
t
)
{\displaystyle y(t)*\delta (t)=\int _{-\infty }^{\infty }y(\tau )\delta (t-\tau )d\tau =y(t)}
The Kronecker delta function is the discrete analog of the Dirac function. It has Energy 1 and only a contribution at
k
=
0
{\displaystyle k=0}
δ
(
k
)
=
{
1
,
k
=
0
0
,
k
≠
0
{\displaystyle \delta (k)={\begin{cases}1,&k=0\\0,&k\neq 0\end{cases}}}
E
=
∑
k
=
−
∞
∞
δ
(
k
)
2
=
1
{\displaystyle E=\sum _{k=-\infty }^{\infty }\delta (k)^{2}=1}
y
(
k
)
∗
δ
(
k
)
=
∑
m
=
−
∞
∞
y
(
k
)
δ
(
k
−
m
)
=
y
(
k
)
{\displaystyle y(k)*\delta (k)=\sum _{m=-\infty }^{\infty }y(k)\delta (k-m)=y(k)}