Küpfmüller's uncertainty principle by Karl
Küpfmüller states that the relation of the
rise time of a bandlimited signal to its bandwidth
is a constant.
Δ
f
Δ
t
≥
k
{\displaystyle \Delta f\Delta t\geq k}
with
k
{\displaystyle k}
either
1
{\displaystyle 1}
or
1
2
{\displaystyle {\frac {1}{2}}}
== Proof ==
A bandlimited signal
u
(
t
)
{\displaystyle u(t)}
with fourier transform
u
^
(
f
)
{\displaystyle {\hat {u}}(f)}
in frequency space is given by the multiplication
of any signal
u
^
_
(
f
)
{\displaystyle {\underline {\hat {u}}}(f)}
with
u
^
(
f
)
=
u
^
_
(
f
)
|
Δ
f
{\displaystyle {\hat {u}}(f)={{\underline
{\hat {u}}}(f)}{{\Big |}_{\Delta f}}}
with a rectangular function of width
Δ
f
{\displaystyle \Delta f}
g
^
(
f
)
=
rect
⁡
(
f
Δ
f
)
=
χ
[
−
Δ
f
/
2
,
Δ
f
/
2
]
(
f
)
:=
{
1
|
f
|
≤
Δ
f
/
2
0
else
{\displaystyle {\hat {g}}(f)=\operatorname
{rect} \left({\frac {f}{\Delta f}}\right)=\chi
_{[-\Delta f/2,\Delta f/2]}(f):={\begin{cases}1&|f|\leq
\Delta f/2\\0&{\text{else}}\end{cases}}}
as (applying the convolution theorem)
g
^
(
f
)
⋅
u
^
(
f
)
=
(
g
∗
u
)
(
t
)
{\displaystyle {\hat {g}}(f)\cdot {\hat {u}}(f)=(g*u)(t)}
Since the fourier transform of a rectangular
function is a sinc function and vice versa,
follows
g
(
t
)
=
1
2
π
∫
−
Δ
f
2
Δ
f
2
1
⋅
e
j
2
π
f
t
d
f
=
1
2
π
⋅
Δ
f
⋅
si
⁡
(
2
π
t
⋅
Δ
f
2
)
{\displaystyle g(t)={\frac {1}{\sqrt {2\pi
}}}\int \limits _{-{\frac {\Delta f}{2}}}^{\frac
{\Delta f}{2}}1\cdot e^{j2\pi ft}df={\frac
{1}{\sqrt {2\pi }}}\cdot \Delta f\cdot \operatorname
{si} \left({\frac {2\pi t\cdot \Delta f}{2}}\right)}
Now the first root of
g
(
t
)
{\displaystyle g(t)}
is at
±
1
Δ
f
{\displaystyle \pm {\frac {1}{\Delta f}}}
, which is the rise time
Δ
t
{\displaystyle \Delta t}
of the pulse
g
(
t
)
{\displaystyle g(t)}
, now follows
Δ
t
=
1
Δ
f
{\displaystyle \Delta t={\frac {1}{\Delta
f}}}
Equality is given as long as
Δ
t
{\displaystyle \Delta t}
is finite.
Regarding that a real signal has both positive
and negative frequencies of the same frequency
band,
Δ
f
{\displaystyle \Delta f}
becomes
2
⋅
Δ
f
{\displaystyle 2\cdot \Delta f}
,
which leads to
k
=
1
2
{\displaystyle k={\frac {1}{2}}}
instead of
k
=
1
{\displaystyle k=1
