Küpfmüller's uncertainty principle by Karl Küpfmüller in the year 1924 states that the relation of the rise time of a bandlimited signal to its bandwidth is a constant.[1]
![{\displaystyle \Delta f\Delta t\geq k}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with
either
or ![{\displaystyle {\frac {1}{2}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Proof
A bandlimited signal
with fourier transform
is given by the multiplication of any signal
with a rectangular function of width
in frequency domain:
![{\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}}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
This multiplication with a rectangular function acts as a Bandlimiting filter and results in
Applying the convolution theorem, we also know
![{\displaystyle {\hat {g}}(f)\cdot {\hat {u}}(f)={\mathcal {F}}((g*u)(t))}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Since the fourier transform of a rectangular function is a sinc function
and vice versa, it follows directly by definition that
![{\displaystyle g(t)={\mathcal {F}}^{-1}({\hat {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)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Now the first root
is at
. This is the rise time
of the pulse
. Since the rise time influences how fast g(t) can go from 0 to its maximum, it affects how fast the bandwidth limited signal transitions from 0 to its maximal value.
We have the important finding, that the rise time is inversely related to the frequency bandwidth:
![{\displaystyle \Delta t={\frac {1}{\Delta f}},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
the lower the rise time, the wider the frequency bandwidth needs to be.
Equality is given as long as
is finite.
Regarding that a real signal has both positive and negative frequencies of the same frequency band,
becomes
,
which leads to
instead of ![{\displaystyle k=1}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
See also
References
- ^ Rohling, Hermann [in German] (2007). "Digitale Übertragung im Basisband" (PDF). Nachrichtenübertragung I (in German). Institut für Nachrichtentechnik, Technische Universität Hamburg-Harburg. Archived from the original (PDF) on 2007-07-12. Retrieved 2007-07-12.
Further reading
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