Küpfmüller's uncertainty principle

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.

$\Delta f\Delta t\geq k$

with $k$ either $1$ or ${\frac {1}{2}}$

Proof

A bandlimited signal $u(t)$ with fourier transform ${\hat {u}}(f)$ is given by the multiplication of any signal ${\underline {\hat {u}}}(f)$ with a rectangular function of width $\Delta f$ in frequency domain:

${\hat {g}}(f)=\operatorname {rect} \left({\frac {f}{\Delta f}}\right)=\chi _{[-\Delta f/2,\Delta f/2]}(f):={\begin{cases}1&|f|\leq \Delta f/2\\0&{\text{else}}\end{cases}}.$

This multiplication with a rectangular function acts as a Bandlimiting filter and results in ${\hat {u}}(f)={\hat {g}}(f){\underline {\hat {u}}}(f)=:{{\underline {\hat {u}}}(f)}{{\Big |}_{\Delta f}}.$

Applying the convolution theorem, we also know

${\hat {g}}(f)\cdot {\hat {u}}(f)={\mathcal {F}}((g*u)(t))$

Since the fourier transform of a rectangular function is a sinc function $\operatorname {si}$ and vice versa, it follows directly by definition that

$g(t)={\mathcal {F}}^{-1}({\hat {g}})(t)={\frac {1}{\sqrt {2\pi }}}\int \limits _{-{\frac {\Delta f}{2}}}^{\frac {\Delta f}{2}}1\cdot e^{j2\pi ft}df={\frac {1}{\sqrt {2\pi }}}\cdot \Delta f\cdot \operatorname {si} \left({\frac {2\pi t\cdot \Delta f}{2}}\right)$

Now the first root $g(\Delta t)=0$ is at $\Delta t=\pm {\frac {1}{\Delta f}}$. This is the rise time $\Delta t$ of the pulse $g(t)$. 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:

$\Delta t={\frac {1}{\Delta f}},$

the lower the rise time, the wider the frequency bandwidth needs to be.

Equality is given as long as $\Delta t$ is finite.

Regarding that a real signal has both positive and negative frequencies of the same frequency band, $\Delta f$ becomes $2\cdot \Delta f$, which leads to $k={\frac {1}{2}}$ instead of $k=1$

See also

• Heisenberg's uncertainty principle
• Nyquist theorem

Further reading

• Küpfmüller, Karl; Kohn, Gerhard (2000). Theoretische Elektrotechnik und Elektronik (in German). Berlin, Heidelberg: Springer-Verlag. ISBN 978-3-540-56500-0.
• Hoffmann, Rüdiger (2005). Grundlagen der Frequenzanalyse - Eine Einführung für Ingenieure und Informatiker (in German) (2 ed.). Renningen, Germany: Expert Verlag. ISBN 3-8169-2447-6.
• Girod, Bernd; Rabenstein, Rudolf; Stenger, Alexander (2007). Einführung in die Systemtheorie (in German) (4 ed.). Wiesbaden, Germany: Teubner Verlag. ISBN 978-3-83510176-0.

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