Küpfmüller's uncertainty principle

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.

${\displaystyle \Delta f\Delta t\geq k}$

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

Proof

A bandlimited signal ${\displaystyle u(t)}$ with fourier transform ${\displaystyle {\hat {u}}(f)}$ in frequency space is given by the multiplication of any signal ${\displaystyle {\underline {\hat {u}}}(f)}$ with ${\displaystyle {\hat {u}}(f)={{\underline {\hat {u}}}(f)}{{\Big |}_{\Delta f}}}$ with a rectangular function of width ${\displaystyle \Delta f}$

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

as (applying the convolution theorem)

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

Since the fourier transform of a rectangular function is a sinc function and vice versa, follows

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

Now the first root of ${\displaystyle g(t)}$ is at ${\displaystyle \pm {\frac {1}{\Delta f}}}$, which is the rise time ${\displaystyle \Delta t}$ of the pulse ${\displaystyle g(t)}$, now follows

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

Equality is given as long as ${\displaystyle \Delta t}$ is finite.

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

References

• Küpfmüller, Karl; Kohn, Gerhard (2000), Theoretische Elektrotechnik und Elektronik, Berlin, Heidelberg: Springer-Verlag, ISBN 978-3-540-56500-0.