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.^[1]

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

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

Proof

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

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

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

Applying the convolution theorem, we also know

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

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

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

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

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

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

Equality is given as long as ${\displaystyle \Delta t}$ 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 ${\displaystyle k=1}$

See also

• Heisenberg's uncertainty principle
• Nyquist theorem