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.


 * $$\Delta f\Delta t \ge k$$

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)



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

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


 * $$ g(t) =\mathcal{F}^{-1}(\hat{g})(t)=\frac1{\sqrt{2\pi}} \int \limits_{-\frac{\Delta f}{2}}^{\frac{\Delta f}{2}} 1 \cdot e^{j 2 \pi f t} df = \frac1{\sqrt{2\pi}} \cdot \Delta f \cdot \operatorname{si} \left( \frac{2 \pi t \cdot \Delta f}{2} \right)$$

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 $$k = 1$$