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.

$\Delta f\Delta t \ge k$

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

Proof

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

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

as (applying the convolution theorem)

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

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

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

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

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

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 $k = 1$

References

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