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 $\frac$

Proof

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

:$\hat(f) = \operatorname \left(\frac \right) =\chi_(f) := \begin1 & , f, \le\Delta f/2 \\ 0 & \text \end$

as (applying the convolution theorem)

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

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

:$g(t) = \frac1 \int \limits_^ 1 \cdot e^ df = \frac1 \cdot \Delta f \cdot \operatorname \left( \frac \right)$

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

:$\Delta t = \frac$

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 $k = \frac$ instead of $k = 1$

See also

* Heisenberg's uncertainty principle

References

Further reading

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{{DEFAULTSORT:Kupfmuller's uncertainty principle
Electronic engineering
1924 in science
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