Engineering: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]
- [math]\displaystyle{ \Delta f\Delta t \ge k }[/math]
with [math]\displaystyle{ k }[/math] either [math]\displaystyle{ 1 }[/math] or [math]\displaystyle{ \frac{1}{2} }[/math]
Proof
 | This engineering may require cleanup to meet Wikipedia's quality standards. The specific problem is: steps missing, why is the rise time involved? Is it related to sampling intervals? Please help improve this engineering if you can; the talk page may contain suggestions. |
A bandlimited signal [math]\displaystyle{ u(t) }[/math] with fourier transform [math]\displaystyle{ \hat{u}(f) }[/math] is given by the multiplication of any signal [math]\displaystyle{ \underline{\hat{u}}(f) }[/math] with a rectangular function of width [math]\displaystyle{ \Delta f }[/math] in frequency domain:
- [math]\displaystyle{ \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}. }[/math]
This multiplication with a rectangular function acts as a Bandlimiting filter and results in [math]\displaystyle{ \hat{u}(f) =\hat{g}(f) \underline{\hat{u}}(f)=:{{\underline{\hat{u}}(f)}}{{\Big|}_{\Delta f}}. }[/math]
Applying the convolution theorem, we also know
- [math]\displaystyle{ \hat{g}(f) \cdot \hat{u}(f) = \mathcal{F}((g * u)(t)) }[/math]
Since the fourier transform of a rectangular function is a sinc function [math]\displaystyle{ \operatorname{si} }[/math] and vice versa, it follows directly by definition that
- [math]\displaystyle{ 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) }[/math]
Now the first root [math]\displaystyle{ g(\Delta t) =0 }[/math] is at [math]\displaystyle{ \Delta t= \pm \frac{1}{\Delta f} }[/math]. This is the rise time [math]\displaystyle{ \Delta t }[/math] of the pulse [math]\displaystyle{ g(t) }[/math]. 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:
- [math]\displaystyle{ \Delta t = \frac{1}{\Delta f}, }[/math]
the lower the rise time, the wider the frequency bandwidth needs to be.
Equality is given as long as [math]\displaystyle{ \Delta t }[/math] is finite.
Regarding that a real signal has both positive and negative frequencies of the same frequency band, [math]\displaystyle{ \Delta f }[/math] becomes [math]\displaystyle{ 2 \cdot \Delta f }[/math], which leads to [math]\displaystyle{ k = \frac{1}{2} }[/math] instead of [math]\displaystyle{ k = 1 }[/math]
See also
- Heisenberg's uncertainty principle
- Nyquist theorem
References
- ↑ "Digitale Übertragung im Basisband" (in de). Nachrichtenübertragung I. Institut für Nachrichtentechnik, Technische Universität Hamburg-Harburg. 2007. http://www.et2.tu-harburg.de/lehre/Nachrichtenuebertragung/vorlesungI/NUE1_kapitel4_2_2sl.pdf.
Further reading
- (in de) Theoretische Elektrotechnik und Elektronik. Berlin, Heidelberg: Springer-Verlag. 2000. ISBN 978-3-540-56500-0.
- (in de) Grundlagen der Frequenzanalyse - Eine Einführung für Ingenieure und Informatiker (2 ed.). Renningen, Germany: Expert Verlag. 2005. ISBN 3-8169-2447-6.
- (in de) Einführung in die Systemtheorie (4 ed.). Wiesbaden, Germany: Teubner Verlag. 2007. ISBN 978-3-83510176-0.
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