Küpfmüller's uncertainty principle
Appearance
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.[1]
with either or
Proof
A bandlimited signal with fourier transform in frequency space is given by the multiplication of any signal with with a rectangular function of width
as (applying the convolution theorem)
Since the fourier transform of a rectangular function is a sinc function and vice versa, follows
Now the first root of is at , which is the rise time of the pulse , now follows
Equality is given as long as is finite.
Regarding that a real signal has both positive and negative frequencies of the same frequency band, becomes , which leads to instead of
See also
References
- ^ Rohling, Hermann [in German] (2007). "Digitale Übertragung im Basisband" (PDF). Nachrichtenübertragung I (in German). Institut für Nachrichtentechnik, Technische Universität Hamburg-Harburg. Archived from the original (PDF) on 2007-07-12. Retrieved 2007-07-12.
Further reading
- Küpfmüller, Karl; Kohn, Gerhard (2000). Theoretische Elektrotechnik und Elektronik (in German). Berlin, Heidelberg: Springer-Verlag. ISBN 978-3-540-56500-0.
- Hoffmann, Rüdiger (2005). Grundlagen der Frequenzanalyse - Eine Einführung für Ingenieure und Informatiker (in German) (2 ed.). Renningen, Germany: Expert Verlag. ISBN 3-8169-2447-6.
- Girod, Bernd; Rabenstein, Rudolf; Stenger, Alexander (2007). Einführung in die Systemtheorie (in German) (4 ed.). Wiesbaden, Germany: Teubner Verlag. ISBN 978-3-83510176-0.