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Küpfmüller's uncertainty principle

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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]

with either or

Proof

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A bandlimited signal with fourier transform is given by the multiplication of any signal with a rectangular function of width in frequency domain:

This multiplication with a rectangular function acts as a Bandlimiting filter and results in

Applying the convolution theorem, we also know

Since the fourier transform of a rectangular function is a sinc function and vice versa, it follows directly by definition that

Now the first root is at . This is the rise time of the pulse . 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:

the lower the rise time, the wider the frequency bandwidth needs to be.

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

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References

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  1. ^ 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

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