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.
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
References
- Küpfmüller, Karl; Kohn, Gerhard (2000), Theoretische Elektrotechnik und Elektronik, Berlin, Heidelberg: Springer-Verlag, ISBN 978-3-540-56500-0.
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