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.
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
si
{\displaystyle \operatorname {si} }
and vice versa, it follows directly by definition that Now the first root
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
Regarding that a real signal has both positive and negative frequencies of the same frequency band,