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 üpfmüa 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