Küpfmüller's uncertainty principle

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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


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