Küpfmüller's uncertainty principle

Proof

A bandlimited signal u ( t ) with fourier transform u ^ ( f ) in frequency space is given by the multiplication of any signal u ^ _ ( f ) with u ^ ( f ) = u ^ _ ( f ) | Δ f with a rectangular function of width Δ f

g ^ ( f ) = rect ⁡ ( f Δ f ) = χ [ − Δ f / 2 , Δ f / 2 ] ( f ) := { 1 | f | ≤ Δ f / 2 0 else

as (applying the convolution theorem)

g ^ ( f ) ⋅ u ^ ( f ) = ( g ∗ u ) ( t )

Since the fourier transform of a rectangular function is a sinc function and vice versa, follows

g ( t ) = 1 2 π ∫ − Δ f 2 Δ f 2 1 ⋅ e j 2 π f t d f = 1 2 π ⋅ Δ f ⋅ si ⁡ ( 2 π t ⋅ Δ f 2 )

Now the first root of g ( t ) is at ± 1 Δ f , which is the rise time Δ t of the pulse g ( t ) , now follows

Δ t = 1 Δ f

Equality is given as long as Δ t is finite.

Regarding that a real signal has both positive and negative frequencies of the same frequency band, Δ f becomes 2 ⋅ Δ f , which leads to k = 1 2 instead of k = 1