Küpfmüller's uncertainty relation

The Küpfmüller uncertainty relation (after Karl Küpfmüller , who formulated it in 1924) or the uncertainty relation of communications engineering is the statement of signal theory (communications engineering ) that the duration or settling duration and the bandwidth of a signal cannot be arbitrarily small at the same time. It delivers a statement that is adapted to the conditions of telecommunications systems and analogous to Heisenberg's uncertainty principle .

The following always applies to the duration Δ t and the bandwidth Δ f of a signal:

{\ displaystyle \ Delta t \ cdot \ Delta f \ geq k}

where k assumes the value 1 or 0.5 depending on the definition of bandwidth and duration.

Well-known basic rules of measurement technology are based on the uncertainty relation with regard to duration and bandwidth :

To measure a signal of 100 ns duration, you need an oscilloscope with a bandwidth of more than 10 MHz.
To determine a frequency of 1 Hz, the signal must be measured for at least 1 s.

Time bandwidth product

Since the definition of a certain bandwidth follows different stipulations, and the duration of signals is also not unambiguous, there are several different stipulations of time-bandwidth products. For example, for signals that are not limited in time, the product is expressed using rectangles of equal area around the maximum value of the signal at the origin:

{\ displaystyle \ Delta t = \ left. {\ frac {X (\ mathrm {j} \ omega)} {x (t)}} \ right \ vert _ {\ omega = 0, t = 0}}

with , the Fourier transform of . The bandwidth is determined analogously to: ${\ displaystyle X (\ mathrm {j} \ omega) = {\ mathcal {F}} \ left \ {{x \ left ({t} \ right)} \ right \}}$ ${\ displaystyle x (t)}$ ${\ displaystyle \ Delta f}$

{\ displaystyle \ Delta f = 2 \ pi \ left. {\ frac {x (t)} {X (\ mathrm {j} \ omega)}} \ right \ vert _ {\ omega = 0, t = 0} }

With this definition of the duration and bandwidth, the time-bandwidth product is constant with the value:

{\ displaystyle \ Delta t \ cdot \ Delta f = 2 \ pi}

literature

Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 .
Rüdiger Hoffmann: Fundamentals of frequency analysis. An introduction for engineers and computer scientists, 2nd edition, Expert Verlag, Renningen 2005, ISBN 3-8169-2447-6 .

Individual evidence

↑ Digital transmission in the baseband. (PDF; 652 kB) Technical University Hamburg-Harburg, 2007, archived from the original on July 12, 2007 ; Retrieved July 12, 2007 .

[1] Digital transmission in the baseband. (PDF; 652 kB) Technical University Hamburg-Harburg, 2007, archived from the original on July 12, 2007 ; Retrieved July 12, 2007 .