Kolmogorov automorphism

2013-03-10T12:22:52Z

76.24.31.52: removed extra apostrophe

{{One source|date=September 2013}}
'''Küpfmüller's uncertainty principle''' states that the relation of the rise time of a bandlimited signal to its bandwidth is a constant.

<math>\Delta f\Delta t \ge k</math>

with <math>k</math> either <math>1</math> or <math>\frac{1}{2}</math>

==Proof==
A bandlimited signal <math>u(t)</math> with [[fourier transform]] <math>\hat{u}(f)</math> in frequency space is given by the multiplication of any signal <math>\underline{\hat{u}}(f)</math> with <math>\hat{u}(f) = {{\underline{\hat{u}}(f)}}{{\Big|}_{\Delta f}}</math> with a [[rectangular function]] of width <math>\Delta f</math>

<math>\hat{g}(f) = \operatorname{rect} \left(\frac{f}{\Delta f} \right) =\chi_{[-\Delta f/2,\Delta f/2]}(f)
:= \begin{cases}1 & |f|\le\Delta f/2 \\ 0 & \text{else} \end{cases} </math>

as (applying the [[convolution theorem]])

<math>\hat{g}(f) \cdot \hat{u}(f) = (g * u)(t) </math>

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

<math> g(t) = \frac1{\sqrt{2\pi}} \int \limits_{-\frac{\Delta f}{2}}^{\frac{\Delta f}{2}} 1 \cdot e^{j 2 \pi f t} df = \frac1{\sqrt{2\pi}} \cdot \Delta f \cdot \operatorname{si} \left( \frac{2 \pi t \cdot \Delta f}{2} \right)</math>

Now the first root of <math> g(t) </math> is at <math> \pm \frac{1}{\Delta f} </math>, which is the rise time <math> \Delta t </math> of the [[Pulse (signal processing)|pulse]] <math> g(t) </math>, now follows

<math> \Delta t = \frac{1}{\Delta f} </math>

Equality is given as long as <math>\Delta t</math> is finite.

Regarding that a real signal has both positive and negative frequencies of the same frequency band, <math>\Delta f</math> becomes <math>2 \cdot \Delta f</math>,
which leads to <math>k = \frac{1}{2}</math> instead of <math>k = 1</math>

==References==
*{{Citation | last1=Küpfmüller | first1=Karl | last2=Kohn | first2=Gerhard | title=Theoretische Elektrotechnik und Elektronik | publisher=[[Springer-Verlag]] | location=Berlin, Heidelberg | isbn=978-3-540-56500-0 | year=2000}}.

{{DEFAULTSORT:Kupfmuller's uncertainty principle}}
[[Category:Electronic engineering]]

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Kolmogorov automorphism