|
|
| Line 1: |
Line 1: |
| {{One source|date=September 2013}}
| | I like to put these in my heel because that is an area of impact for me and because my toes tend to get blisters so any extra room to reduce friction is great for me.<br>I do not write this to disparage hospital treatment in America, which I believe is also excellent, terrible things are said of socialized medicine.<br>http://eastwest-tours.com/style/?p=6 <br /> http://eastwest-tours.com/style/?p=76 <br /> http://eastwest-tours.com/style/?p=57 <br /> http://eastwest-tours.com/style/?p=67 <br /> http://eastwest-tours.com/style/?p=136 <br /><br><br>If you have any inquiries pertaining to where and the best ways to make use of [http://www.bendtrapclub.com/cheap/ugg.asp Cheap Uggs Boots], you could contact us at the webpage. |
| '''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]]
| |
I like to put these in my heel because that is an area of impact for me and because my toes tend to get blisters so any extra room to reduce friction is great for me.
I do not write this to disparage hospital treatment in America, which I believe is also excellent, terrible things are said of socialized medicine.
http://eastwest-tours.com/style/?p=6
http://eastwest-tours.com/style/?p=76
http://eastwest-tours.com/style/?p=57
http://eastwest-tours.com/style/?p=67
http://eastwest-tours.com/style/?p=136
If you have any inquiries pertaining to where and the best ways to make use of Cheap Uggs Boots, you could contact us at the webpage.