File:Stegsvar bikvadratiskt filter olika dfaktor.png

Summary[edit]

DescriptionStegsvar bikvadratiskt filter olika dfaktor.png	Svenska: Stegsvar hos bikvadratiskt filter för olika dämpfaktorer.
Date	25 September 2009
Source	Own work by uploader   This diagram was created with Mathematica.
Author	Calle

Additional info[edit]

The function plotted is:

${\displaystyle H_{k}(x)={\begin{cases}1-({\frac {k}{\sqrt {1-k^{2}}}}\sin({{\sqrt {1-k^{2}}}2\pi x})+\cos({{\sqrt {1-k^{2}}}2\pi x}))e^{-k2\pi x}&k<1\\1-(1+2\pi x)e^{-2\pi x}&k=1\\1-{\frac {1}{2{\sqrt {k^{2}-1}}}}({\frac {1}{k_{1}}}e^{-k_{1}2\pi x}-{\frac {1}{k_{2}}}e^{-k_{2}2\pi x})&k>1\end{cases}}}$

where

${\displaystyle {\begin{matrix}k_{1}=&k-{\sqrt {k^{2}-1}}\\k_{2}=&k+{\sqrt {k^{2}-1}}\end{matrix}}}$

Mathematica source code:

H[x_, k_] :=
 If[k == 1,
  1 - (1 + 2 \[Pi] x) Exp[-2 \[Pi] x],
  If[k > 1,
   Module[{k1 = k - Sqrt[k^2 - 1], k2 = k + Sqrt[k^2 + 1]},
    1 - 1/(
      2 Sqrt[k^2 - 1]) (1/k1 Exp[-2 \[Pi] k1 x] - 
        1/k2 Exp[-2 \[Pi] k2 x])],
   1 - (k/Sqrt[1 - k^2] Sin[Sqrt[1 - k^2] 2 \[Pi] x] + 
       Cos[Sqrt[1 - k^2] 2 \[Pi] x]) Exp[-2 \[Pi] k x]]]

Needs["PlotLegends`"]

Plot[{H[t, .2], H[t, .4], H[t, .6], H[t, .8], H[t, 1], H[t, 2], 
  H[t, 3]}, {t, 0, 3}, AxesLabel -> {t/Subscript[T, 0], H}, 
 PlotLegend -> {.2, .4, .6, .8, 1, 2, 3}, LegendShadow -> False, 
 LegendPosition -> {.7, .1}, LegendSize -> .5]

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following licenses:

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https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

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This diagram was created with Mathematica.