Mathematica 9 is now available
Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2004
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2004

[Date Index] [Thread Index] [Author Index]

Search the Archive

a new sort of Gaussian noise

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49251] a new sort of Gaussian noise
  • From: "Roger L. Bagula" <rlbtftn at netscape.net>
  • Date: Fri, 9 Jul 2004 02:26:23 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Last night I noticed that the second derivative of the projective line
was a parametric cardioid which has genus one. I realized that I might be able to
get a new noise effect by making a random projection from the
cardioid to the real line and from there to a Gaussian noise.
It works and it apparently gives an effect much like shot noise/ 
tunneling effects in transistors.
These also seems to be a cut off effect in the amplitudes which divides 
them into two distinct parts.
I'm attaching both the notebook striped of pictures and pictures of the 
notebook( deleted for newsgroup posts: I posted the pictures to alt.fractals) 
I call the noise a martingale as that is the traditional name for 
functionally random noises
different than the standard probability distributions (pdf, I hate such 
abrivations).
Respectfully, Roger L. Bagula

tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
URL :  http://home.earthlink.net/~tftn
URL :  http://victorian.fortunecity.com/carmelita/435/ 

(***********************************************************************

                    Mathematica-Compatible Notebook

This notebook can be used on any computer system with Mathematica 3.0,
MathReader 3.0, or any compatible application. The data for the notebook 
starts with the line of stars above.

To get the notebook into a Mathematica-compatible application, do one of 
the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the application;

* Copy the data starting with the line of stars above to the
  clipboard, then use the Paste menu command inside the application.

Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode.  Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing the 
word CacheID, otherwise Mathematica-compatible applications may try to 
use invalid cache data.

For more information on notebooks and Mathematica-compatible 
applications, contact Wolfram Research:
  web: http://www.wolfram.com
  email: info at wolfram.com
  phone: +1-217-398-0700 (U.S.)

Notebook reader applications are available free of charge from 
Wolfram Research.
***********************************************************************)

(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[      5498,        191]*)
(*NotebookOutlinePosition[      6395,        220]*)
(*  CellTagsIndexPosition[      6351,        216]*)
(*WindowFrame->Normal*)



Notebook[{
Cell[BoxData[
    \(Clear[x, a]\)], "Input"],

Cell[BoxData[
    \( (*\ 
      adding\ a\ genus\ \((\ g = 1\ Cardioid)\)\ effect\ to\ a\ noise\ as\ a\ 
        martingale*) \)], "Input"],

Cell[BoxData[
    \( (*\ seems\ to\ simulate\ effects\ like\ shot\ noise\ in\ transisters*) 
      \)], "Input"],

Cell[BoxData[
    \( (*\ by\ Roger\ L . \ Bagula\ 8\ july\ 2004  \[Copyright]\ *) \)], 
  "Input"],

Cell[BoxData[
    \( (*\ second\ derivative\ of\ projective\ line\ is\ a\ Cardioid*) \)], 
  "Input"],

Cell[BoxData[
    \(\(x2[s_] = \(4\ \((\(-1\) + 3\ s\^2)\)\)\/\((1 + s\^2)\)\^3; \)\)], 
  "Input"],

Cell[BoxData[
    \(\(y2[s_] = \(4\ s\ \((\(-3\) + s\^2)\)\)\/\((1 + s\^2)\)\^3; \)\)], 
  "Input"],

Cell[BoxData[
    \(\(r4 = Simplify[x2[s]^2 + y2[s]^2]; \)\)], "Input"],

Cell[BoxData[
    \(\(g3 = ParametricPlot[{x2[s], y2[s]}, {s, \(-10\), 10}]; \)\)], "Input"],

Cell[BoxData[
    \(\(g4 = 
      ParametricPlot[{2*\((1 - 1.25*Cos[s])\)*Cos[s] + 0.5, 
          2*\((1 - 1.25*Cos[s])\)*Sin[s]}, {s, \(-10\), 10}]; \)\)], "Input"],

Cell[BoxData[
    \(Show[g3, g4]\)], "Input"],

Cell[BoxData[
    \( (*\ solution\ of\ approximate\ cardioid\ in\ s*) \)], "Input"],

Cell[BoxData[
    \(Solve[
      2*\((1 - 1.25*a)\)*b*\((1 + s\^2)\)\^3 - 4\ s\ \((\(-3\) + s\^2)\) == 
        0, s]\)], "Input"],

Cell[BoxData[
    \( (*\ 
      function\ made\ from\ solutions\ as\ the\ projective\ line\ of\ a\ 
        \(cardioid : \ 
          second\ function\ taken\ as\ closer\ to\ the\ cardioid\)*) \)], 
  "Input"],

Cell[BoxData[
    \(\(s[a_, b_] = 
      Root[\(-0.8`\)\ b + 1.`\ a\ b - 4.80000000000000071`\ #1 - 
            2.40000000000000035`\ b\ #1\^2 + 3.`\ a\ b\ #1\^2 + 
            1.60000000000000008`\ #1\^3 - 2.40000000000000035`\ b\ #1\^4 + 
            3.`\ a\ b\ #1\^4 - 0.8`\ b\ #1\^6 + 1.`\ a\ b\ #1\^6&, 1]; \)\)], 
  "Input"],

Cell[BoxData[
    \(Plot[s[Cos[2*Pi*t], Sin[2*Pi*t]], {t, \(-1\), 1}]\)], "Input"],

Cell[BoxData[
    \( (*\ 
      plot\ of\ line\ random\ to\ Gaussian\ distribution\ height\ at\ that\ 
          point\ on\ the\ line\ of\ a\ Cardioid\ random\ instead\ of\ a\ 
          \(circle\  : \ a\) = \ Cos[2*Pi*Random[]], b = Sin[2*Pi*Random[]]*) 
      \)], "Input"],

Cell[BoxData[
    \( (*\ Gauss_dist[] = Exp[\(-s[a, b]^2\)/2]/Sqrt[2*Pi]\ *) \)], "Input"],

Cell[BoxData[
    \(SeedRandom[]\)], "Input"],

Cell[BoxData[
    \(\(noise = 
      Table[Re[Exp[\(-s[Cos[2*Pi*Random[]], Sin[2*Pi*Random[]]]^2\)/2]/
            Sqrt[2*Pi]], {n, 1, 500}]; \)\)], "Input"],

Cell[BoxData[
    \(ListPlot[noise, PlotRange -> All, \ PlotJoined -> \ True]\)], "Input"],

Cell[BoxData[
    \(\(noiseI = 
      Table[Im[Exp[\(-s[Cos[2*Pi*Random[]], Sin[2*Pi*Random[]]]^2\)/2]/
            Sqrt[2*Pi]], {n, 1, 500}]; \)\)], "Input"],

Cell[BoxData[
    \(ListPlot[noiseI, PlotRange -> All, \ PlotJoined -> \ True]\)], "Input"],

Cell[BoxData[
    \(\(noiseAbs = 
      Table[Abs[
          Exp[\(-s[Cos[2*Pi*Random[]], Sin[2*Pi*Random[]]]^2\)/2]/
            Sqrt[2*Pi]], {n, 1, 500}]; \)\)], "Input"],

Cell[BoxData[
    \(ListPlot[noiseAbs, PlotRange -> All, \ PlotJoined -> \ True]\)], "Input"],

Cell[BoxData[
    \(\(ba = Table[Floor[2500*noiseAbs[\([n]\)]], {n, 1, 500}]; \)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(b0 = \(Dimensions[ba]\)[\([1]\)]\)], "Input"],

Cell[BoxData[
    \(500\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(bmax = Max[ba]\)], "Input"],

Cell[BoxData[
    \(4252\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(bmin = Min[ba]\)], "Input"],

Cell[BoxData[
    \(0\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(c = Table[Count[ba, n], {n, Floor[bmin], bmax}]; \)\)], "Input"],

Cell[BoxData[
    \(ListPlot[c, PlotJoined -> True]\)], "Input"]
},
FrontEndVersion->"Macintosh 3.0",
ScreenRectangle->{{0, 1920}, {0, 1060}},
WindowSize->{1294, 878},
WindowMargins->{{72, Automatic}, {Automatic, 13}},
PrintingCopies->1,
PrintingPageRange->{1, Automatic},
MacintoshSystemPageSetup->"\<\
00/0004/0B`000003509H?ocokD"
]


(***********************************************************************
Cached data follows.  If you edit this Notebook file directly, not using
Mathematica, you must remove the line containing CacheID at the top of 
the file.  The cache data will then be recreated when you save this file 
from within Mathematica.
***********************************************************************)

(*CellTagsOutline
CellTagsIndex->{}
*)

(*CellTagsIndex
CellTagsIndex->{}
*)

(*NotebookFileOutline
Notebook[{
Cell[1709, 49, 44, 1, 27, "Input"],
Cell[1756, 52, 139, 3, 27, "Input"],
Cell[1898, 57, 112, 2, 27, "Input"],
Cell[2013, 61, 98, 2, 27, "Input"],
Cell[2114, 65, 101, 2, 27, "Input"],
Cell[2218, 69, 99, 2, 47, "Input"],
Cell[2320, 73, 99, 2, 47, "Input"],
Cell[2422, 77, 71, 1, 27, "Input"],
Cell[2496, 80, 92, 1, 27, "Input"],
Cell[2591, 83, 166, 3, 27, "Input"],
Cell[2760, 88, 45, 1, 27, "Input"],
Cell[2808, 91, 83, 1, 27, "Input"],
Cell[2894, 94, 130, 3, 32, "Input"],
Cell[3027, 99, 209, 5, 27, "Input"],
Cell[3239, 106, 331, 6, 66, "Input"],
Cell[3573, 114, 82, 1, 27, "Input"],
Cell[3658, 117, 275, 5, 43, "Input"],
Cell[3936, 124, 90, 1, 27, "Input"],
Cell[4029, 127, 45, 1, 27, "Input"],
Cell[4077, 130, 157, 3, 27, "Input"],
Cell[4237, 135, 90, 1, 27, "Input"],
Cell[4330, 138, 158, 3, 27, "Input"],
Cell[4491, 143, 91, 1, 27, "Input"],
Cell[4585, 146, 172, 4, 27, "Input"],
Cell[4760, 152, 93, 1, 27, "Input"],
Cell[4856, 155, 93, 1, 27, "Input"],

Cell[CellGroupData[{
Cell[4974, 160, 65, 1, 27, "Input"],
Cell[5042, 163, 37, 1, 26, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[5116, 169, 47, 1, 27, "Input"],
Cell[5166, 172, 38, 1, 26, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[5241, 178, 47, 1, 27, "Input"],
Cell[5291, 181, 35, 1, 26, "Output"]
}, Open  ]],
Cell[5341, 185, 86, 1, 27, "Input"],
Cell[5430, 188, 64, 1, 27, "Input"]
}
]
*)




(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)







-- 
Respectfully, Roger L. Bagula
tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
URL :  http://home.earthlink.net/~tftn
URL :  http://victorian.fortunecity.com/carmelita/435/ 


  • Prev by Date: Re: Excel formulas --> Mathematica
  • Next by Date: Fit : complex data to complex function, coefficients must be real
  • Previous by thread: Re: what kind of a programming language is Mathematica?
  • Next by thread: Fit : complex data to complex function, coefficients must be real