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Re: Finding a sine wave

  • To: mathgroup at smc.vnet.net
  • Subject: [mg95512] Re: Finding a sine wave
  • From: dh <dh at metrohm.com>
  • Date: Tue, 20 Jan 2009 05:48:56 -0500 (EST)
  • References: <gkppq1$dmn$1@smc.vnet.net>


Hi Hug,

I think a direct method is much simpler than numerical iterations.

Your model is: c0 Sin[c1 t+c2]

You have 3 data points for 3 times. Therefore we can get rid of c0 by 

division e.g. the first 2 data points by the last. This leaves 2 data 

points and the model: Sin[c1 t+c2]. The data values must be between 

-1..1, this shows that there is a constrain on the original values. Now 

we take the ArcSin of the data values, this gives a model: c1 t + c2. 

Now the fit is trivial. c0 can be obtained from the original data 

values. Here is an example:

d0 = {0.45, 0.86};

t = {0.5, 1.2}

d = ArcSin /@ d0;

cs = {c1, c2} /.

    Solve[{c1 t[[1]] + c2 == d[[1]], c1 t[[2]] + c2 == d[[2]]}, {c1,

       c2}][[1]];

Plot[Sin[cs.{t, 1}], {t, 0, 2 Pi/cs[[1]]},

  Prolog -> {Red, Point[Transpose[{t, d0}]]}]

hope this helps, Daniel



Hugh Goyder wrote:

> An experiment will give me three coordinates which lie on a sine wave.

> I have to find the sine wave efficiently. There are three unknowns the

> sine wave amplitude, A, the wavelength of the sine wave, L,  and the

> phase, ph. I also know that the wavelength is larger than the interval

> containing my measurement points. I think this condition removes

> possible multiple solutions.

> 

>  Below I give two methods which need improving. The first method uses

> FindRoot. This methods works about 60% of the time. I give six

> examples where it fails. The failure may be due to poor initial

> guesses. In the second method I use FindFit. This is not quite the

> correct method because I have an equal number of equations and

> unknowns. Some of the failures here, I think, are due to there being

> no error for the algorithm to work with. I give examples of failures.

> I have also tried FindInstance, Reduce and NSolve but I don't think

> these are  appropriate.

> 

> Here are some questions

> 

> 1. I would really like a symbolic solution rather than an iterative

> one. Is such a solution possible?

> 2. Can anyone improve on the methods below to make them more robust?

> 3. I have some control over my x-locations. How can I work out best x

> locations given an estimate of the wavelength L?

> 

> Many thanks for all answers.

> 

> 

> xx = Sort[Join[{0}, RandomReal[{0, 10}, 2]]]

> 

> yy = RandomReal[{-25, 25}, 3]

> 

> sol = FindRoot[{yy[[1]] - A*Sin[ph],

>    yy[[2]] - A*Sin[2*Pi*(xx[[2]]/L) + ph],

>        yy[[3]] - A*Sin[2*Pi*(xx[[3]]/L) + ph]}, {{A,

>     Max[Abs[yy]]}, {L, Max[xx]}, {ph, 0}}]

> 

> Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10},

>    Epilog -> {PointSize[0.02], (Point[#1] & ) /@

>     Transpose[{xx, yy}]}, PlotRange -> {{0, 10}, {-25, 25}}]

> 

> FindRootFailures = {{{0, 3.781264462608982,

>      3.797055100562352}, {-22.948348087068737, 1.4581744078038472,

>            -14.242676740704574}}, {{0, 4.424069570670131,

>      8.861716396743098}, {-6.444538773775843, -10.787309362608688,

>            15.579334094330942}}, {{0, 4.424069570670131,

>      8.861716396743098}, {-5.944536471563289, 10.627536107497393,

>            -21.294232497202316}}, {{0, 1.3680556047878967,

>      8.546115267250002}, {4.128528863849845, 2.9017293848933923,

>            -22.51610539815371}}, {{0, 1.3738869718371616,

>      5.689309423462079}, {20.553993773523437, -16.972841620064592,

>            16.61185061676568}}, {{0, 1.0408828831133632,

>      8.484645515699821}, {-3.7267589861478045, 4.1016610850387,

>            -24.600635061804443}}};

> 

> (({xx, yy} = #1;

>     sol = FindRoot[{yy[[1]] - A*Sin[ph],

>        yy[[2]] - A*Sin[2*Pi*(xx[[2]]/L) + ph],

> 

>        yy[[3]] - A*Sin[2*Pi*(xx[[3]]/L) + ph]}, {{A,

>         Max[Abs[yy]]}, {L, Max[xx]}, {ph, 0}}];

>         Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10},

> 

>      Epilog -> {PointSize[0.02], (Point[#1] & ) /@

>         Transpose[{xx, yy}]},

>      PlotRange -> {{0, 10}, {-25, 25}}]) & ) /@

>    FindRootFailures

> 

> xx = Sort[Join[{0}, RandomReal[{0, 10}, 2]]]

> 

> yy = RandomReal[{-25, 25}, 3]

> 

> sol = FindFit[

>   Transpose[{xx, yy}], {A*Sin[2*Pi*(x/L) + ph],

>    Max[Abs[xx]] < L}, {A, L, ph}, x]

> 

> Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10},

>    Epilog -> {PointSize[0.02], (Point[#1] & ) /@

>     Transpose[{xx, yy}]}, PlotRange -> {{0, 10}, {-25, 25}}]

> 

> FindFitFailures = {{{0, 2.463263668380835,

>      4.3190892163093615}, {-1.8407827676541144, -8.736574079785198,

>            12.661520984622932}}, {{0, 3.894521446091823,

>      9.937403619870642}, {12.381369822165155, 15.840399165432128,

>            -6.914634137727327}}, {{0, 8.087369725271945,

>      8.343899312282815}, {24.73795103895976, -13.396248713970305,

>            7.150470311065216}}, {{0, 1.5480031866178834,

>      9.575255260205617}, {-8.163720246784278, 13.373468882892958,

>            -18.018462091502098}}, {{0, 0.6152784485896601,

>      0.818296772602134}, {-1.858698140836046, 3.695113783904491,

>            -1.3186989232026658}}, {{0, 4.743093154057316,

>      6.028314583406327}, {-16.60893277597446, -17.413392198343093,

>            -18.54081837965986}}};

> 

> (({xx, yy} = #1;

>     sol = FindFit[

>       Transpose[{xx, yy}], {A*Sin[2*Pi*(x/L) + ph],

>        Max[Abs[xx]] < L}, {A, L, ph}, x];

>         Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10},

> 

>      Epilog -> {PointSize[0.02], (Point[#1] & ) /@

>         Transpose[{xx, yy}]},

>      PlotRange -> {{0, 10}, {-25, 25}}]) & ) /@

>    FindFitFailures

> 

> 

> 

> 




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