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Re: Want a smooth function from Arg[ ]

Ryan Scott wrote:
> I need to be able to fit an nth degree polynomial to the phase function
> for  a list of data.  However, the Arg[x] function wraps the phase at
> +/- Pi  creating discontinuities in what would be a smooth function.
> To clearify, let me give the following example:
> Create a list of data with a  gaussian profile and quadratic phase:
> nn=128;
> a=0.005; b=0.003;
> data=Table[N[Exp[-a*(t-nn/2)^2+I*b*(t-nn/2)^2]], {t,nn}];
> Plot the magnitude and phase:
> ListPlot[Abs[data],PlotRange->{0,1},PlotJoined->True,
>   PlotLabel->FontForm[
>       "Absolute value of waveform",{"Helvetica-Bold",10}]]
> ListPlot[Arg[data], PlotJoined->True, PlotRange->All,
>    PlotLabel->FontForm[ "Phase of time waveform",
> {"Helvetica-Bold",10}]]
> I would like to now fit a curve to this phase (which should be 0.003*x^2
> ).
>  However, the wrapping of the phase would seem to make this difficult to
> do  directly.  Any ideas on how to manipulate the results of Arg to
> give a  continuous function of data?
> Thanks,
> Ryan
> --
> ________________________________________________
>   Ryan P. Scott
>   Laser and Electro-Optics Research Group
>   UC Davis - Department of Applied Science
>   Tel: (530)754-4358  Fax: (530)752-1652
>   Email:
> ________________________________________________

One approach might be to renormalize the arguments. You can do this by
comparing Arg[point] to Arg[previous point]. Any time you see a jump
larger than, say, Pi, then shift successive arg values by Pi.

args = Arg[data];
renormalize[args_] := Module[
	{pairs, diffs, j, len=Length[args], corr=0},
	pairs = Partition[args,2,1];
	diffs = Map[#[[1]]-#[[2]]&,pairs];
	PrependTo[diffs, 0];
	diffs = 2*Pi*Sign[Chop[diffs,Pi]];
		corr += diffs[[j]];
newargs = renormalize[args];
ListPlot[newargs, PlotJoined->True, PlotRange->All,
	PlotLabel->FontForm["Phase of time waveform"]]

This gives a smooth parabola as desired.

Daniel Lichtblau
Wolfram Research

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