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Curve Fitting

  1. Aug 25, 2004 #1
    hi,
    I face the following problem.
    I need to find the best values of the parameters [itex]a,b,c[/itex]
    of the complex function [itex]f(x)=a+\frac{b-a}{1+j x c}[/itex] of the real
    variable [itex]x[/itex] where ([itex]j^2=-1[/itex])
    such that
    [itex]f(2 \pi 10^6)=2.33-j 1.165 10^{-3}[/itex] and
    [itex]f(2 \pi 10^{10})=2.347-j 3.7552 10^{-3}[/itex].

    It seems to be a curve fitting problem but the function [itex]f(x)[/itex] is complex!
     
  2. jcsd
  3. Aug 25, 2004 #2

    arildno

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    True enough; you have three constants to optimize to 4 restraints.
    What's your problem?
     
  4. Aug 25, 2004 #3

    HallsofIvy

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    Oh, those engineers and their jmaginary numbers!
     
  5. Aug 26, 2004 #4

    arildno

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    To get you started:
    1. Define:
    [tex]x_{0}=2\pi{10}^{6}[/tex]
    [tex]x_{1}=2\pi{10}^{10}[/tex]
    Ask yourself:
    Why have you been given so huge arguments?
    In particular, can I use that fact to my advantage later on?

    2.Rewrite:
    [tex]f(x)=a+\frac{b-a}{1+jxc}=a+\frac{1-jxc}{1-jxc}\frac{b-a}{1+jxc}=\frac{ax^{2}c^{2}+b}{1+x^{2}c^{2}}+j\frac{(a-b)xc}{1+x^{2}c^{2}}[/tex]
    3. Requirements of curve fitting:
    [tex]\frac{ax_{0}^{2}c^{2}+b}{1+x_{0}^{2}c^{2}}\approx{2.33}[/tex]
    [tex]\frac{ax_{1}^{2}c^{2}+b}{1+x_{1}^{2}c^{2}}\approx{2.347}[/tex]
    [tex]\frac{(a-b)x_{0}c}{1+x_{0}^{2}c^{2}}\approx{-1.16510*10^{-3}}[/tex]
    [tex]\frac{(a-b)x_{1}c}{1+x_{1}^{2}c^{2}}\approx{-3.755210*10^{-3}}[/tex]
    4. Define:
    [tex]y_{0r}=2.33,y_{1r}=2.347,y_{0i}=-1.16510*10^{-3},y_{1i}=-3.755210*10^{-3}[/tex]
    5. Define:
    [tex]\hat{y}_{0r}=\frac{ax_{0}^{2}c^{2}+b}{1+x_{0}^{2}c^{2}}[/tex]
    [tex]\hat{y}_{1r}=\frac{ax_{1}^{2}c^{2}+b}{1+x_{1}^{2}c^{2}}[/tex]
    [tex]\hat{y}_{0i}=\frac{(a-b)x_{0}c}{1+x_{0}^{2}c^{2}}[/tex]
    [tex]\hat{y}_{1i}=\frac{(a-b)x_{1}c}{1+x_{1}^{2}c^{2}}[/tex]

    6. Construct:
    [tex]S(a,b,c)=(y_{0r}-\hat{y}_{0r})^{2}+(y_{0i}-\hat{y}_{0i})^{2}+(y_{1r}-\hat{y}_{1r})^{2}+(y_{1i}-\hat{y}_{1i})^{2}[/tex]

    Clearly, S>=0, and S=0 if and only if the curve fitting is exact.
    We are interested in the choice of (a,b,c) such that a minimum of S is found.
    Hence, we should consider the system of 3 equations:
    [tex]\frac{\partial{S}}{\partial{a}}=0[/tex]
    [tex]\frac{\partial{S}}{\partial{b}}=0[/tex]
    [tex]\frac{\partial{S}}{\partial{c}}=0[/tex]

    This system can (theoretically, at least!) be solved for minimizing values
    [tex](a_{m},b_{m},c_{m})[/tex]
    To find a simple, approximate solution to the system of equations, I suggest that you utilize your knowledge that [tex](x_{0},x_{1})[/tex] are huge numbers.
    Good luck!
    NOTE:
    This is just one of many techniques to derive curve-fitting coefficients.
    It is by no means clear that this technique provides the simplest system to solve for coefficients (a,b,c). Look up in a numerical analysis book (or something like that) to get other ideas..
     
    Last edited: Aug 26, 2004
  6. Aug 26, 2004 #5

    Hahaha :rofl:
     
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