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Isolating variables from sine-function, I'm stuck

  1. Jul 2, 2013 #1
    Hey everybody

    I'm developing a computer program that can take a series of data-points and estimate the next local maximum(top) of a sine-curve.

    My approach to this is to take the formula for a sine-function([itex]f(x)=Asin(\omega x - \phi) + B[/itex]), and isolate all the variables, and that way I'll get a series of (more or less) simple equations, that I can use in my program.
    (I'm not entirely sure this is the right approach, let me know otherwise...)

    To do this we're using 4 equations with 4 unknown like this:
    • (1) [itex]y_1 = Asin(\omega x_1 - \phi) + B[/itex]
    • (2) [itex]y_2 = Asin(\omega x_2 - \phi) + B[/itex]
    • (3) [itex]y_3 = Asin(\omega x_3 - \phi) + B[/itex]
    • (4) [itex]y_4 = Asin(\omega x_4 - \phi) + B[/itex]

    Then we isolate B in (1) and substitute into (2), (3) & (4):
    [itex]y_1 = Asin(\omega x_1 - \phi) + B \Leftrightarrow[/itex]
    [itex]B = y_1 - Asin(\omega x_1 - \phi)[/itex]
    and
    • (2.2) [itex]y_2 = Asin(\omega x_2 - \phi) + y_1 - Asin(\omega x_1 - \phi)[/itex]
    • (3.2) [itex]y_3 = Asin(\omega x_3 - \phi) + y_1 - Asin(\omega x_1 - \phi)[/itex]
    • (4.2) [itex]y_4 = Asin(\omega x_4 - \phi) + y_1 - Asin(\omega x_1 - \phi)[/itex]

    Next, we isolate A from (2.2) and substitute into the other 2:
    [itex]y_2 = Asin(\omega x_2 - \phi) + y_1 - Asin(\omega x_1 - \phi) \Leftrightarrow[/itex]
    [itex]y_2 - y_1= Asin(\omega x_2 - \phi) - Asin(\omega x_1 - \phi) \Leftrightarrow[/itex]
    [itex]y_2 - y_1= A(sin(\omega x_2 - \phi) - sin(\omega x_1 - \phi)) \Leftrightarrow[/itex]
    [itex]A = \frac{y_2 - y_1}{sin(\omega x_2 - \phi) - sin(\omega x_1 - \phi)}[/itex]
    and
    • (3.3) [itex]y_3 = \frac{y_2 - y_1}{sin(\omega x_2 - \phi) - sin(\omega x_1 - \phi)}sin(\omega x_3 - \phi) + y_1 - \frac{y_2 - y_1}{sin(\omega x_2 - \phi) - sin(\omega x_1 - \phi)}sin(\omega x_1 - \phi)[/itex]
    • (4.3) [itex]y_4 = \frac{y_2 - y_1}{sin(\omega x_2 - \phi) - sin(\omega x_1 - \phi)}sin(\omega x_4 - \phi) + y_1 - \frac{y_2 - y_1}{sin(\omega x_2 - \phi) - sin(\omega x_1 - \phi)}sin(\omega x_1 - \phi)[/itex]

    Next we want to isolate [itex]\omega[/itex] from (3.3):
    [itex]y_3 = \frac{y_2 - y_1}{sin(\omega x_2 - \phi) - sin(\omega x_1 - \phi)}sin(\omega x_3 - \phi) + y_1 - \frac{y_2 - y_1}{sin(\omega x_2 - \phi) - sin(\omega x_1 - \phi)}sin(\omega x_1 - \phi) \Leftrightarrow[/itex]
    [itex]y_3 - y_1= \frac{y_2 - y_1}{sin(\omega x_2 - \phi) - sin(\omega x_1 - \phi)}sin(\omega x_3 - \phi) - \frac{y_2 - y_1}{sin(\omega x_2 - \phi) - sin(\omega x_1 - \phi)}sin(\omega x_1 - \phi)[/itex]

    And this is where I get stuck...
    I would think I could take [itex]sin^{-1}()[/itex] of everything, but I'm not sure if it's that simple.

    Any help is appreciated
    Thanks
    \\Batnas
     
  2. jcsd
  3. Jul 2, 2013 #2
    Hi !

    it shoud be slightly simpler if you start with the form :
    y = a*sin(w*x) +b*cos(w*x) +c
    where a = A*cos(phi) , b = -A*sin(phi) and c=B
    You will have a linear system of 4 equations considering (a, b, c) only.
    Then you could linearly combine those equations so that (a,b,c) be eliminated. The result will be an equation with the only remaining unknown w.
    But this equation is very big and non-linear relatively to w (it includes many sin and cos fonctions of many different linear functions of w.
    You cannot sovle it on an analytical form.
    Anyway, you will need a computer maths-package for numerical solving of non-linear équations.
    As a consequence, I think that it is simpler to use a computer maths-package able to solve numerical systems of equations and directly input with the original system of 4 equations, instead of first reducing the number of equations (which leads to more complicated formulas).
    Remark : If the whole problem is to find the optimised parametrers A,B,Phi,w (or a,b,B,w) from a large number of experimental data (x,y), there are some sinusoidal regression methods (seach on the WEB). For example, a non-recursive method is published in the pdf (algorithm pp.35-36): "Régressions et équations intégrales" : http://www.scribd.com/JJacquelin/documents
     
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