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Checking for interference equation - two wave mixing

  1. Sep 14, 2010 #1
    1. The problem statement, all variables and given/known data
    Not too sure if my problem falls under this category. Hope I have done all formatting correctly

    Basically I am seeking some confirmation on my maths derivation and would like to enquire why there is discrepancies when we expressed the result in Fourier Spectrum.


    2. Relevant equations

    The fundamental equation I am working on is
    [tex]\left|Z_{1}+Z_{2}exp\left(-jw_{c}t\right)\right|^{2}[/tex]

    By expanding out the equation, I would get
    [tex]\left|Z_{1}+Z_{2}exp\left(-jw_{c}t\right)\right|^{2}[/tex]
    [tex]=\left(Z_{1}+Z_{2}exp\left(-jw_{c}t\right)\right)\left(Z_{1}+Z_{2}exp\left(-jw_{c}t\right)\right)^{*}[/tex]

    [tex]=\left|Z_{1}\right|^{2}+ \left|Z_{2}\right|^{2}+Z_{1}Z_{2}^{*}exp\left(jw_{c}t\right) +Z_{2}Z_{1}^{*}exp\left(-jw_{c}t\right) [/tex]

    Using [tex]AB^{*}+A^{*}B = 2\Re\left(AB^{*}\right)[/tex],
    [tex]=\left|Z_{1}\right|^{2}+\left|Z_{2}\right|^{2}+2Re\left(Z_{1}Z_{2}^{*}exp\left(jw_{c}t\right)\right)[/tex]

    Let's Call this equation 1,

    Expanding out, we have
    [tex]=\left|Z_{1}\right|^{2}+ \left|Z_{2}\right|^{2}+2\left[Re\left(Z_{1}Z_{2}^{*}\right)cos\left(w_{c}t\right)-Im\left(Z_{1}Z_{2}^{*}\right)sin\left(w_{c}t\right)\right][/tex]

    For the 3rd Equation, we use another property,
    [tex]Acos\left(w_{c}t\right)-Bsin\left(w_{c}t\right)[/tex]
    [tex]= \sqrt{A^{2}+B^{2}}cos\left(w_{c}t + \theta\right)[/tex]

    we have the following 3rd equation
    [tex]=\left|Z_{1}\right|^{2}+ \left|Z_{2}\right|^{2}+2Re\left|Z_{1}Z_{2}^{*}\right|cos\left(w_{c}t+\theta\right)[/tex]

    Hope my derivation is correct


    3. The attempt at a solution
    However, if we convert them by applying Fourier Transform, they seem to be unequal

    [tex]F\left[ Eq1\right] = 2F\left(Re\left( Z_{1}Z_{2}^{*}\right)\right)\otimes \delta\left(f+f_{c}\right)[/tex]

    [tex]F\left[ Eq2\right] = \frac{2}{2}F\left(Re\left( Z_{1}Z_{2}^{*}\right)\right)\otimes \left[\delta\left(f+f_{c}\right)+\delta\left(f-f_{c}\right) \right] +\frac{2}{j}F\left(Im\left( Z_{1}Z_{2}^{*}\right)\right)\otimes \left[\delta\left(f+f_{c}\right)-\delta\left(f-f_{c}\right) \right][/tex]

    [tex]F\left[ Eq3\right] = 2F\left(\left| Z_{1}Z_{2}^{*}\right|\right)\otimes \left[\delta\left(f+f_{c}+\theta\right) + \delta\left(f-f_{c}+\theta\right)\right][/tex]
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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