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Finding the magnitude of a complex exponential function

  1. Apr 20, 2013 #1
    1. The problem statement, all variables and given/known data
    I want to know the steps involved in finding the magnitude of a complex exponential function. An example of the following is shown in this picture:

    T2ASrR5.png


    2. Relevant equations
    |a+jb|=sqrt(a^2+b^2)

    |x/y|=|x|/|y|

    3. The attempt at a solution

    For the denominator, I replaced z with e^jw and used euler identity to expand the terms.

    1-2rcos(w)e^(-jw)+r^2e^(-2jw)

    1-rcos(w)*(cos(w)-jsinw)+.5r^2(cos(2w)-jsin(2w))

    After simplifying I get:

    1-rcos^2(w)+.5r^2cos(2w) + j(rsin(w)cos(w)-.5r^2sin(2w)

    from there I let a=1-r(cos^2(w)-.5r^2cos(2w)) and b=r(sin(w)cos(w)-.5rsin(2w))

    and using wolfram alpha to solve for sqrt(a^2+b^2) I don't get the simplified expression shown in the picture above. Am I approaching the problem correctly?
     
  2. jcsd
  3. Apr 20, 2013 #2

    mfb

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    Staff: Mentor

    If z=e^(jw), you can use this directly in the first fraction, to get the factor of (1-r). The other factor is easy to handle then.
     
  4. Apr 20, 2013 #3
    Ah, I see. Thank you very much. I got the answer =)
     
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