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Fourier Series coefficient question

  1. Sep 15, 2012 #1
    I'm learning about the Fourier Series and have a few question.

    First, we have 2 methods to find the Fourier coefficient of the Fourier Series.

    1. Use the formula Cn below.

    9d7f73fbcba87cbff485e66646aa541d.png

    note that: x=2∏ft

    2. Expand the function.

    For Example I have y(t)=3cos(2∏9t)+4cos(2∏8t), let use the Expansion method

    y(t)=3cos(2∏9t)+4cos(2∏8t) <---- Already expand

    we have

    y(j2∏9t)=3 at f=9 Hz Fourier coefficient=3/2
    y(j2∏8t)=4 at f=8 Hz Fourier coefficient=4/2

    Note:I devided by 2 for two side analysis.

    This method makes sense to me.

    but when it comes to use of formula Cn, it becomes nonsense.

    9d7f73fbcba87cbff485e66646aa541d.png

    note that: x=2∏ft

    My question is how come multiply the whole function with sine & cosine at some

    particular frequency and integrate it. it turns into the Fourier coefficient.

    I mean why is that actually work? It's an area, how can that related to the coefficient.

    I want some intuition.

    Thank you!
     
  2. jcsd
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