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Math Help - Fourier analysis

  1. #1
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    Fourier analysis

    Was given this proof to do for a tutorial:

    Prove that the non-zero frequency spectral lines of the periodic unipolar square-wave pulse train, figure 1 where V = 1 volt, are indeed at the same amplitude as the amplitude of the Fourier Transform of a single pulse from within the pulse train.

    Thanks in advance!
    Attached Thumbnails Attached Thumbnails Fourier analysis-image2.jpg  
    Last edited by MathGuru; February 20th 2006 at 03:25 PM.
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    anyone??
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  3. #3
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    Quote Originally Posted by Yaris
    Was given this proof to do for a tutorial:

    Prove that the non-zero frequency spectral lines of the periodic unipolar square-wave pulse train, figure 1 where V = 1 volt, are indeed at the same amplitude as the amplitude of the Fourier Transform of a single pulse from within the pulse train.

    Thanks in advance!
    Depending how you define things, the n-th harmonic of the square wave
    (I will assume unit amplitude for the square wave
    multiplying by a positive constant has no effect on what follows - other
    than increasing every thing by a constant factor that is )
    has complex amplitude :

    <br />
A_n=\int_0^2 \chi_{[0,1]}(t) \exp(2 \pi\  \bold{i}\  n f_0 t) dt<br />
,

    where \chi_{[0,1]} is the characteristic function of the interval
    [0,1].

    So:

    <br />
A_n=\int_0^1  \exp(2 \pi\  \bold{i}\  n f_0 t) dt<br />
,

    and this corresponds to a nominal frequency nf_0, where f_0 is 1/2 Hz.

    Now the Ft of a single pusle (starting at t=0) is:

    <br />
\mathcal{F}\chi_{[0,1]}(f)=\int_{-infty}^{infty} \chi_{[0,1]}(t) \exp(2 \pi\  \bold{i}\  f t) dt<br />
.

    So

    <br />
\mathcal{F}\chi_{[0,1]}(f)=\int_0^1 \exp(2 \pi\  \bold{i}\  f t) dt<br />
.

    Hence:

    <br />
\mathcal{F}\chi_{[0,1]}(n f_0)=A_n<br />

    Now this is for the single pulse starting from t=0, if it starts elsewhere we
    have a translation which corresponds to multiplying the Ft by a complex
    number of unit modulus so the moduli of the amplitudes are still equal.

    RonL

    Note I say depending on how you define things because the definition of
    what is the forward and backward transform varies from author to author,
    as does where one places the normalising constants and so does the
    convention of angular frequency of normal frequency.
    Last edited by CaptainBlack; February 28th 2006 at 10:02 PM.
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