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Discrete Fourier transform of sampled continuous signal

  1. Aug 25, 2011 #1
    1. The problem statement, all variables and given/known data
    Let a system that converts a continuos-time signal to a discrete-time signal. The input x(t) is periodic with period of 0.1 second. The Fourier series coefficients of x(t) are [tex]X_k = \displaystyle\left(\frac{1}{2}\right)^{|k|}[/tex]. The ideal lowpass filter [itex]H(\omega)[/itex] is equal to 0 for [itex]\left|\omega\right| > 205\pi[/itex]. The sampling period is T = 0.005 seconds.

    Determine the Fourier series coefficients of x[n].


    2. Relevant equations

    [itex]X\left(\Omega\right) = X_s\left(\omega\right), \omega = \Omega/Ts[/itex]

    3. The attempt at a solution
    The Fourier transform of [itex]X\left(\omega\right) = 2\pi\displaystyle\sum_{k=-\infty}^{\infty} X_k\delta(\omega-20\pi k)[/itex].

    The output of the filter is [itex]X_c\left(\omega\right) = 2\pi\displaystyle\sum_{k=-10}^{10} X_k\delta(\omega-20\pi k)[/itex] and the last impulse has [itex]\omega = 200\pi[/itex].

    When [itex]X_c(\omega)[/itex] is multiplied by [itex]P(\omega)[/itex], I obtain [itex]X_s(\omega) = \displaystyle\frac{1}{T_s}\sum_{k=-\infty}^{\infty} X_c\left(\omega-\omega_s k\right)[/itex]

    With the expression written in 2., I've the Fourier transform of x[n].

    So, I think that [itex]X_k = \displaystyle\frac{1}{T_s}\left(\frac{1}{2}\right)^{|k|},\;|k| = 0, 1, 2,...10[/itex].

    Is this correct?

    Thank you.
     
  2. jcsd
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