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The Fourier Series: Exercises

Module by: Carlos E. Davila. E-mail the author

  1. Show that an even-symmetric periodic signal has Fourier Series coefficients bn=0bn=0 while an odd-symmetric signal has an=0an=0.
  2. Find the trigonometric form of the Fourier Series of the periodic signal shown in Figure 1.
  3. Find the trigonometric form of the Fourier Series for the periodic signal shown in Figure 2.
  4. Find the trigonometric form of the Fourier Series for the periodic signal shown in Figure 3 for τ=1τ=1, T=10T=10.
  5. Suppose that x(t)=5+3cos(5t)-2sin(3t)+cos(45t)x(t)=5+3cos(5t)-2sin(3t)+cos(45t).
    1. Find the period of this periodic signal.
    2. Find the trigonometric form of the Fourier Series.
  6. Find the complex form of the Fourier Series of the periodic signal shown in Figure 1.
  7. Find the complex form of the Fourier Series of the periodic signal shown in Figure 2.
  8. Find the complex form of the Fourier Series for the signal in Figure 3 using:
    1. τ=1τ=1, T=10T=10.
    2. τ=1τ=1, T=100T=100.
    For each case plot the magnitude of the Fourier Series coefficients. You may use Matlab or some other programming language to do this.
  9. Show that
    1Tt0t0+Tnmcncm*ej(n-m)Ω0tdt=01Tt0t0+Tnmcncm*ej(n-m)Ω0tdt=0
    (1)
Figure 1: Signal for problems Item 2 and Item 6.
Figure 1 (vh2_1.png)
Figure 2: Signal for problem Item 3 and Item 7.
Figure 2 (ch2_2.png)
Figure 3: Pulse train signal for problems Item 4 and Item 8.
Figure 3 (hw2.png)

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