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2.6 The fourier series: exercises

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  1. Show that an even-symmetric periodic signal has Fourier Series coefficients b n = 0 while an odd-symmetric signal has a n = 0 .
  2. Find the trigonometric form of the Fourier Series of the periodic signal shown in [link] .
  3. Find the trigonometric form of the Fourier Series for the periodic signal shown in [link] .
  4. Find the trigonometric form of the Fourier Series for the periodic signal shown in [link] for τ = 1 , T = 10 .
  5. Suppose that x ( t ) = 5 + 3 cos ( 5 t ) - 2 sin ( 3 t ) + cos ( 45 t ) .
    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 [link] .
  7. Find the complex form of the Fourier Series of the periodic signal shown in [link] .
  8. Find the complex form of the Fourier Series for the signal in [link] using:
    1. τ = 1 , T = 10 .
    2. τ = 1 , T = 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
    1 T ∫ t 0 t 0 + T ∑ n ≠ m c n c m * e j ( n - m ) Ω 0 t d t = 0
Signal for problems [link] and [link] .
Signal for problem [link] and [link] .
Pulse train signal for problems [link] and [link] .
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Source:  OpenStax, Signals, systems, and society. OpenStax CNX. Oct 07, 2012 Download for free at http://cnx.org/content/col10965/1.15
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