Fourier Series

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Fourier Series

• 3.1-3.3
• Kamen and Heck

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3.1 Representation of Signals in Terms of
Frequency Components

• x(t) ?k1,N Ak cos(?kt ?k), -?lt t lt ?
• Fourier Series Representation
• Amplitudes
• Frequencies
• Phases
• Example 31. Sum of Sinusoids
• 3 sinusoidsfixed frequency and phase
• Different values for amplitudes (see Fig.3.1-3.4)

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3.2 Trigonometric Fourier Series

• Let T be a fixed positive real number.
• Let x(t) be a periodic continuous-time signal
  with period T.
• Then x(t) can be expressed, in general, as an
  infinite sum of sinusoids.
• x(t)a0?k1,? akcos(k?0t)bksin(k?0t) ?lt t
  lt ? (Eq. 3.4)

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3.2 Trig Fourier Series (p.2)

• a0, ak, bk are real numbers.
• ?0 is the fundamental frequency (rad/sec)
• ?0 2?/T, where T is the fundamental period.
• ak 2/T ?0T x(t) cos(k?0t) dt, k1,2, (3.5)
• bk 2/T ?0T x(t) sin(k?0t) dt, k1,2,... (3.6)
• a0 1/T ?0T x(t) dt, k 1,2, (3.7)
• The with phase form3.8,3.9,3.10.

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3.2 Trig Fourier Series (p.3)

• Conditions for Existence (Dirichlet)
• 1. x(t) is absolutely integrable over any period.
• 2. x(t) has only a finite number of maxima and
  minima over any period.
• 3. x(t) has only a finite number of
  discontinuities over any period.

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3.2 Trig Fourier Series (p.4)

• Example 3.2 Rectangular Pulse Train
• 3.2.1 Even or Odd Symmetry
• Equations become 3.13-3.18.
• Example 3.3 Use of Symmetry (Pulse Train)
• 3.2.2 Gibbs Phenomenon
• As terms are added (to improve the approximation)
  the overshoot remains approximately 9.
• Fig. 3.6, 3.7, 3.8

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3.3 Complex Exponential Series

• x(t) ?k-?,? ck exp(j?0t), -?lt t lt ? (3.19)
• Equations for complex valued coefficients 3.20
  3.24.
• Example 3.4 Rectangular Pulse Train
• 3.3.1Line Spectra
• The magnitude and phase angle of the complex
  valued coefficients can be plotted vs.the
  frequency.
• Examples 3.5 and 3.6

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3.3 Complex Exponential Series (p.2)

• 3.3.2 Truncated Complex Fourier Series
• As with trigonometric Fourier Series, a truncated
  version of the complex Fourier Series can be
  computed.
• 3.3.3 Parsevals Theorem
• The average power P, of a signal x(t), can be
  computed as the sum of the magnitude squared of
  the coefficients.