Signal Representation by Fourier Series FS

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• Chapter 3
• Signal Representation by Fourier Series (FS)
• 3-1 Trigonometric Fourier Series (TFS)
• 3-1-1 Basic Form
• A signal f(t) can be expressed by FS over an
  interval

• In the following, the subscript 0 may be
  ignored

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• Coefficients can be determined by using the
  classic method (see the following slides, or p.
  223-224, textbook)

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• Geometrical explanation

• If executes one cycle during any
  interval of T seconds, then
• executes (nm) cycles in T
  seconds (recall the scaling concept)

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• For the same reason, we have

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• Using the above results, we obtain

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• 3-1-2 Compact TFS
• The sine and cosine terms can be combined into a
  single term

• The compact form is

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• We have shown that an arbitrary f(t) may be
  expressed as a TFS over any interval of T seconds
  
• Example 3.3 (pp. 191 - 192)
• The found FT is equal to over the
  interval only
• What happens to the FT outside this interval?

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• 3-1-3 Periodicity of the TFS
• Recall

• Let

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• 3-1-4 TFS of Periodic Signals
• How about if f(t) itself is periodic with T?
• The FT of f(t) over T will also represent f(t)
  for all t

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• 3-1-5 Fourier Spectra of Periodic Signals
• Amplitude spectrum
• Phase spectrum
• These two plots together are the frequency
  spectra of f(t)
• This is called the frequency domain identity of
  signals

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• 3-1-6 FS Convergence at Jump Points
• If there is a jump point in f(t), then the
  FT of f(t) converges to an average, namely

• 3-1-7 Existence of FS
• Weak Dirichlet condition f(t) is absolutely
  integrable, i.e.
• Strong Dirichlet condition f(t) a finite number
  of
• finite discontinuities and
• maxima and minima

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• Example 3.4 (pp. 195 - 196)
• 3-1-8 Effect of Symmetry
• Recall for periodic signal

• How about if f(t) is even?
• How about if f(t) is odd?

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• A periodic signal f(t) has a half-wave symmetry
  (H. S.) if
• If f(t) has an H.S., then
• All the even-numbered terms of its FT will
  vanish
• There are simplified formulas for the
  coefficients (see Problem 3.4-7)
• Some relations