Fourier Series

1
Fourier Series
2
Course Outline
Roberts, ch. 1-3

• Time domain analysis (lectures 1-10)
• Signals and systems in continuous and discrete
  time
• Convolution finding system response in time
  domain
• Frequency domain analysis (lectures 11-16)
• Fourier series
• Fourier transforms
• Frequency responses of systems
• Generalized frequency domain analysis (lectures
  17-26)
• Laplace and z transforms of signals
• Tests for system stability
• Transfer functions of linear time-invariant
  systems

Roberts, ch. 4-7
Roberts, ch. 9-12
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Periodic Signals

• For some positive constant T0
• f(t) is periodic if f(t) f(t T0) for all
  values of t ? (-?, ?)
• Smallest value of T0 is the period of f(t)
• A periodic signal f(t)
• Unchanged when time-shifted by one period
• May be generated by periodically extending one
  period
• Area under f(t) over any interval of duration
  equal to the period is same e.g., integrating
  from 0 to T0 would give the same value as
  integrating from T0/2 to T0 /2

4
Sinusoids

• Fundamental f1(t) C1 cos(2 p f0 t q1)
• Fundamental frequency in Hertz is f0
• Fundamental frequency in rad/s is w0 2 p f0
• Harmonic fn(t) Cn cos(2 p n f0 t qn)
• Frequency, n f0, is nth harmonic of f0
• Magnitude/phase and Cartesian representations
• Cn cos(n w0 t qn) Cn cos(qn) cos(n w0 t) -
  Cn sin(qn) sin(n w0 t) an cos(n w0 t)
  bn sin(n w0 t)

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Fourier Series

• General representationof a periodic signal
• Fourier seriescoefficients
• Compact Fourierseries

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Existence of the Fourier Series

• Existence
• Convergence for all t
• Finite number of maxima and minima in one period
  of f(t)
• What about periodic extensions of

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Example 1

• Fundamental period
• T0 2
• Fundamental frequency
• f0 1/T0 1/2 Hz
• w0 2p/T0 p rad/s

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Example 2

• Fundamental period
• T0 2p
• Fundamental frequency
• f0 1/T0 1/(2p) Hz
• w0 2p/T0 1 rad/s