Signals, Fourier Series

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Signals, Fourier Series

• Analog and Digital Communications
• Autumn 2005-2006

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Review of Signals and Spectra

• A generic sinusoidal signal
• Phasor representation
• Frequency domain representation
• Rotating phasors
• Frequency plots
• Amplitude
• Phase

Amplitude
Phase
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Review of Signals and Spectra

• Two sided spectra can be seen from
• This represents two rotating phasors
• Amplitude and phase spectrum (two sided)

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Periodic Signals

• A signal is periodic if there exists such
  that
• Smallest such is called fundamental period
• Any integer multiple of is also a period

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Average signal and Power

• Average signal
• Average power
• For periodic signals, limit goes away

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Fourier Series Representation

• Projection of periodic signals onto basis
  functions
• Periodic signal is a weighted sum of these basis
  functions
• Exponentials are used as basis functions for
  writing Fourier series
• Any periodic signal can be expressed as a sum of
  infinite number of exponentials (or sinusoids for
  real signals)

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Fourier Series
(Each term represents one frequency)
(fundamental frequency)
Line spectra at frequencies that are integer
multiple of fundamental frequency
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Sinc Function
Note the points where this function becomes zero
Where does this appear?
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Fourier Series Example
Amplitude normalized by A/4
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Sinusoidal Fourier Series

• For real signals
• and can be expressed in terms of