Interpretation of Fourier series as an expansion on an orthonormal basis'

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Lecture 13

• Interpretation of Fourier series as an expansion
  on an orthonormal basis.
• RMS value of a periodic function
• Parsevals relation
• Application to power conversion.

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Fourier Series - Recap
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Interpretation of Fourier series expansion in
an orthonormal basis.
Analogy orthonormal basis in 3-dimensional space
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Expanding a vector in an orthonormal basis.
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Analogy between vector and Fourier expansions
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Example sawtooth function, of period T 2.
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2
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-1
-2
0
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• It is one of many possible norms
• (i.e. notions of size) of a function. Why this
  choice?
• Mathematically, it has nice properties, like
  vector length.
• Physical interpretation based on power

Example f(t) I(t), current going through a unit
resistor R1.
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One more ingredient in the analogy
PARSEVALS RELATION
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Example Square Wave
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Application of Fourier series power conversion.
DC
AC

Rectifier

• A rectifier is a circuit that converts the power
  to DC
• (used by electronic equipment such as
  computers, audio, ...).
• Ideally, y(t) should be a perfectly flat,
  constant DC voltage.
• In practice, one gets an approximation to DC,
  with some
• remaining oscillations (AC component).

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Nonlinear, time invariant and memoryless
system. Can be approximately implemented by a
diode circuit.
Not quite flat, but lets see how much of y is
DC.
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From the graph, the AC part looks significant.
Power analysis by Parseval