Fourier series expansion

1. Fourier series expansion

2. Spectral analysis
Most part of signals involved in systems
working, are time-varying quantities.
Although a signal physically exists in
time domain, we can represent it in the
so called frequency domain, in which it
consists of a series of sinusoidal
components at various frequencies.
The frequency domain description is
called spectral analysis.
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3. LTI systems
The spectral analysis of signals, coupled with
frequency response characteristics of systems,
allows us to have a good approach in design
work.
In fact, when we can study the behavior of a
linear time-invariant (LTI) system in presence of
a particular sinusoidal signal, we can also
study the behavior in presence of all the
sinusoidal signals, and therefore in presence of
all the signals which we can consider
composed by a series of sinusoidal signals.
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4. Jean Baptiste Joseph Fourier
The mathematical methods
which help us in this work
are based on the studies of
a French physicist and
mathematician, Jean Bap-
tiste Joseph Fourier who
lived between XVIII and XIX
century.
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5. Methods
There are two kinds of methods:
• Series expansion of periodic signals
• Transform of non-periodic signals
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6. Series expansion
Every periodic function can be represented as
the expansion of a series of sinusoidal functions:
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where

7. Example 1 - 1
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8. Example 1 - 2
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periodic odd function

9. Example 1 - 3
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in the end:
If b=0, the function represents the restriction of function sgn(t)
into the ] interval, periodically extended outside. In the
figure below (left) are represented the first five Fourier
polynomials of this function.
The amplitude spectrum is a line spectrum (right figure).

10. Example 2
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periodic even function

11. Example 3
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12. Example 4 - 1
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  

13. Example 4 - 2
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

14. Example 4 - 3
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 

15. Example 4 - 4
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16. Example 5
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17. Example 6
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