Leo Lam 2010-2012 Signals and Systems EE235 Lecture 23
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Leo Lam 2010-2012 Todays menu Fourier Series Example Fourier
Transform
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Leo Lam 2010-2012 Motivation
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Fourier Series: Quick exercise Leo Lam 2010-2012 4 Given: Find
its exponential Fourier Series: (Find the coefficients d n and 0
)
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Fourier Series: Fun examples Leo Lam 2010-2012 5 Rectified
sinusoids Find its exponential Fourier Series: t 0 f(t) =|sin(t)|
Expand as exp., combine, integrate
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Fourier Series: Circuit Application Leo Lam 2010-2012 6
Rectified sinusoids Now we know: Circuit is an LTI system: Find
y(t) Remember: +-+- sin(t) full wave rectifier y(t) f(t) Where did
this come from? S Find H(s)!
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Fourier Series: Circuit Application Leo Lam 2010-2012 7 Finding
H(s) for the LTI system: e st is an eigenfunction, so Therefore:
So: Shows how much an exponential gets amplified at different
frequency s
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Fourier Series: Circuit Application Leo Lam 2010-2012 8
Rectified sinusoids Now we know: LTI system: Transfer function: To
frequency: +-+- sin(t) full wave rectifier y(t) f(t)
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Fourier Series: Circuit Application Leo Lam 2010-2012 9
Rectified sinusoids Now we know: LTI system: Transfer function:
System response: +-+- sin(t) full wave rectifier y(t) f(t)
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Leo Lam 2010-2012 Summary Fourier Series circuit example
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Fourier Series: Dirichlet Conditon Leo Lam 2010-2012 11
Condition for periodic signal f(t) to exist has exponential series:
Weak Dirichlet: Strong Dirichlet (converging series): f(t) must
have finite maxima, minima, and discontinuities in a period All
physical periodic signals converge
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End of Fourier Series Leo Lam 2010-2012 12 We have
accomplished: Introduced signal orthogonality Fourier Series
derivation Approx. periodic signals: Fourier Series Properties
Next: Fourier Transform
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Fourier Transform: Introduction Leo Lam 2010-2012 13 Fourier
Series: Periodic Signal Fourier Transform: extends to all signals
Recall time-scaling:
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Fourier Transform: Leo Lam 2010-2012 14 Recall time-scaling: 0
Fourier Spectra for T, Fourier Spectra for 2T,
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Fourier Transform: Leo Lam 2010-2012 15 Non-periodic signal:
infinite period T 0 Fourier Spectra for T, Fourier Spectra for
2T,
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Fourier Transform: Leo Lam 2010-2012 16 Fourier Formulas: For
any arbitrary practical signal And its coefficients (Fourier
Transform): F( ) is complex Rigorous math derivation in Ch. 4 (not
required reading, but recommended) Time domain to Frequency
domain
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Fourier Transform: Leo Lam 2010-2012 17 Fourier Formulas
compared: Fourier transform coefficients: Fourier transform
(arbitrary signals) Fourier series (Periodic signals): Fourier
series coefficients: and
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Fourier Transform (example): Leo Lam 2010-2012 18 Find the
Fourier Transform of What does it look like? If a