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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