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Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 23
Leo Lam © 2010-2012 Today’s menu Fourier Series Example Fourier Transform
Leo Lam © 2010-2012 Motivation
Fourier Series: Quick exercise Leo Lam © 2010-2012 4 Given: Find its exponential Fourier Series: (Find the coefficients d n and 0 )
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
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)!
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
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)
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)
Leo Lam © 2010-2012 Summary Fourier Series circuit example
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
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
Fourier Transform: Introduction Leo Lam © 2010-2012 13 Fourier Series: Periodic Signal Fourier Transform: extends to all signals Recall time-scaling:
Fourier Transform: Leo Lam © 2010-2012 14 Recall time-scaling: 0 Fourier Spectra for T, Fourier Spectra for 2T,
Fourier Transform: Leo Lam © 2010-2012 15 Non-periodic signal: infinite period T 0 Fourier Spectra for T, Fourier Spectra for 2T,
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
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
Fourier Transform (example): Leo Lam © 2010-2012 18 Find the Fourier Transform of What does it look like? If a <0, blows up magnitude varies with phase varies with
Fourier Transform Periodicity of Fourier series
Signals and Systems Fall 2003 Lecture #5 18 September Complex Exponentials as Eigenfunctions of LTI Systems 2. Fourier Series representation of.
Leo Lam © Signals and Systems EE235 Leo Lam.
Lecture 7: Basis Functions & Fourier Series
Leo Lam © Signals and Systems EE235 Lecture 16.
Leo Lam © Signals and Systems EE235. Transformers Leo Lam ©
Leo Lam © Signals and Systems EE235. Fourier Transform: Leo Lam © Fourier Formulas: Inverse Fourier Transform: Fourier Transform:
Lecture 8: Fourier Series and Fourier Transform
Lecture 14: Laplace Transform Properties
Lecture 8 Topics Fourier Transforms –As the limit of Fourier Series –Spectra –Convergence of Fourier Transforms –Fourier Transform: Synthesis equation.
Lecture 12: Laplace Transform
Leo Lam © Signals and Systems EE235. Leo Lam © Speed of light.
Leo Lam © Signals and Systems EE235. Leo Lam © Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to.
Leo Lam © Signals and Systems EE235 Lecture 27.
Leo Lam © Signals and Systems EE235. So stable Leo Lam ©
Leo Lam © Signals and Systems EE235. Leo Lam © x squared equals 9 x squared plus 1 equals y Find value of y.
Leo Lam © Signals and Systems EE235 Leo Lam © Stanford The Stanford Linear Accelerator Center was known as SLAC, until the big earthquake,
Leo Lam © Signals and Systems EE235. Leo Lam © Merry Christmas! Q: What is Quayle-o-phobia? A: The fear of the exponential (e).
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