Signals and Systems EE235 Lecture 23 Leo Lam © 2010-2012

Today’s menu Fourier Series Example Fourier Transform Leo Lam © 2010-2012

Motivation Leo Lam © 2010-2012

Fourier Series: Quick exercise Given: Find its exponential Fourier Series: (Find the coefficients dn and w0) 4 Leo Lam © 2010-2012

Fourier Series: Fun examples Rectified sinusoids Find its exponential Fourier Series: t f(t) =|sin(t)| Expand as exp., combine, integrate 5 Leo Lam © 2010-2012

Fourier Series: Circuit Application 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? Find H(s)! S 6 Leo Lam © 2010-2012

Fourier Series: Circuit Application Finding H(s) for the LTI system: est is an eigenfunction, so Therefore: So: Shows how much an exponential gets amplified at different frequency s 7 Leo Lam © 2010-2012

Fourier Series: Circuit Application Rectified sinusoids Now we know: LTI system: Transfer function: To frequency: + - sin(t) full wave rectifier y(t) f(t) 8 Leo Lam © 2010-2012

Fourier Series: Circuit Application Rectified sinusoids Now we know: LTI system: Transfer function: System response: + - sin(t) full wave rectifier y(t) f(t) 9 Leo Lam © 2010-2012

Summary Fourier Series circuit example Leo Lam © 2010-2012

Fourier Series: Dirichlet Conditon 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 Weak Dirichlet: Otherwise you can’t solve for the coefficients! 11 Leo Lam © 2010-2012

End of Fourier Series We have accomplished: Next: Fourier Transform 12 Introduced signal orthogonality Fourier Series derivation Approx. periodic signals: Fourier Series Properties Next: Fourier Transform 12 Leo Lam © 2010-2012

Fourier Transform: Introduction Fourier Series: Periodic Signal Fourier Transform: extends to all signals Recall time-scaling: 13 Leo Lam © 2010-2012

Fourier Transform: Recall time-scaling: 14 Fourier Spectra for T, Fourier Spectra for T, for 2T, 14 Leo Lam © 2010-2012

Fourier Transform: Non-periodic signal: infinite period T 15 Fourier Spectra for T, for 2T, 15 Leo Lam © 2010-2012

Fourier Transform: Fourier Formulas: For any arbitrary practical signal And its “coefficients” (Fourier Transform): F(w) is complex Rigorous math derivation in Ch. 4 (not required reading, but recommended) Time domain to Frequency domain Weak Dirichlet: Otherwise you can’t solve for the coefficients! 16 Leo Lam © 2010-2012

Fourier Transform: Fourier Formulas compared: 17 Fourier transform Fourier transform coefficients: Fourier transform (arbitrary signals) Fourier series (Periodic signals): Fourier series coefficients: and 17 Leo Lam © 2010-2012

Fourier Transform (example): Find the Fourier Transform of What does it look like? If a <0, blows up phase varies with  magnitude varies with  18 Leo Lam © 2010-2012