Leo Lam © Signals and Systems EE235 Leo Lam

Leo Lam © Today’s menu Fourier Series (Exponential form)

Fourier Series Table Leo Lam © Added constant only affects DC term Linear ops Time scale Same d k, scale  0 reverse Shift in time –t 0 Add linear phase term –jk   t 0 Fourier Series Properties:

Fourier Series: Quick exercise Leo Lam © Given: Find its exponential Fourier Series: (Find the coefficients d n and  0 )

Fourier Series: Fun examples Leo Lam © Rectified sinusoids Find its exponential Fourier Series: t 0 f(t) =|sin(t)| Expand as exp., combine, integrate

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

Fourier Series: Dirichlet Conditon Leo Lam © 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 © We have accomplished: –Introduced signal orthogonality –Fourier Series derivation –Approx. periodic signals: –Fourier Series Properties Next: Fourier Transform

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

Fourier Transform: Leo Lam © Recall time-scaling: 0 Fourier Spectra for T, Fourier Spectra for 2T,

Fourier Transform: Leo Lam © Non-periodic signal: infinite period T 0 Fourier Spectra for T, Fourier Spectra for 2T,

Fourier Transform: Leo Lam © Fourier Formulas: (Rigorous derivation in Ch.4) 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 © Fourier Formulas compared: Fourier transform coefficients: Fourier transform (arbitrary signals) Fourier series (Periodic signals): Fourier series coefficients: and

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

Fourier Transform (example): Leo Lam © Fourier Transform of Real-time signals magnitude: even phase: odd magnitudephase

Fourier Transform (Symmetry): Leo Lam © Real-time signals magnitude: even – why? magnitude Even magnitude Odd phase Useful for checking answers

Fourier Transform/Series (Symmetry): Leo Lam © Works for Fourier Series, too! Fourier transform (arbitrary practical signal) Fourier series (periodic functions) Fourier coefficients Fourier transform coefficients magnitude: even & phase: odd

Fourier Transform (example): Leo Lam © Fourier Transform of F(  ) is purely real F() for a=1

Fourier Transform (delta function): Leo Lam © Fourier Transform of Standard Fourier Transform pair notation

Leo Lam © Summary Fourier Transform intro Inverse etc.