1. Leo Lam © 2010-2011 Signals and Systems EE235 Leo Lam

2. Leo Lam © 2010-2011 Today’s menu Fourier Series (Exponential form)

3. Fourier Series Table Leo Lam © 2010-2011 3 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:

4. Fourier Series: Quick exercise Leo Lam © 2010-2011 4 Given: Find its exponential Fourier Series: (Find the coefficients d n and  0 )

5. Fourier Series: Fun examples Leo Lam © 2010-2011 5 Rectified sinusoids Find its exponential Fourier Series: t 0 f(t) =|sin(t)| Expand as exp., combine, integrate

6. Fourier Series: Circuit Application Leo Lam © 2010-2011 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)!

7. Fourier Series: Circuit Application Leo Lam © 2010-2011 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

8. Fourier Series: Circuit Application Leo Lam © 2010-2011 8 Rectified sinusoids Now we know: LTI system: Transfer function: To frequency: +-+- sin(t) full wave rectifier y(t) f(t)

9. Fourier Series: Circuit Application Leo Lam © 2010-2011 9 Rectified sinusoids Now we know: LTI system: Transfer function: System response: +-+- sin(t) full wave rectifier y(t) f(t)

10. Fourier Series: Dirichlet Conditon Leo Lam © 2010-2011 10 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

11. End of Fourier Series Leo Lam © 2010-2011 11 We have accomplished: –Introduced signal orthogonality –Fourier Series derivation –Approx. periodic signals: –Fourier Series Properties Next: Fourier Transform

12. Leo Lam © 2010-2011 Summary Fourier series Examples Dirichlet condition Examples galore on Monday