1. Leo Lam © 2010-2012 Signals and Systems EE235

2. Leo Lam © 2010-2012 Today’s menu Fourier Series (Exponential form) Fourier Transform!

3. Example: Sped up delta-train Leo Lam © 2010-2012 3 Sped-up by 2, what does it do? Fundamental frequency doubled d n remains the same So: where time T/2 0 m=1 2 3 T’ is the new period

4. Lazy ways: re-using Fourier Series Leo Lam © 2010-2012 4 Example: Time scaling (last example we did): Given that: and New signal: What are the new coefficients in terms of d k ? Use the Synthesis equation: k is the integer multiple of the fundamental frequency corresponding to coefficient d k.

5. Graphical: Time scaling: Fourier Series Leo Lam © 2010-2012 5 Example: Time scaling up (graphical) New signal based on f(t): Using the Synthesis equation: Fourier spectra 0 Twice as far apart as f(t)’s

6. Graphical: Time scaling: Fourier Series Leo Lam © 2010-2012 6 Spectra change (time-scaling up): f(t) g(t)=f(2t) Does it make intuitive sense? 0 1 0 1

7. Graphical: Time scaling: Fourier Series Leo Lam © 2010-2012 7 Spectra change (time scaling down): f(t) g(t)=f(t/2) 0 1 0 1

8. Fourier Series Table Leo Lam © 2010-2012 8 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:

9. Fourier Series: Quick exercise Leo Lam © 2010-2012 9 Given: Find its exponential Fourier Series: (Find the coefficients d n and  0 )

10. Fourier Series: Fun examples Leo Lam © 2010-2012 10 Rectified sinusoids Find its exponential Fourier Series: t 0 f(t) =|sin(t)| Expand as exp., combine, integrate

11. Fourier Series: Circuit Application Leo Lam © 2010-2012 11 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)!

12. Fourier Series: Circuit Application Leo Lam © 2010-2012 12 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

13. Fourier Series: Circuit Application Leo Lam © 2010-2012 13 Rectified sinusoids Now we know: LTI system: Transfer function: To frequency: +-+- sin(t) full wave rectifier y(t) f(t)

14. Fourier Series: Circuit Application Leo Lam © 2010-2012 14 Rectified sinusoids Now we know: LTI system: Transfer function: System response: +-+- sin(t) full wave rectifier y(t) f(t)

15. Leo Lam © 2010-2012 Summary Fourier Series circuit example