1. Fourier Series

2. is the “fundamental frequency”

3. Fourier Series is the “fundamental frequency”

4. Fourier Series Integration limits: when, then so we get:

5. Fourier Series Euler:

6. Fourier Series We can show ; recall that

7. Phasors: Phasors

8. Symmetry Odd f(-t) =-f(t) Fourier: sine terms only Even f(t) = f(-t) Fourier: cosine terms only Neither

9. Half-wave symmetry: has no even harmonics | t t+T/2

10. Example of non-symmetric waveform:

11. Fundamental Signals Unit Step:

12. Fundamental Signals Unit Step: Unit Impulse

13. (“deivative” of Unit Step)

14. Unit Impulse (“derivative” of Unit Step)

17. Exponential A=1 a=-1 A=1 a=1

18. Exponential A=1 a=-1

19. Sinusoid For

21. Damped Sinusoid

22. For Here,

23. For Here, Recall (Euler) phase shift

24. For Here,

25. For Here, Recall (Euler) phase shift

26. Also note that, in phasors:

27. Recall the identity:

28. Time Invariance:

29. Integrator:

33. “Sifting” property of

38. Again, in the limit, as Summing the components (Superposition): Let and

39. Again, in the limit, as Summing the components (Superposition): Let and CONVOLUTION

40. or

41. Proof: Change of variables. Let so

42. Convolution is commutative, associative and distributive

43. Convolution is commutative, associative and distributive =

44. Convolution is commutative, associative and distributive =

45. Convolution is commutative, associative and distributive Proof: