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Introduction to Fourier Series: Inner Product, Orthogonal Representation, and Complex Form

This lecture covers the concepts of inner product, orthogonal representation, and complex form in Fourier series. It also includes examples and a homework assignment.

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Introduction to Fourier Series: Inner Product, Orthogonal Representation, and Complex Form

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  1. 38655 BMED-2300-02 Lecture 5: Fourier Series Ge Wang, PhD Biomedical Imaging Center CBIS/BME, RPI wangg6@rpi.edu January 30, 2018

  2. BB Schedule for S18 Office Hour: Ge Tue & Fri 3-4 @ CBIS 3209 | wangg6@rpi.edu Kathleen Mon 4-5 & Thurs 4-5 @ JEC 7045 | chens18@rpi.edu

  3. Outline

  4. Inner Product: Sum of Products of Paired Data

  5. As DNA for Data Science

  6. Inner Product from Linear System x(n) h y(n) δ(n) h h(n) δ(n-k) h h(n-k) x(k)δ(n-k) h x(k)h(n-k) x(k)δ(n-k) h x(k)h(n-k) x(n) h x(k)h(n-k)

  7. Inner Product from Geometry

  8. Inner Product in C–S Inequality

  9. Geometric Proof (Easy)

  10. RN Space • Points • Distance • Projection (Min Distance from V to the OW line) • Angle/Orthogonality • Basis/Dimensionality V W θ O

  11. 3D Orthogonal Representation

  12. ND Orthogonal Representation

  13. Basis Not Unique http://www.ccp14.ac.uk/ccp/web-mirrors/klaus_eichele_software/klaus/nmr/conventions/euler/euler.html

  14. Vector Space

  15. Point = Vector

  16. Vector = Function

  17. Functional Space Can we find a basis? δ(x-a), a in R

  18. Function As a Sum of Impulses

  19. Pixel: Picture Element

  20. Sine Wave Comes Naturally

  21. Sinusoidal Basis: Even & Odd

  22. Function As Sum of Waves

  23. Demo: Spectral Synthesis

  24. Fourier Analysis

  25. Duality of Information Particle Wave

  26. Periodic Function

  27. Fourier Series (Unit Period)

  28. Orthogonal Representation

  29. Harmonics

  30. Fourier Series (Unit Period) Direct Component Even Parts Odd Parts

  31. Even & Odd Functions

  32. Fourier Coefficients (Unit Period)

  33. Fourier Series (Real Form)

  34. Complex Representation Conjugate

  35. Two Popular Systems

  36. Complex Inner Product

  37. Complex Inner Product

  38. Euler’s Formula

  39. Real Form to Complex Form

  40. Orthonormal Basis

  41. Fourier Series (Complex Form)

  42. Fourier Series Visualized

  43. Example: Square

  44. Example: Square

  45. Only Odd Terms

  46. Error Due to Finite Terms

  47. Errors Due to Gibbs Effect

  48. When Period Isn’t Unit

  49. 2D Fourier Series

  50. Homework for BB05 Derive the formulas for the coefficients of the Fourier series in real notations (become familiar with the ideas I explained to you today). Watch https://www.youtube.com/watch?v=05jUkVtIwVs, and implement the square wave example. Due date: One week from now (by midnight next Tuesday). Please upload your report to MLS, including both the script and the figures in a word file.

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