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Fourier Series

Fourier Series

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Fourier Series

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  1. Fourier Series Engineering Mathematics – I Prepared By G.VANMATHI, M.Sc.,M.Phil., Assistant Professor / Mathematics

  2. Content • Periodic Functions • Fourier Series

  3. Fourier Series Periodic Functions

  4. The Mathematic Formulation • Any function that satisfies where T is a constant and is called the period of the function.

  5. Example: Find its period. Fact: smallest T

  6. Example: Find its period. must be a rational number

  7. Example: Is this function a periodic one? not a rational number

  8. Fourier Series Fourier Series

  9. A periodic sequence f(t) t T 2T 3T Introduction • Decompose a periodic input signal into primitive periodic components.

  10. Synthesis T is a period of all the above signals Even Part Odd Part DC Part Let 0=2/T.

  11. Orthogonal Functions • Call a set of functions {k}orthogonal on an interval a < t < b if it satisfies

  12. Orthogonal set of Sinusoidal Functions Define 0=2/T. We now prove this one

  13. Proof m  n 0 0

  14. Proof m = n 0

  15. Define 0=2/T. Orthogonal set of Sinusoidal Functions an orthogonal set.

  16. Decomposition

  17. Proof Use the following facts:

  18. f(t) 1 -6 -5 -4 -3 -2 -  2 3 4 5 Example (Square Wave)

  19. f(t) 1 -6 -5 -4 -3 -2 -  2 3 4 5 Example (Square Wave)

  20. f(t) 1 -6 -5 -4 -3 -2 -  2 3 4 5 Example (Square Wave)

  21. Harmonics T is a period of all the above signals Even Part Odd Part DC Part

  22. Define , called the fundamental angular frequency. Define , called the n-th harmonicof the periodic function. Harmonics

  23. Harmonics

  24. harmonic amplitude phase angle Amplitudes and Phase Angles