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Fourier Series

Fourier Series. Outline. Property # 6 of Fourier coefficient System Analysis Fourier Series Transformation. Property 6. If the nth derivative of x(t) is the first derivative that contains a discontinuity,.., then the Fourier coefficients approach zero as 1/k n+1 , for sufficiently large k.

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Fourier Series

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  1. Fourier Series

  2. Outline • Property # 6 of Fourier coefficient • System Analysis • Fourier Series Transformation

  3. Property 6 • If the nth derivative of x(t) is the first derivative that contains a discontinuity,.., then the Fourier coefficients approach zero as 1/kn+1, for sufficiently large k. • Note: This property is not applicable for impulse function since it is not a bounded function (i.e. it does not satisfy the Dirichlet condition)!

  4. Example (1) Discontinuity in the 0th derivative Therefore, Ck decreases as 1/k as k increases.

  5. Example (2) Discontinuity in the 0th derivative Therefore, Ck decreases as 1/k as k increases. Maximum peak value of sinc X for |X| less than 5π Key: |sinc x| can be approximated using 1/x

  6. Example 3 Discontinuity in the 1th derivative Therefore, Ck decreases as 1/k1+1 as k increases.

  7. Important Results from Chapter 3 Steady state behavior becomes important after the natural response disappears. Let y(t)=i(t) and x(t)=v(t) Differential Equation & ES 220

  8. Important Results from Chapter 3 How do you find h(t)? How do you find H(s)?

  9. Important Results from Chapter 3

  10. System Analysis • Represent a period input x(t) by its Fourier series • Utilize the Transfer Function • Determine the output in terms of its Fourier series

  11. Evaluate the Frequency Response of a Transfer Function in Matlab

  12. Example 1

  13. Example 2 • Calculate Cky and Ckx using Matlab

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