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Convergence of Fourier Series

Convergence of Fourier Series. Can we get Fourier Series representation for all periodic signals. I.e. are the coefficients from eqn 3.39 finite or in other words the integrals do not diverge.

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Convergence of Fourier Series

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  1. Convergence of Fourier Series • Can we get Fourier Series representation for all periodic signals. • I.e. are the coefficients from eqn 3.39 finite or in other words the integrals do not diverge. • If the coefficients are finite and substituted in the synthesis eqn, will the resulting infinite series converge or not to the original signal x(t).

  2. Convergence of Fourier Series

  3. Convergence of Fourier Series

  4. Convergence of Fourier Series If x(t) has a Fourier Series representation, the best approximation is obtained by truncating the Fourier series to the desired term

  5. Dirichlet Conditions

  6. Gibbs Phenomenon • The partial sum in the vicinity of the discontinuity exhibits ripples. • The peak amplitudes of these ripples does not seem to decrease with increasing N. • As N increases, the ripples in the partial sums become compressed towards the discontinuity, but for any finite value of N, the peak amplitude of the ripples remains constant. Fig 3.9 pg 201 illustrates this phenomenon.

  7. Properties of Continuous-time Fourier Series.

  8. Properties of Continuous-time Fourier Series.

  9. Properties of Continuous-time Fourier Series.

  10. Properties of Continuous-time Fourier Series.

  11. Properties of Continuous-time Fourier Series.

  12. Properties of Continuous-time Fourier Series.

  13. Properties of Continuous-time Fourier Series. The total average power in a periodic signal equals the sum of the average powers in all of its harmonic components.

  14. The Discrete-time Fourier Series In the previous lectures you have learned about:- Fourier Series Pair Equations for the continuous-time periodic signals. And also their properties. The derivation is through using:- Signals as represented by linear combinations of basic signals with the following 2 properties. The set of basic signals can be used to construct a broad and useful class of signals. The response of an LTI system is a combination of the responses to these basic signals at the input.

  15. Parallel Between The Continous-time & The Discrete-time. x(t) y(t) LTI x[n] y[n]

  16. Parallel Between The Continous-time & The Discrete-time.

  17. Discrete-time

  18. Linear Combinations of Harmonically Related Complex Exponentials,

  19. Linear Combinations of Harmonically Related Complex Exponentials.

  20. Discrete-time Fourier Series

  21. Discrete-time Fourier series pair

  22. Continuous-time & Discrete-time Fourier Series

  23. True for Discrete-time case True also for C-T Not true for C-T

  24. Convergence of Fourier Series • Continuous time: • x(t) square integrable • or Dirichlet conditions.

  25. Fourier Series Representation of Discrete-time Periodic Signals • Fourier series of continuous-time periodic signals are infinite series. • Fourier series of discrete-time periodic signals are of finite series in nature. • So for the fourier series of discrete-time periodic signals, mathematical issues of convergence do not arise.

  26. Re(ak) Example 3.11 3/2 1 0 -N N k Im (ak) 1/2 0 N -N k

  27. Example 3.11 |(ak)| 1 1/2 0 -N N k p/2 phase(ak) 0 N -N k -p/2

  28. Example 3.12 | 1 0 -N N -N1 n N1

  29. Example 3.12 | 1 0 -N N1 N -N1 n

  30. Example 3.12 1/2 1/2 | Case N=10, 2N1+1=5. 2 4 3 7 0 9 1 8 -1 5 6 10

  31. Example 3.12 1/4 1/4 | Case N=20, 2N1+1=5. 5 12 6 8 7 4 3 0 11 2 9 10 1 20 19 -1 -2

  32. Example 3.12 1/8 1/8 | Case N=40, 2N1+1=5. 9 7 3 0 5 6 2 1 8 4 40 -2 -1

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