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Convergence of the Fourier Series

Module by: Carlos E. Davila. E-mail the author

Consider the trigonometric form of the Fourier series

x ( t ) = a 0 + n = 1 a n cos ( n Ω 0 t ) + n = 1 b n sin ( n Ω 0 t ) x ( t ) = a 0 + n = 1 a n cos ( n Ω 0 t ) + n = 1 b n sin ( n Ω 0 t )
(1)

It is important to state under what conditions this series (the right-hand side of Equation 1) will actually converge to x(t)x(t). The nature of the convergence also needs to be specified. There are several ways of defining the convergence of a series.

  1. Uniform convergence: define the finite sum:
    xN(t)=a0+n=1Nancos(nΩ0t)+n=1Nbnsin(nΩ0t)xN(t)=a0+n=1Nancos(nΩ0t)+n=1Nbnsin(nΩ0t)
    (2)
    where NN is finite. Then the series converges uniformly if the absolute value of x(t)-xN(t)x(t)-xN(t) satisfies
    x(t)-xN(t)<ϵx(t)-xN(t)<ϵ
    (3)
    for all values of tt and some small positive constant ϵϵ.
  2. Point-wise convergence: as with uniform convergence, we require that
    x(t)-xN(t)(t)<ϵx(t)-xN(t)(t)<ϵ
    (4)
    for all tt. The main difference between uniform and point-wise convergence is that for the latter, the number of terms in the summation N(t)N(t) needed to get the error below ϵϵ may vary for different values of t.
  3. Mean-squared convergence: here, the series converges if for all tt:
    limNt0t0+Tx(t)-xN(t)(t)2dt=0limNt0t0+Tx(t)-xN(t)(t)2dt=0
    (5)
    Gibb's phenomenon, mentioned in some of the examples above, is an example of mean-squared convergence of the series. The overshoot in Gibb's phenomenon occurs only at abrupt discontinuities. Moreover, the height of the overshoot stays the same independently of the number of terms in the series, NN. The overshoot merely becomes less noticeable because it becomes more and more narrow as NN increases.

Dirichlet has given a series of conditions which are necessary for a periodic signal to have a Fourier series. If these conditions are met, then

  • the Fourier series has point-wise convergence for all tt at which x(t)x(t) is continuous.
  • where x(t)x(t) has a discontinuity, then the series converges to the midpoint between the two values on either side of the discontinuity.

The Dirichlet Conditions are:

  1. x(t)x(t) has to be absolutely integrable on any period:
    t0t0+Tx(t)dt<t0t0+Tx(t)dt<
    (6)
  2. x(t)x(t) can have only a finite number of discontinuities on any period.
  3. x(t)x(t) can have only a finite number of extrema on any period.

Most periodic signals of practical interest satisfy these conditions.

References

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