The square wave alternates between a low interval of length five and a high interval of length five; its integral average is three. In the real Fourier series convention
(1)
integration over one period gives
(2)
and
(3)
Thus only odd harmonics occur:
(4)
The Dirichlet-Jordan convergence theorem makes the sum equal to the function at its continuity points and to the mean of its one-sided limits at a jump. Consequently the sum is 3 at each of , regardless of any separately assigned values of the original function at those endpoints.
Figure 1.
A periodic square wave and its Fourier approximations
. Original plot of the alternating low and high intervals and two partial Fourier sums. The jump values of the Fourier series are their midpoint; finite sums exhibit Gibbs oscillations. Details.

Ancestors (12)

  1. 2
  2. Paper 2
  3. 2004
  4. Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge
  5. Mathematics course of the year 1 of the natural sciences course of the University of Cambridge
  6. Year 1 of the natural sciences course of the University of Cambridge
  7. Year of the natural sciences course of the University of Cambridge
  8. Natural sciences course of the University of Cambridge
  9. Course of the University of Cambridge
  10. University of Cambridge
  11. List of universities
  12. Home

View article source

Discussion (0) New discussion

There are no discussions about this article yet.

Articles by others on the same topic (0)

There are currently no matching articles.
See all articles in the same topic Create my own version