Here is a Maple worksheet to illustrate the Fourier Series approximation of a Square Wave. If you don't want to copy the lines below, you can click here to download the source file (very small, and zipped). On the page below, the red lines are the crutial Maple lines .. the green lines are just comments to help clarify what is being done.
> restart;
with(plots):
> m:=40: #
Number of terms to keep
> # calculate the array of f(x) terms -
one for each term up to M (successively smaller!)
> # myseq[1] = the first term of the
fourier series
> # myseq[2] = the second term of the
fourier series (smaller than the first) .. etc.
> myseq := seq(
(2/((2*n-1)*Pi))*sin((2*n-1)*Pi*x),n=1..m): # notice only
odd terms calculated
> # calculate the array of the sums of
each terms
> myseqsum :=
seq(sum(myseq[nn],nn=1..n),n=1..m):
> # myseqsum[1] = the sum of only ONE
term in series
> # myseqsum[2] = the sum of the first
TWO terms in the series (combined)
> # myseqsum[3] = the sum of the first
THREE terms - looking more like square wave
> # now plot the above sums individually
- animated - shows evolution to the squarewave
> display(seq(plot(myseqsum[i],
x=-0.2..2.2,color=blue, thickness=2),i=1..m), insequence=true, title=`Evolution
of Fourier series to Square Wave`);
Here is an animation of the sequence as you add more terms to the series .. notice that it starts approaching an square wave, but there is "ringing" at the corners. Because of the discontinuities in the square wave, we can't completely remove those artifacts. (Thus a square wave is never a perfect square wave!) Look at these other pages :
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> # Note: Watch the corners .. the series will never fully match the square wave because of the discontinuity ... that "ringing" at the corners is called the Gibb's Phenomena!
![[Maple Plot]](seq_square.gif)