Fourier Expansion

[Davide Sandona']

Hello everyone,

Today I decided to refresh my knowledge of Fourier Expansion. I tried to use Sympy but I'm not sure if the result is correct or if I did something wrong, so I need a fresh pair of eyes on this :)

I was following the "Fourier Series of a Square Wave" in the Wolfram documentation [1], and I compare the result with Sympy. Here is the code and the picture.

x, n = symbols("x, n")
N = 10
expr = 2 * (Heaviside(x / 2) - Heaviside(x / 2 - 1)) - 1
fs = expr.fourier_series()
wolfram = 4 / pi * sin(n * pi * x / 2) / n
wolfram_expr = 0
for i in range(1, N + 1):
    if i % 2 != 0:
        wolfram_expr += (4 / pi * sin(n * pi * x / 2) / n).subs(n, i)
plot(expr, fs.truncate(N), wolfram_expr, (x, -5, 5))

Did I do something wrong?

Thanks for you time! :)

[1] https://mathworld.wolfram.com/FourierSeriesSquareWave.html

[mikhae...@umontpellier.fr]

Dear Sandona,

  I am not a skilled Sympy user. However, I guess the fact that you do not give the periodicity does not allow sympy to compute the Fourier Series. I did this that seems satisfactory :

from sympy import *

x, n = symbols("x, n")

N = 10

expr = 2 * (Heaviside(x / 2) - Heaviside(x / 2 -1)) - 1

fs = fourier_series(expr,(x,-2,2))

wolfram = 4 / pi * sin(n * pi * x / 2) / n

wolfram_expr = 0

for i in range(1, N + 1):

    if i % 2 != 0:

        wolfram_expr += (4 / pi * sin(n * pi * x / 2) / n).subs(n, i)

plot(expr, fs.truncate(N/2), wolfram_expr, (x, -5, 5))

Moreover, I guess (I did not check in Sympy documentation) that Sympy only displays non-zero harmonics on "truncate" ?

Hope this helps,

   Mike

[Davide Sandona']

Hello Mike,

thank you very much, you really helped me! I was too focused exploring the different methods, and the solution was just under my nose. Thank you! :)

Of course, the example I've chosen (the square wave) takes a tremendous amount of time to be solved by Sympy. Anyway, I tried a sawtooth wave and it works great! Thank you again!

Davide.

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[mikhae...@umontpellier.fr]

Happy it helped ;-)