Fourier series of an arbitrary periodic piecewise function

Question

Let's take a switch function as an example:
$ u(t)= \begin{cases} 1, & \text{$0<t<dT_s$} \\ 0, & \text{$dT_s<t<T_s$} \end{cases} $
How can I get Fourier series of it in Mathematica?
Is there a way to get Fourier series of arbitrary periodic piecewise function?

I would start by having a look at Piecewise and Fourier. — b.gates.you.know.what, Feb 26 '19 at 09:09
@b.gatessucks You probably meant FourierSeries, right? Fourier is the discrete fourier transform (FFT). I am also puzzeld by the many different Fourier-related commands in Mathematica. — Henrik Schumacher, Feb 26 '19 at 10:11
Strongly related, if not duplicate: https://mathematica.stackexchange.com/q/149468/1871 — xzczd, Feb 26 '19 at 10:39
People here generally like users to post code as Mathematica code instead of just images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find this this meta Q&A helpful — Michael E2, Feb 26 '19 at 14:29

score 2 · Answer 1 · answered Feb 26 '19 at 10:18

2

for example:

u[t_] = Piecewise[{{1, 0 < t < dT}, {0, dT < t < T}}];
FourierTrigSeries[u[t], t, 3, FourierParameters -> {1, 2 \[Pi]/T}, 
 Assumptions -> 0 < dT < T && 2 dT == T]

answered Feb 26 '19 at 10:18

rmw

1,950
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score 1 · Answer 2 · answered Feb 26 '19 at 09:11

1

Try u[t_] := UnitStep[Sin[t]]; FourierSeries[u[x], x, 3] 3 is series

answered Feb 26 '19 at 09:11

XinBae

617
3
9

1

One might want to be mindful of the FourierParameters setting when using FourierSeries[] and other sundry functions, lest Mathematica's chosen normalization might not be the same as your preferred one. – J. M.'s missing motivation Feb 26 '19 at 09:15

Fourier series of an arbitrary periodic piecewise function

2 Answers2