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Here are different ways to write a Fourier series:

In Wikipedia:

$$f(x) =\sum_{n=-N}^N c_n\cdot \mathrm e^{i\frac{2\pi nx}{P}}\tag 1$$

On a lecture by Prof. Strang:

$$f(x) =\sum_{n=-\infty}^\infty c_n\cdot \mathrm e^{i\; nx} \tag 2$$

Or on a lecture by Prof. Brad Osgood:

$$f(x) =\sum_{n=-N}^N c_n\cdot \mathrm e^{i\; 2\pi n x} \tag 3$$

I see that the difference between (1) and (3) is the premise that $P=1$ in the Stanford lecture.

However, I want to ask for some insight about when $2\pi$ is needed - one obvious difference is the sum limits with $-N$ to $N$ calling for $2 \pi$ versus $-\infty$ to $\infty$ in (2) - the only formula without $2\pi.$ But what is the reason for this different formulation?

Antoni Parellada
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    I suspect that Professor Strang conveniently chose a period equal to $2\pi$,, which in (1) would give us$\Sigma_{n=-N}^{N} c_n \exp(i \frac{2\pi n x}{2\pi})$, which is the finite fourier series version of (2). I will read the provided link and update accordingly. EDIT: He chose $f(x+2\pi)=f(x)$ so $P=2\pi$ as suspected. – Shinaolord Aug 28 '18 at 00:28
  • @Shinaolord Is this choosing of a $2\pi$ period something you can only do in the abstract, or does it have some basis in "real-life" applications? – Antoni Parellada Aug 28 '18 at 00:31
  • we can always scale the variable we are using such that, for example, we go from $f(x+k)=f(x)$ to$ f(y+2\pi)=f(y)$, for appropriate choice of y with relation to x. – Shinaolord Aug 28 '18 at 00:31
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    This link may help you understand the particulars. – Shinaolord Aug 28 '18 at 00:33

1 Answers1

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Just for the sake of closing the question (thanks to @Shinaolord for the comments), the equation is predictably the same in the three instances:

$$f(x) =\sum_{n=-N}^N c_n\cdot \mathrm e^{i\;2\pi\;\frac{ n}{P}\;x}$$

This would be the finite or partial-sum FS of period $P$ (equivalent to the discrete Fourier transform (DFT)), as opposed to the infinite sum or Fourier series representation of $f(x)$ in equation (2) (equivalent to the discrete-time Fourier transform (DTFT)).

In addition, equation (2) uses a $P=2\pi.$ In the case of equation (3), $P=1.$

Antoni Parellada
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