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I'm just beginning to learn about Fourier series and I'm trying to figure out how to find the Fourier series coefficients for $x(t) = e^{i100\pi t}$

I know also that $$x(t) = \sum_{-\infty}^{\infty} a_{k} e^{ik(2\pi/T)t}$$

How do I find these coefficients? I'm not even sure where to begin.

dmtri
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Austin
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    Do you have some formulas for computing the coefficients? If so, please add them to your question. Different authors sometimes use slightly different conventions. –  Oct 29 '15 at 16:25
  • I do, but I'm not very familiar with the math typesetting and I believe I'm supposed to be able to get the coefficients using a short-cut without having to compute the integral anyway. – Austin Oct 29 '15 at 16:28
  • What is the general form for a Fourier series? Also, I think you are missing an $i$ in the exponent of your function. –  Oct 29 '15 at 16:30
  • Oops, yes missing the j. I don't have my notebook with me at the moment, but as soon as I get home I'll update it. – Austin Oct 29 '15 at 16:32
  • is this $j$ a $i$ ? i.e., $j^2=-1$ ? or is it just some real-number parameter ? If this $j$ is a $i$, then, do you know the formula giving the coefficient for frequency f of the Fourier transform of f ? :-) – Fabrice NEYRET Oct 29 '15 at 16:54
  • Sorry yes j = i, we just use j in engineering classes because we use i for current – Austin Oct 29 '15 at 17:03

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We can make the $k$th term in the series $$\sum_{k=-\infty}^{\infty}a_k e^{ik(2\pi /T)t}$$ equal to $e^{i100\pi t}$ if $$k(2\pi/T) = 100\pi$$ or equivalently $$k = 50T$$ Note that if $T$ is the fundamental period of the series. If $T$ is not specified in advance, then there are infinitely many solutions. We require $k$ to be an integer, which will be true as long as $T$ is any integer multiple of $1/50$. So, for any positive integer $n$, the following gives us a perfectly valid representation of $x(t)$ as a Fourier series: $$a_k = \delta(k,n) \text{ and }T = n/50$$ where the $\delta$ is the Kronecker delta: $$\delta(k,n) = \begin{cases}1 & \text{if }k = n \\ 0 & \text{otherwise} \\ \end{cases}$$ We exclude integers $n \leq 0$ because the fundamental period $T = n/50$ should be positive.

  • If I wanted to graph this function, would I solve for T = 1/50 then plug this into T=n/50 to get n = 1, then just graph δ(t-1)? – Austin Oct 30 '15 at 01:24
  • @AustinMW: What exactly do you want to plot? The function $x(t) = e^{i100\pi t}$? It looks the same no matter which Fourier representation you choose: its real part is $\cos(100\pi t)$ and its imaginary part is $\sin(100\pi t)$. If you mean you want to plot the Fourier series coefficients versus $k$, you would have a single nonzero coefficient at $k=n$, and the other coefficients are all zero. –  Oct 30 '15 at 01:26
  • Yep, the Fourier series coefficients, but I missed your last part about for any integer n, so I guess I can just make n = 1 and plot that. Thanks! Not necessary, but if you have the time--do you happen to know how to calculate the coefficients with any applications to check my answers? Matlab/Mathematica/web app? Thanks again. – Austin Oct 30 '15 at 01:37
  • @AustinMW: WolframAlpha seems to have a FourierSeries function, but the documentation is not very insightful and I wasn't able to make it work correctly. Maybe you will have better luck: http://reference.wolfram.com/language/ref/FourierSeries.html –  Oct 30 '15 at 02:00
  • Thanks, I tried that a while ago and couldn't make it work either, but if I figure it out I'll add a comment. – Austin Oct 30 '15 at 02:07