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I am looking for a formula (Fourier series) to generate an impulse train waveform - a spike-wave with amplitude and period both $1$ – so that $f(x)$ has value $1$ at $x = 1,2,3,4...$ and $f(x)$ has value $0$ at all non-integer values of $x$.

Someone very helpfully gave me:

$$S = \frac 1 N \sum_{k=0}^{N-1} e ^{j2\pi\frac{kn}{N}}$$

The same equation can be found on this site: Equation for impulse train as sum of complex exponentials. However...

I have two questions:

  1. Is there an equivalent trigonometric function? If so, what is it?

  2. Sadly, my maths A level is such ancient history that I am struggling with what the terms in the above equation mean. Specifically:

    • $S$ = series, i.e. the equivalent of $f(x)$ - yes?

    • $e$ = famous irrational number - yes?

    • $j$ = square root of $-1$ - yes?

    • $N$ and $n$ ... Now, here I get muddled. 30 years of doing no maths at all has left me less than fluent... I assume the lowercase ($n$) is the period/frequency of the impulse train. So that leaves uppercase ($N$) as... The number on the $x$-axis that we are solving for...? The duration of the signal...? Some factor that compensates for $\pi$ to create integer values...?

Sorry. I know this is basic stuff. But I'd really appreciate some help...

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  • $\begingroup$ I think this question and its answers should solve your problem. $\endgroup$
    – Matt L.
    Jan 31, 2018 at 10:58
  • $\begingroup$ Be careful, you state that you want some function of $x$, which seems to be a continuous variable according to your first paragraph, while the formula you show represents a function of $n$, which is a discrete variable. $\endgroup$
    – Tendero
    Jan 31, 2018 at 13:00
  • $\begingroup$ Regarding your questions: 1. Equivalent trigonometric function to what? I didn't get that. 2. $S$ is the Discrete Fourier Series (in this case, the impulse train); you are right with $e$ and $j$; $n$ is the discrete-time variable and $N$ is the period of the impulse train. $\endgroup$
    – Tendero
    Jan 31, 2018 at 13:02
  • $\begingroup$ Thanks for this. I suspect I'm in way out of my depth... Although I realise that Fourier analysis is largely used in signal processing, I had been hoping that a Fourier Series could be used as a continuous function, in the same way as f(x) = 2x+1... is this wrong / naive? I can get that N is the period of the impulse train, but not sure what "n is a discrete time variable" means... $\endgroup$ Jan 31, 2018 at 17:41
  • $\begingroup$ And my search for a trigonometric function comes from the fact that many Fourier series can be expressed both in terms of e and also in terms of a sum (to infinity) of sin's and cos's. $\endgroup$ Jan 31, 2018 at 17:45

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