# The Fourier Transform of a periodic function and it's series

Let $$X(f)$$ be the Fourier transform of $$x(t)$$:

$$X(f) \triangleq \mathscr{F}\Big\{ x(t) \Big\} = \int\limits_{-\infty}^{\infty} x(t)\,e^{-j 2 \pi f t} \ \mathrm{d}t$$

$$x(t) \triangleq \mathscr{F}^{-1}\Big\{ X(f) \Big\} = \int\limits_{-\infty}^{\infty} X(f)\,e^{j 2 \pi f t} \ \mathrm{d}f$$

The question here it's how can I mathematically relate the Fourier transform of a function with its Fourier coefficients $$c_n$$ (complex form)

$$x(t) = \sum\limits_{n=-\infty}^{\infty} c_n e^{j 2 \pi (n/T)t}$$ where $$c_n = \frac{1}{T} \int\limits_{t_0}^{t_0+T} x(t)\,e^{-j 2 \pi (n/T)t} \ \mathrm{d}t \qquad \qquad -\infty < t_0 < +\infty$$

I mean, it's quite obvious that for a non periodic function $$T\cdot c_n = \mathscr{F}\Big\{ x(t) \Big\}$$

However, if I have $$x(t)$$ periodic and take $$x_m(t)=x(t) \cdot \operatorname{rect}\left( \tfrac{t}{T} \right)$$

(that means we take a period of $$x(t)$$)

$$\mathscr{F}\Big\{ x_m(t) \Big\} = X(\nu) * T \operatorname{sinc}(T \nu)$$

($$\nu$$ is the frequency variable in Hz)

On the other side we may consider that

$$x_m(t) = \sum\limits_{n=-\infty}^{\infty} c_n e^{j 2 \pi (n/T)t} \longleftrightarrow \mathscr{F}\Big\{ x_m(t) \Big\} = \sum\limits_{n=-\infty}^{\infty} c_n \delta(\nu-\tfrac{n}{T})$$

So what I've seen people doing was saying that we could replace $$c_n$$ for $$\frac{1}{T}\mathscr{F}\Big\{ x_m(t) \Big\}$$ and then if I have a repeated spectrum in time I can only calculate it's transform and I'll have the amplitudes on the frequency spectrum for each dirac.

I cannot see it mathematically nor can say if it's true.

• you need to fix your mathematical notation. you are using "$f$" for two different mathematical objects. – robert bristow-johnson Mar 8 '19 at 22:33
• also, do you know how to do $\LaTeX$ markup here? would you like me to edit your question and show you? – robert bristow-johnson Mar 8 '19 at 22:34
• did i express your question correctly (as you intended), Lucas? you just can't have two $f$ symbols meaning different things, so i left $f$ as frequency and changed your function to $x(t)$. – robert bristow-johnson Mar 8 '19 at 23:41
• Oh, come on, Robert, what's the harm if Lucas uses $f$ to mean two different things? After all, even you (and Lucas too, I think) use $x_m(t)$ to denote both one period of $x(t)$ and all of $x(t)$ since you equate $x_m(t)$ with the Fourier series expression for $x(t)$. – Dilip Sarwate Mar 9 '19 at 3:56
• @DilipSarwate i saw that too. but i was trying to translate Lucas's statement of the problem faithfully, but with eliminating one of the two uses of $f$. – robert bristow-johnson Mar 9 '19 at 4:43