Trasformata di Fourier di un prolungamento periodico

[Oiram92]

Ciao a tutti, ho appena trovato tra i vecchi compiti del mio prof un esercizio che non ho mai visto (in quelli di questi anni) e sinceramente non saprei come procedere..

Determinare la trasformata di Fourier del prolungamento periodico a \(\displaystyle ]-\infty,\infty[ \) di periodo \(\displaystyle 2 \) di :

\(\displaystyle f(t) = t + \left|t-\frac{1}{2}\right| \;\;\;\;\;\;\;\;\;\;\; t\in [-1,1[ \)

da vecchi ricordi di teoria dei segnali il segnale \(\displaystyle f(t) \) è dato dalla composizione di un segnale costante ed un segnale triangolare (troncato nell'intervallo in cui è definita \(\displaystyle t \)) e infatti graficando mi trovo :

[fcd="Segnale periodico"][FIDOCAD]
LI 160 40 160 150 0
FCJ 1 0 3 2 0 0
LI 10 125 325 125 0
FCJ 2 0 3 2 0 0
TY 130 125 4 3 0 0 0 * -1
TY 185 125 4 3 0 0 0 * 1
LI 135 110 160 110 0
TY 150 105 4 3 0 0 0 * 0.5
LI 160 110 185 70 0
TY 160 65 4 3 0 0 0 * 1.5
LI 185 110 210 110 0
LI 210 110 235 70 0
LI 185 70 185 110 0
FCJ 0 0 3 2 3 0
LI 85 110 110 110 0
LI 110 110 135 70 0
LI 135 70 135 110 0
FCJ 0 0 3 2 3 0
LI 35 110 60 110 0
FCJ 0 0 3 1 1 0
LI 60 110 85 70 0
LI 85 70 85 110 0
FCJ 0 0 3 2 3 0
LI 235 110 260 110 0
LI 260 110 285 70 0
LI 285 70 285 110 0
FCJ 0 0 3 2 3 0
LI 235 70 235 110 0
FCJ 0 0 3 2 3 0
LI 285 110 310 110 0
FCJ 0 0 3 1 1 0
TY 235 125 4 3 0 0 0 * 3
TY 285 125 4 3 0 0 0 * 5
TY 80 125 4 3 0 0 0 * -3[/fcd]

che ho già periodicizzato graficamente con periodo \(\displaystyle 2 \). Si nota anche che la periodicizzazione soddisfa il teorema di Shannon (perchè non abbiamo overlap e quindi niente aliasing). Adesso se non ricordo male dovrei scriverne la serie di Fourier e poi il testo chiede di calcolarne la trasformata. La cosa mi stupisce un pò perchè nel nostro programma di analisi 3 non abbiamo la serie di Fourier (anche se effettivamente è stata fatta in altri corsi), quindi per questo motivo non so bene come procedere. Mi sembra ti intuire che c'entrano in qualche modo le distribuzioni però..tanti dubbi a riguardo..qualcuno potrebbe spiegarmi un pò? Grazie mille

[Oiram92]

nessuno? sarebbe sufficiente anche solo un aiutino per indirizzarmi sulla logica dell'esercizio

[javicemarpe]

What do you mean by Fourier transform? I think maybe you want to compute the Fourier series...

[Oiram92]

Sincerely I don't understand very well this exercise. I think that I've to do the Fourier series of \(\displaystyle f(t) \) and then calculate the Fourier transform of the Fourier series. Is this possible?

Maybe, for the first Poisson summation formula :

\(\displaystyle y(t) = \frac{1}{T} \; \sum_{k=-\infty}^{+\infty} F\left(\frac{k}{T}\right) e^{i\;2\pi\;k\frac{t}{T}} = \frac{1}{2} \; \sum_{k=-\infty}^{+\infty} F\left(\frac{k}{2}\right) e^{i\;2\pi\;k\frac{t}{2}} \)

where \(\displaystyle y(t) \) is the periodization of \(\displaystyle f(t) \) and \(\displaystyle F\left(\frac{k}{2}\right) \) is the Fourier transform of \(\displaystyle f(t) \) evaluated in \(\displaystyle \frac{k}{2} \). Now we can use the Fourier transform so :

\(\displaystyle Y(f) = \frac{1}{2} \; \sum_{k=-\infty}^{+\infty} \left( \frac{1}{2}\; \int_{-1}^{1} F\left(\frac{k}{2}\right)\; e^{-i\;2 \pi \;k \frac{t}{2}} dt \right) \; \delta\left(f-\frac{k}{2}\right) \)

so the final step is to calculate :

\(\displaystyle \int_{-1}^{1} \left[\int_{-\infty}^{+\infty} \left(t + \left|t-\frac{1}{2}\right|\right)\;e^{-i \;2\pi\;t\;\xi} \;dt \right]_{\xi=\frac{k}{2}} \; e^{-i\;\pi \;k t} dt \)

correct?

[javicemarpe]

I think you are right, because it is not possible to calculate the Fourier transform of the periodic prolongation of $f$. So, if we are not mistaken, you want to calculate the Fourier coefficients (and, then, the Fourier series) of your function $f$, which are these integrals you wrote at the end of your post.

[gio73]

Hi javicemarpe
welcome among us
where are you from?

[javicemarpe]

"gio73":Hi javicemarpe
welcome among us
where are you from?

Hi. Thank you. I'm from Spain.

[Oiram92]

"javicemarpe":I think you are right, because it is not possible to calculate the Fourier transform of the periodic prolongation of $f$. So, if we are not mistaken, you want to calculate the Fourier coefficients (and, then, the Fourier series) of your function $f$, which are these integrals you wrote at the end of your post.

Thank you for your reply but I'm still doubtful regarding the last step..prof has never done exercise like that and also he never mentioned the Poisson's formulas..mmm I don't know

@gio73 tu avresti qualche idea a riguardo?

[javicemarpe]

To be honest, I don't know what do you have to do. But, as far as I know about Fourier things, you cannot compute the Fourier transform of a function which is not integrable (at least, not as a function).

So, on one hand, the periodic extension of your function is not an integrable function in $\mathbb{R}$ because, for example, its limit when $|t|\to\infty$ is not zero. Then, I'm sure that you don't have to compute the Fourier transform of the periodic extension.

On the other hand, maybe your professor wanted to ask you for the Fourier series of the periodic extension of your function (which is the same as calculating the Fourier series of your function as a function defined in the circle centered at the origin with perimeter 2). So you only have to calculate the Fourier coefficients of $f$ and sum.

In any case, I'm not sure at all. Maybe you should ask your professor what does he/she want.