1
$\begingroup$

As the title mentioned, I've not known exactly about Fourier series and when I was reading an digital communication textbook, I wondered about below equation derivation of Fourier series like $$\alpha(f)=\frac1T\sum^{\infty}_{m=-\infty}\delta\left(f-\frac mT\right)$$ which is periodic with period $\frac1T$ and $\delta$ is the Dirac-Delta function. From Fourier series, we have $$\alpha(f)=\frac1{1/T}\sum^\infty_{m=-\infty}c_ne^{-i2\pi nf/(1/T)}=\sum^\infty_{m=-\infty}e^{-i2\pi nfT}$$ where $$c_n=\int^\frac1{2T}_{-\frac1{2T}}\alpha(f)e^{i2\pi nf/(1/T)}df=\int^\frac1{2T}_{-\frac1{2T}}\frac1T\delta(f)e^{i2\pi nf/(1/T)}df=\frac1T$$ As I read the Fourier series information on Wiki, but I don't know how that equation is changed simply although Fourier series is very complicated. Please help me understand detail procedure about that. Thank you.

$\endgroup$
1
$\begingroup$

What you need from the theory of Fourier series is the following statement:

Let $f$ be a function that periodic with period $1/T$, i.e. $f(t)=f(t+1/T)$, then this function can be expanded in the following form $$ f(t)=\sum_{n=-\infty}^{\infty} c_n \mathrm{e}^{2 \pi \mathrm{i}n\omega t}, $$ where $$ c_n =\frac1T\int_{-1/(2T)}^{1/(2T)} f(t) \mathrm{e}^{-2 \pi \mathrm{i}n\omega t} dt. $$

I will not go into detail on the convergence results, i.e. under which circumstances does the series converge to the function $f$. However, it is instructive to note that this tells us that it is possible to expand a periodic function into building blocks of the form $\mathrm{e}^{2 \pi \mathrm{i}n\omega t}$.

The function that is of interest to you, i.e. $\frac{1}{T} \sum_{m=-\infty}^\infty \delta(f-m/T)$ indeed has the period $1/T$, which is easy to see. Therefore you can use the statement that I quoted above to compute the coefficients of the Fourier series to write the function as a superposition of complex exponential functions. This is what happens in the text you quoted.

Should you require more information on Fourier series I would suggest you to read the section on the subject in Stephane Mallats book A wavelet tour of signal processing, page 50. It contains an excellent and compact explanation of Fourier series and shows the connections to the standard Fourier transform.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.