Are the period of signal and its fourier coefficient the same?i mean if the the period of $x[n]$ is $5$,will the period of its fourier coefficient ,$a_k$, also $5$? how to prove it?
Because the definition of a periodic is $x[n]=x[n+N]$,but if we do fourier series at the same time ,it will become $a_k=a_k \times e^{jwkN}$,and in this formula,$w=\frac{2 \pi}{N}$,so i it will become $a_k=a_k \times e^{j \pi k}$ ,i can't know the period of $a_k$ from here.
if $a_k$ is periodic,then $a_k=a_{k+p}$,$p$ is the period of $a_k$
by definition
$a_k=\frac{1}{N}\sum\limits_{n=-\infty}^{\infty}e^{-j\frac{2 \pi}{N}kn}$,and $a_{k+p}=\frac{1}{N}\sum\limits_{n=-\infty}^{\infty}e^{-j\frac{2 \pi}{N}(z-p)n}=\frac{1}{N}\sum\limits_{n=-\infty}^{\infty}e^{-j\frac{2 \pi}{N}zn}e^{j\frac{2 \pi}{N}pn}$,if $a_{k+p}$ wants to be equal to $a_k$,then $p$ must be equal to $N$,so that $e^{j\frac{2 \pi}{N}pn}=e^{j2 \pi n} = 1$.
$p$ must be equal to $N$.that is the period of the $a_k = $ the period of the $x[n]$