Question

A discrete-time signal
$x\left[n\right]=e^{j\left(\frac{5\pi }{3}\right)n}+e^{j\left(\frac{\pi }{4}\right)n}$ is down-sampled to the signal $x_d\left[n\right]$ such that $x_d\left[n\right]=x\left[4n\right]$. The fundamental period of the down-sampled signal $x_d\left[n\right]$ is _______

Answer 1

$x\left(n\right)=e^{j\left(5\pi /3\right)n}+e^{j\left(\pi /4\right)n}$

${\omega }_1=\frac{5\pi }{3}$

${\omega }_2=\frac{\pi }{4}$

$\text{GCD}({\omega }_1,{\omega }_2),{\omega }_0=\frac{\pi }{12}$

$x_d\left(n\right)=x\left(4n\right)$

Apply time scaling so,

${\omega }_d=4{\omega }_0=\frac{4\pi }{12}=\frac{\pi }{3}$

$\frac{{\omega }_d}{2\pi }=\frac{m}{N}$

$\Rightarrow \frac{\pi /3}{2\pi }=\frac{m}{N}$

$N=6$