Question

Let x (t)= cos (10$\pi$t)+cos(30$\pi$t) be sampled at 20 Hz and reconstructed using an ideal low-pass filter with cut-off frequency of 20 Hz. The frequency/frequencies present in the reconstructed signal is/are

Answer 1

$x\left(t\right)=cos\left(10\pi t\right)+cos\left(30\pi t\right)$

$\text{Given sampling frequency,}$ $f_s=20Hz$

${\omega }_s=40\pi rad/sec$


$Y\left(\omega \right)=\sum _{K=-\infty }^{\infty }X(\omega -K{\omega }_s)$
After sampling, waveform will be


Applying an ideal low-pass filter of cut-off frequency of 20 Hz or 40$\pi$ rad/sec, we get the frequencies in reconstructed signal as 10$\pi$ and 30$\pi$ rad/sec or 5 Hz and 15 Hz