Consider the following sinusoidal signal with the fundamental frequency ${f}_{0}$f0 of 4 kHz:

$g\left(t\right)=5\mathrm{cos}\left(2\pi {f}_{0}t\right)=5\mathrm{cos}\left(800\pi t\right)\mathrm{.}$

(i) The sinusoidal signal is sampled at a sampling rate ${f}_{s}$fs of 6000 samples/s and reconstructed with an ideal LPF with the following transfer function:

${H}_{1}\left(\omega \right)=\{\begin{array}{cc}1\mathrm{/}6000& \mathrm{\mid}\omega \mathrm{\mid}\le 6000\pi \\ 0& elsewhere\mathrm{.}\end{array}$

Determine the reconstructed signal.

(ii) Repeat (i) for a sampling rate ${f}_{s}$fs of 12 000 samples/s and an ideal LPF with the following transfer function:

${H}_{2}\left(\omega \right)=\{\begin{array}{cc}1\mathrm{/}12000& \mathrm{\mid}\omega \mathrm{\mid}\le 12000\pi \\ 0& elsewhere\mathrm{.}\end{array}$