(Solved): 1. Consider the analog signal \[ x_{a}(t)=3 \cos 2000 \pi t+5 \sin 6000 \pi t+10 \cos 12000 \pi t \ ...

1. Consider the analog signal \[ x_{a}(t)=3 \cos 2000 \pi t+5 \sin 6000 \pi t+10 \cos 12000 \pi t \] (a) What is the Nyquist rate for this signal? (b) Assume now that we sample this signal using a sampling rate \( F_{s}=5000 \frac{\text { samples }}{\text { sec }} \). What is the discrete-time signal obtained after sampling? (c) What is the analog signal \( y_{a}(t) \) that we can reconstruct from the samples if we use ideal interpolation? 2. An Analog signal has frequency components at \( 1 \mathrm{kHz}, 12 \mathrm{kHz}, 28 \mathrm{kHz} \), and \( 30 \mathrm{kHz} \). The signal is sampled at a sampling rate of \( 20 \mathrm{kHz} \), then the resulting signal goes through an ideal reconstruction low pass filter (LPF) with a cutoff of \( 9 \mathrm{kHz} \). What frequencies are present in the output?