Sampling is x[n]=x_c(nT_s) with f_s=1/T_s=8000 Hz and t in seconds

Sampling is \(x[n]=x_c(nT_s)\) with \(f_s=1/T_s=8000\,\mathrm{Hz}\) and \(t\) in seconds. Ideal reconstruction means the impulse train \(\sum_n x[n]\delta(t-nT_s)\) is passed through \[ H_r(j\Omega)=\begin{cases} T_s,& |\Omega|<\pi/T_s,\\ 0,& |\Omega|>\pi/T_s, \end{cases} \] with the two Nyquist-frequency values left unspecified. For each analog cosine \(x_c(t)=\cos(2\pi f_0 t)\), the discrete-time angular frequency is \(\omega=2\pi f_0 T_s\), reduced modulo \(2\pi\) into \((-\pi,\pi]\). (a) For each of \(f_0=1200\,\mathrm{Hz}\), \(f_0=4800\,\mathrm{Hz}\), and \(f_0=9200\,\mathrm{Hz}\), determine the principal \(\omega\in(-\pi,\pi]\) and the equivalent analog frequency \(f_{\mathrm{eq}}\in[0,f_s/2]\) of the cosine produced by ideal reconstruction. (b) Determine every analog frequency \(f_0\ge 0\) at which \(\cos(2\pi f_0 t)\) produces the same sample sequence as \(\cos(2\pi\cdot 1200\,t)\). Write the family as an explicit set. (c) Among the three tones of (a), decide which (if any) lie strictly below the Nyquist frequency \(4000\,\mathrm{Hz}\), which lie strictly above it, and which (if any) lie exactly at a multiple of \(f_s\). (d) If the three analog cosines are summed and then sampled, write the reconstructed analog signal \(x_r(t)\) as a sum of at most three real cosines with frequencies in \([0,4000]\,\mathrm{Hz}\). Combine any duplicate equivalent frequencies.

Fold each analog frequency into \((-f_s/2,f_s/2]\) by subtracting integer multiples of \(f_s\) and using that cosine is even.