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$x(t)$ is a signal of bandwidth $4 \mathrm{~kHz}$. It was sampled at a rate of $16 \mathrm{~kHz}$.
\[
x_{n}=x(n T), \quad n \text { integer, } \quad T=\frac{1}{16} \mathrm{~ms} .
\]
Due to a data handling error alternate samples were erased and set to $0.$
\[
y_{n}=\left\{\begin{array}{ll}
x_{n}, & n \text { even, } \\
0, & n \text { odd }
\end{array}\right.
\]
Without realizing this error an engineer uses $\operatorname{sinc}$ interpolation to try to reconstruct $x(t)$. She obtains
\[
y(t)=\sum_{n=-\infty}^{\infty} y_{n} \operatorname{sinc}\left(\frac{t-n T}{T}\right), T=\frac{1}{16} \mathrm{~ms} .
\]
Using which of the procedures below can she recover $c x(t)$ from $y(t)$ where $c$ is some non-zero scaling factor?

  1. resample $y(t)$ at $16 \mathrm{~kHz}$ and sinc interpolate using $T=\frac{1}{8} \mathrm{~ms}$
  2. resample $y(t)$ at $8 \mathrm{~kHz}$ and sinc interpolate using $T=\frac{1}{16} \mathrm{~ms}$
  3. send $y(t)$ over a low pass filter of bandwidth $4\text{ KHz}$
  4. any of the above
  5. none of the above
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