$($Sampling$)$ A continuous$-$time signal $(t)$ is obtained at the output of an ideal low$-$pass filter with cutoff frequency ${}_c=2000.$

$($a$)(5$ pts $)$ What is the maximum frequency ${}_M$ of $x(t)?$

$($b$)(5$ pts$)$ What is the Nyquist rate of the signal $x(t)?$

$($c$)(10$ pts$)$ What is the requirement on sampling period $T_s$ such that no aliasing will occur when recovering $x(t)$ from its samples?

If impulse$-$train sampling is performed on $(t),$ would the following sampling periods guarantee that $x(t)$ can be recovered from its sampled version using an appropriate lowpass filter and why?

$($d$)(5pts)T_s=0\mathrm{.}0001s$

$($e$)(5pts)T_s=0\mathrm{.}00025s$

$($f$)(5$ pts$)T_s=0\mathrm{.}00075s$

$($g$)(5pts)T_s=0\mathrm{.}001s$

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