QuestionDrill Problem 5.7 Consider a continuous-time signal defined by $$g(t)=\frac{\sin (\pi t)}{\pi t}$$
The signal $g(t)$ is uniformly sampled to produce the infinite sequence $\left\{g\left(n T_{s}\right)\right\}_{n=-\infty}^{\infty}$. Determine the condition that the sampling period $T_{s}$ must satisfy so that the signal $g(t)$ is uniquely recovered from the sequence $\left\{g\left(n T_{s}\right)\right\}$.

Studdy Solution
Determine the condition on the sampling period $T_s$.
The sampling frequency $f_s$ is related to the sampling period $T_s$ by $f_s = \frac{1}{T_s}$. Therefore, the condition on $T_s$ is:
$$\frac{1}{T_s} > 1$$
Solving for $T_s$, we get:
$$T_s < 1$$
Thus, the sampling period $T_s$ must be less than 1 second to ensure that the signal $g(t)$ can be uniquely recovered from its samples.
The condition that the sampling period $T_s$ must satisfy is $\boxed{T_s < 1}$.