Math / Calculus

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

STEP 1

1. The signal $g(t) = \frac{\sin(\pi t)}{\pi t}$ is a sinc function.
2. The sinc function is a bandlimited signal with a specific bandwidth.
3. To recover a continuous-time signal from its samples, the sampling theorem (Nyquist-Shannon) must be satisfied.
4. The sampling period $T_s$ must be chosen such that the sampling frequency $f_s = \frac{1}{T_s}$ is greater than twice the maximum frequency of the signal.

STEP 2

1. Identify the bandwidth of the signal $g(t)$ .
2. Apply the Nyquist-Shannon sampling theorem.
3. Determine the condition on the sampling period $T_s$ .

STEP 3

Identify the bandwidth of the signal $g(t) = \frac{\sin(\pi t)}{\pi t}$ .
The sinc function $\frac{\sin(\pi t)}{\pi t}$ is known to be bandlimited with a cutoff frequency of $\frac{1}{2}$ Hz. This is because the Fourier transform of a sinc function is a rectangular function in the frequency domain.

STEP 4

Apply the Nyquist-Shannon sampling theorem.
According to the Nyquist-Shannon sampling theorem, to perfectly reconstruct a bandlimited signal from its samples, the sampling frequency $f_s$ must be greater than twice the maximum frequency of the signal.
For $g(t)$ , the maximum frequency is $\frac{1}{2}$ Hz, so the sampling frequency must satisfy:
$f_s > 2 \times \frac{1}{2} = 1 \text{ Hz}$

STEP 5

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 $\boxed{T_s < 1}$ .

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