Question

Determine the conditions on the sampling interval \(T\) such that \[ x(t)=\frac{\cos(2\pi t)\sin(\pi t)}{\pi t} + \frac{2\sin(6\pi t)\sin(2\pi t)}{\pi t} \] is uniquely represented by the discrete-time sequence \(x[n]=x(nT)\).

• A. \(1\)
• B. \(\frac16\)
• C. \(\frac18\)
• D. \(\frac1{12}\)

Hint:

Always determine the highest frequency component first and then use \[ f_s \ge 2f_{\max}. \]

Answer 1

The Correct Option is D

Concept: For unique reconstruction, \[ f_s \ge 2f_{\max}. \]

Step 1: Find the highest frequency component.
Using spectral shifting properties, the first term occupies frequencies up to \[ 1.5~Hz. \] The second term occupies frequencies up to \[ 6~Hz. \] Hence, \[ f_{\max}=6Hz. \]

Step 2: Apply Nyquist criterion.
\[ f_s\ge 2f_{\max} = 12Hz. \]

Step 3: Determine sampling interval.
\[ T\le \frac1{12}. \] Among the given options, \[ \boxed{\frac1{12}} \] is the correct choice.