Question

Let $x(t)=\cos(10\pi t)+\cos(30\pi t)$ be sampled at $20$ Hz and reconstructed using an ideal low–pass filter with cut-off frequency of $20$ Hz. The frequency/frequencies present in the reconstructed signal is/are

• A. $5$ Hz and $15$ Hz only
• B. $10$ Hz and $15$ Hz only
• C. $5$ Hz, $10$ Hz and $15$ Hz only
• D. $5$ Hz only

Hint:

Frequencies above Nyquist fold back into the baseband as aliases.

Answer 1

The Correct Option is D

Step 1: Find frequencies of the given signal.
\[ x(t)=\cos(10\pi t)+\cos(30\pi t) \] \[ f_1=\frac{10\pi}{2\pi}=5\text{ Hz}, \quad f_2=\frac{30\pi}{2\pi}=15\text{ Hz} \]
Step 2: Identify sampling frequency.
\[ f_s=20\text{ Hz}, \quad \text{Nyquist frequency}=10\text{ Hz} \]
Step 3: Check aliasing.
The $5$ Hz component is below Nyquist and remains unchanged.
The $15$ Hz component is above Nyquist and aliases to:
\[ |f_s-f_2|=|20-15|=5\text{ Hz} \]
Step 4: Effect of ideal reconstruction filter.
The ideal low-pass filter passes frequencies up to $20$ Hz, so the aliased $5$ Hz component passes through.
Step 5: Final conclusion.
Both components appear at $5$ Hz after reconstruction. Hence, only $5$ Hz is present.