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
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
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Frequencies above Nyquist fold back into the baseband as aliases.
Updated On: Feb 9, 2026
- $5$ Hz and $15$ Hz only
- $10$ Hz and $15$ Hz only
- $5$ Hz, $10$ Hz and $15$ Hz only
- $5$ Hz only
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The Correct Option is D
Solution and Explanation
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.
\[ x(t)=\cos(10\pi t)+\cos(30\pi t) \] \[ f_1=\frac{10\pi}{2\pi}=5\text{ Hz}, \quad f_2=\frac{30\pi}{2\pi}=15\text{ Hz} \]
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.
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