Digital Signal Processing

Step 1: Identify frequency components of .
Given:

Angular frequencies are and rad/s.
Step 2: Convert angular frequency to frequency in Hz.

Maximum frequency in is Hz.
Step 3: Effect of time scaling in .
Time scaling by a factor of increases all frequencies by a factor of .

Step 4: Apply Nyquist sampling theorem.

Step 5: Final conclusion.
The Nyquist sampling rate is 16 samples per second.

Step 1: Identify carrier frequency.
Given carrier term .

Step 2: Identify message bandwidth.
Baseband signal bandwidth:

Step 3: Determine highest frequency in .
For amplitude modulation:

Step 4: Apply Nyquist sampling criterion.

Step 5: Minimum practical sampling rate.
To avoid aliasing and ensure recovery of , the closest higher option is 60 kHz.
Step 6: Final conclusion.
Minimum required sampling rate is 60 kHz.

Step 1: Find frequencies of the given signal.

Step 2: Identify sampling frequency.

Step 3: Check aliasing.
The Hz component is below Nyquist and remains unchanged.
The Hz component is above Nyquist and aliases to:

Step 4: Effect of ideal reconstruction filter.
The ideal low-pass filter passes frequencies up to Hz, so the aliased Hz component passes through.
Step 5: Final conclusion.
Both components appear at Hz after reconstruction. Hence, only Hz is present.