Question E7F05:
How frequently must an analog signal be sampled to be accurately reproduced?  

Choices:  
A: At least half the rate of the highest frequency component of the signal  
B: At least twice the rate of the highest frequency component of the signal  
C: At the same rate as the highest frequency component of the signal  
D: At four times the rate of the highest frequency component of the signal    

Correct Answer is  B: At least twice the rate of the highest frequency component of the signal  

To be accurately reproduced (perfectly reconstructed), an analog signal must be sampled at least twice the highest frequency present. This is the Nyquist-Shannon sampling theorem.

Key formula:
$$f_s \ge 2 f_{max}$$
- $f_s$ = sampling rate
- $f_{max}$ = highest frequency component of the signal
- The Nyquist frequency is $f_N = \frac{f_s}{2}$

Why B is correct:
- Sampling at $f_s \ge 2 f_{max}$ prevents aliasing (overlap of spectral replicas) and allows exact reconstruction with an ideal low-pass filter.

Why the others are incorrect:
- A: “Half the rate” misreads the theorem. $f_N = \frac{f_s}{2}$ must be at least $f_{max}$, not the other way around.
- C: “Same rate” ($f_s = f_{max}$) gives $f_N = \frac{f_s}{2} = 0.5 f_{max}$, which is too low; aliasing occurs.
- D: “Four times” is not required. Oversampling (e.g., $4 f_{max}$) can ease filter design and reduce noise, but the minimum is $2 f_{max}$.

Rules of thumb:
- Use an anti-aliasing filter so the input is band-limited to $f_{max} \le \frac{f_s}{2}$.
- In practice, choose $f_s$ slightly above $2 f_{max}$ to allow for real filter roll-off (e.g., voice band to 3 kHz → sample at 8 kHz).