Question E8A10:
What is the purpose of a low-pass filter used at the output of a digital-to-analog converter?  

Choices:  
A: Lower the input bandwidth to increase the effective resolution  
B: Improve accuracy by removing out-of-sequence codes from the input  
C: Remove spurious sampling artifacts from the output signal  
D: All these choices are correct    

Correct Answer is  C: Remove spurious sampling artifacts from the output signal  

To reconstruct a smooth analog signal from a DAC, you must remove the high-frequency “images” created by sampling.

Why C is correct:
- A DAC output is a staircase that contains the desired baseband plus spectral replicas at multiples of the sampling rate. A low-pass “reconstruction” filter passes the desired band and suppresses those replicas (spurious sampling artifacts).
- In frequency terms, sampling creates copies of the spectrum at $f = k f_s \pm f_{\text{baseband}}$ for $k=1,2,\dots$. The reconstruction LPF removes these images.

Why the other choices are incorrect:
- A: “Lower the input bandwidth to increase the effective resolution” — The filter is at the output, not the input, and it does not change the DAC’s bit resolution. While oversampling plus filtering can improve in-band SNR, the LPF’s primary role here is not to increase resolution.
- B: “Remove out-of-sequence codes from the input” — That is a digital-coding issue (sometimes seen in ADC “bubble” errors). An analog LPF at the output cannot reorder or correct input codes.
- D: “All these choices are correct” — Only removing spurious sampling artifacts is the intended purpose in this context.

Useful formulas and rules:
- Nyquist limit: $$f_{\text{max}} < \frac{f_s}{2}$$
- Spectral images at: $$f = k f_s \pm f_{\text{signal}},\quad k=1,2,\dots$$
- Ideal reconstruction filter magnitude: 
  $$|H(f)| = \begin{cases}1, & |f| \le B \\ 0, & |f| > B \end{cases}$$
  where $B$ is the highest desired output frequency.
- Zero-order-hold (DAC) droop (envelope of the stair-step output):
  $$|H_{\text{ZOH}}(f)| = \left|\frac{\sin(\pi f / f_s)}{\pi f / f_s}\right|$$
- Rule of thumb: Choose the LPF cutoff just above the highest desired output frequency and well below $f_s - B$ to provide strong attenuation of the first image near $f_s$.