Question E7F09:
Why is an anti-aliasing filter required in a decimator?  

Choices:  
A: It removes high-frequency signal components that would otherwise be reproduced as lower frequency components  
B: It peaks the response of the decimator, improving bandwidth  
C: It removes low-frequency signal components to eliminate the need for DC restoration  
D: It notches out the sampling frequency to avoid sampling errors    

Correct Answer is  A: It removes high-frequency signal components that would otherwise be reproduced as lower frequency components  

To decimate (downsample) by an integer factor M, you reduce the sampling rate from $F_s$ to:
$$F_s' = \frac{F_s}{M}$$
By the Nyquist criterion, the signal must be band-limited to:
$$B \le \frac{F_s'}{2} = \frac{F_s}{2M}$$
An anti-aliasing filter is a low-pass filter applied before downsampling to remove components above this new Nyquist limit so they do not fold (alias) into baseband.

Why the correct answer is correct (A):
- When you drop samples, any spectral content above $F_s'/2$ folds into lower frequencies. The anti-aliasing filter removes those high-frequency components so they cannot be reproduced as spurious low-frequency components after decimation.
- Aliasing mapping (conceptually): a tone at frequency $f$ will appear after decimation near
  $$f_{\text{alias}} = |f - nF_s'| \quad \text{(for some integer n)}$$
  so anything above $F_s'/2$ can land inside the passband if not removed.

Why the other choices are incorrect:
- B: “Peaks the response of the decimator, improving bandwidth”  
  Anti-aliasing filters are low-pass and intentionally limit bandwidth; peaking the response does not prevent aliasing and is counter to the goal.
- C: “Removes low-frequency signal components to eliminate the need for DC restoration”  
  That’s a high-pass function unrelated to aliasing. Anti-aliasing is about removing high-frequency content before downsampling.
- D: “Notches out the sampling frequency to avoid sampling errors”  
  Aliasing is not caused by a tone at exactly the sampling frequency; it’s caused by all content above the new Nyquist limit. A single notch at $F_s$ (or harmonics) will not prevent aliasing; you need a low-pass filter.

Rules of thumb:
- Before decimating by $M$, use a low-pass with cutoff
  $$f_c \lesssim \frac{F_s}{2M}$$
  with some guard band to allow for transition width.
- Practical example: If $F_s = 96$ kHz and $M=3$, then $F_s' = 32$ kHz and you must low-pass to about $16$ kHz (or slightly less) before dropping samples.