Questions: Decimation, Anti-Aliasing, and Downsampling

Question 1

A digital signal sampled at 48 kHz contains a tone at 10 kHz. It is downsampled by factor M = 4 (to 12 kHz) without any prior filtering. What appears in the output?

• A. A 10 kHz tone — downsampling preserves the signal content faithfully
• B. Nothing — the 10 kHz tone is above the new Nyquist rate and is silently discarded
• C. A spurious 2 kHz tone — the 10 kHz component folds into the baseband through aliasing
• D. A 6 kHz tone — the component wraps to the new Nyquist boundary

Answer: (C)

After downsampling by 4, the new sample rate is 12 kHz and the new Nyquist is 6 kHz. The 10 kHz tone exceeds the new Nyquist and aliases: it folds to |12 − 10| = 2 kHz. A 2 kHz tone now appears in the output that was never in the original signal — this is aliasing corruption, not mere information loss. This example shows why unfiltered downsampling is destructive: it does not just discard high frequencies, it injects false low-frequency components into the band you care about.

Question 2

In a properly designed decimator with downsampling factor M, where should the anti-aliasing lowpass filter's cutoff frequency be set?

• A. At fs/2 — the original Nyquist rate, to reject any frequencies above that
• B. At fs/(2M) — the new Nyquist rate after downsampling
• C. At fs/M — the new sample rate itself
• D. At 2fs/M — twice the new sample rate to provide margin

Answer: (B)

The new Nyquist rate after downsampling to fs/M is fs/(2M). Any signal energy above fs/(2M) will alias into the 0 to fs/(2M) band in the decimated output. The anti-aliasing filter must attenuate all energy above fs/(2M) before the downsampling step occurs. Setting the cutoff at the original Nyquist (fs/2) would fail to remove the components between fs/(2M) and fs/2 that cause aliasing — exactly the wrong answer and the most common design error.

Question 3

Downsampling without prior anti-aliasing filtering can introduce frequency components into the output that were not present in the original signal.

• T. True
• F. False

Answer: True

This is the essence of aliasing: high-frequency components fold into lower frequencies when the sampling rate is reduced. A tone at frequency f that exceeds the new Nyquist rate fs/(2M) reappears at a lower aliased frequency |k·(fs/M) − f| for some integer k. These aliased tones were never in the original baseband signal — they are fabricated artifacts of the undersampling process. This is why the filter must precede the downsampler: once aliasing occurs, there is no way to undo it.

Question 4

The purpose of the anti-aliasing filter in a decimation system is primarily to remove high-frequency noise and improve signal quality before storage.

• T. True
• F. False

Answer: False

Noise removal is a side benefit, but the primary purpose is to prevent aliasing — specifically, to eliminate signal energy between fs/(2M) and fs/2 that would otherwise fold into the baseband and corrupt the low-frequency content the decimator is trying to preserve. Without the filter, a clean signal with no noise can still be badly corrupted by its own high-frequency components aliasing into the passband. The filter is not optional cleanup; it is a required step for mathematically correct decimation.

Question 5

A colleague claims: 'Downsampling just throws away samples, so it can only lose information — it can't add anything or corrupt the remaining signal.' What is wrong with this reasoning, and what actually happens to a high-frequency tone during unfiltered downsampling?

Model answer: The error is treating downsampling as simple deletion. Downsampling is equivalent to re-sampling the signal at a lower rate. When a signal contains energy above the new Nyquist rate, that energy is not simply absent from the output — it folds back (aliases) into the lower frequency band through spectral wrapping. A tone at frequency f aliases to |f − k·fs_new| for the nearest integer k. This creates a new tone at a completely different frequency that was never present in the original, corrupting the passband content the decimator was supposed to preserve faithfully.

The intuition for why this happens: keeping every Mth sample is mathematically equivalent to multiplying the signal by a pulse train with period M, which in the frequency domain convolves with a train of impulses at multiples of fs/M. This convolution copies the spectrum at every multiple of fs/M — and these copies overlap with the baseband if the signal isn't bandlimited to fs/(2M) first. The overlap is the aliasing, and it is additive corruption, not mere truncation.