DSP 101
M2 · L2
Module 2 — Sampling & Quantization
Aliasing —
What Happens When You Under-sample

Breaking the Nyquist rule has consequences. High frequencies fold back, disguise themselves, and corrupt your signal permanently. Let’s see exactly how.

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DSP 101
M2 · L2
The Core Problem
What Is Aliasing?

When you sample below the Nyquist rate (fs < 2fmax), the spectrum’s periodic copies collide and overlap. High-frequency components masquerade as low-frequency impostors — and once it happens, it’s irreversible.

Key property
Aliasing cannot be undone. No filter, no AI, no post-processing can recover the lost information once sampling has occurred.
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DSP 101
M2 · L2
The Mechanism
Frequency Folding

Sampling at fs creates spectral copies spaced fs apart. The folding frequency is fs/2. Any signal energy above this limit reflects back symmetrically.

Alias Frequency
f_{\text{alias}} = \left| f - \operatorname{round}\!\left(\tfrac{f}{f_s}\right) \cdot f_s \right|

Example: a 7 kHz tone sampled at 10 kHz folds to 10 − 7 = 3 kHz.

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DSP 101
M2 · L2
In the Wild
The Wagon-Wheel Effect

A film camera shoots at 24 frames/s (Nyquist: 12 Hz). One marked spoke turning at 23 rev/s appears to spin backwards at 1 Hz; at 25 rev/s it crawls forwards at 1 Hz — temporal aliasing made visible.

  • Moiré patterns: spatial aliasing from fine textures exceeding pixel Nyquist limit
  • Camera OLPFs (optical low-pass filters) prevent sensor-level aliasing
  • Video compression can introduce temporal aliasing in fast motion
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DSP 101
M2 · L2
Phantom Tones
Audio Aliasing

A 22.5 kHz harmonic sampled at 44.1 kHz (Nyquist: 22.05 kHz) aliases to 44.1 − 22.5 = 21.6 kHz. An inharmonic ghost tone corrupts the recording.

22.05
CD Nyquist kHz
0 dB
Alias power loss
∞
Post-fix difficulty
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DSP 101
M2 · L2
The Solution
Anti-Aliasing Filter

Apply an analog lowpass filter before the ADC. Attenuate everything above fs/2. Whatever reaches the sampler satisfies Nyquist by construction.

  • Real filters have transition bands — gradual roll-off, not brick walls
  • Steeper roll-off requires more complex, expensive analog stages
  • 7th-order Butterworth provides ~60 dB rejection at 1.1× Nyquist
  • Every professional ADC includes an anti-aliasing filter
Interactive: switch a real Butterworth filter in and watch the folds shrink but never vanish — AI-generated
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DSP 101
M2 · L2
Modern Approach
Oversampling + Digital Filtering

Sample at 4× or 64× the Nyquist rate. Use a simple analog filter for gross rejection, then a sharp digital FIR filter and decimate to the target rate.

Oversampling Ratio
\text{OSR} = \dfrac{f_s}{2\,f_{\max}}

Sigma-delta ADCs use OSR = 64–256. Sharp digital filters cost nothing in silicon.

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DSP 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
Score 0/0

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DSP 101
M2 · L2
Key Takeaways
What You Learned
  • Aliasing folds high frequencies below fs/2, creating phantom signals
  • It is permanent — no post-processing can undo it
  • Anti-aliasing filters bandlimit the signal before sampling
  • Oversampling moves the alias zone out, simplifying the analog front end
  • Aliasing appears in video, audio, radar, MRI, and wireless systems
Next Lesson
Quantization & Bit Depth — continuous amplitude meets discrete levels.
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