Aliasing¶

Formula¶

\[ \omega_{\text{alias}} = \omega + 2\pi k\quad (k\in\mathbb{Z}) \]

Parameters¶

\(\omega\): angular frequency
\(k\): integer shift due to sampling

What it means¶

High-frequency components become indistinguishable from lower frequencies after sampling.

What it's used for¶

Explaining artifacts from undersampling.
Choosing safe sampling rates.

Key properties¶

Caused by sampling below the Nyquist rate
Spectra repeat every \(2\pi\) in discrete time

Common gotchas¶

Anti-alias filtering is required before sampling.
Aliasing can appear even with noisy measurements if bandwidth is not controlled.

Example¶

A 9 Hz sine sampled at 10 Hz appears as a 1 Hz sine (alias).

How to Compute (Pseudocode)¶

Input: continuous frequency f, sampling rate f_s
Output: aliased frequency observed after sampling (baseband representative)

# Fold frequency into the principal interval around 0 (or [0, f_s/2] by convention)
f_alias <- f modulo f_s
if f_alias > f_s / 2:
  f_alias <- f_s - f_alias

return f_alias

Complexity¶

Time: \(O(1)\) for a single frequency alias-mapping calculation
Space: \(O(1)\)
Assumptions: Scalar-frequency alias mapping shown; conventions differ for signed frequency, angular frequency, and FFT-bin indexing