Nyquist-Shannon Sampling Theorem¶

Formula¶

\[ \text{If } f_s \ge 2B,\quad x(t)=\sum_{n=-\infty}^{\infty} x(nT)\,\operatorname{sinc}\left(\frac{t-nT}{T}\right) \]

Parameters¶

\(B\): signal bandwidth (Hz)
\(f_s=1/T\): sampling frequency

What it means¶

Bandlimited signals can be perfectly reconstructed from uniform samples.

What it's used for¶

Choosing sampling rates for bandlimited signals.
Preventing aliasing in data acquisition.

Key properties¶

Nyquist rate is \(2B\)
Reconstruction uses sinc interpolation

Common gotchas¶

Bandlimiting is an assumption; practical signals need anti-alias filters.
Sampling below Nyquist causes irreversible aliasing.

Example¶

If a signal is bandlimited to 3 kHz, any \(f_s>6\) kHz (e.g., 8 kHz) is sufficient.

How to Compute (Pseudocode)¶

Input: estimated signal bandwidth B, candidate sampling rate f_s
Output: sampling-rate check (and reconstruction workflow note)

nyquist_rate <- 2 * B
if f_s >= nyquist_rate:
  report "sampling theorem condition satisfied (ideal bandlimited case)"
else:
  report "aliasing risk: below Nyquist rate"

# Practical workflow note:
# apply anti-alias filtering before sampling; ideal reconstruction uses sinc interpolation

Complexity¶

Time: \(O(1)\) for the Nyquist-rate check itself
Space: \(O(1)\)
Assumptions: This is a design/check workflow for the theorem condition; practical reconstruction/filtering cost depends on the implementation and signal length