Explaination on Nyquist criterion

[amh]

Nyquist criterion is a fundamental principle for sampling continuous-time signals to avoid aliasing. In brief, to perfectly reconstruct a bandlimited signal after sampling, the sampling frequency must be at least twice the highest frequency present in the signal.

Direct answer:

- Critical condition: f_s > 2 f_max where f_s is the sampling frequency and f_max is the maximum frequency component of the signal content.
- Practical interpretation: To safely capture dynamics up to a maximum frequency f_max without aliasing, choose a sample rate f_s that satisfies f_s ≥ 2 f_max (often with a safety margin, e.g., f_s = 2.5–4 times f_max in practice).
- Anti-aliasing prefilter: Before sampling, apply an anti-aliasing filter with a cutoff below f_s/2 to attenuate content above the Nyquist frequency f_N = f_s/2.
- Real-world considerations: If the signal is not strictly bandlimited or if higher-frequency content exists due to transients, spectral leakage and aliasing can occur despite precautions; choose windowing, pre-filtering, and sampling rate to mitigate these effects.

Key formulas:

- Nyquist frequency: $$ f_N = \frac{f_s}{2} $$
- Sampling theorem condition: $$ f_s > 2 f_{\max} $$ (strictly, $$ f_s \ge 2 f_{\max} $$ for ideal reconstruction)
- Alternative conservative guideline: select $$ f_s $$ so that $$ f_s $$ is at least 4× the highest frequency of interest to provide robust anti-aliasing and facilitate processing.

Practical tips for vibration analysis:

- Identify the highest vibration mode of interest (e.g., seat-supported modes up to 100 Hz). Set f_s to at least 200 Hz, preferably 400 Hz or more depending on transients and measurement bandwidth.
- Use an anti-aliasing low-pass filter with a cutoff below f_N to ensure minimal leakage from frequencies above Nyquist.
- If re-sampling or decimation is required, apply proper digital filtering to avoid aliasing.