Peak Picking (PP)¶

This page expands 2.1 PP from the SHM roadmap: how to quickly estimate modal frequencies and mode shapes using peak picking.

Basic idea¶

Frequency: Compute the FFT or power spectrum of each channel; the frequencies of the dominant peaks correspond to the damped natural frequencies of the modes (for light damping, approximately the undamped natural frequency \(\omega_n\)), and are used as the modal frequency estimates.
Mode shape: At each peak frequency \(\omega_r\), form the ratio of each channel’s amplitude to the reference; that ratio approximates the relative mode shape. Physically the quantity used must be “amplitude” (same as the vibration amplitude at that frequency component), not raw power—the ratio of FFT magnitudes and the ratio of PSD values correspond to different physical quantities, so mode-shape estimation must treat them differently (below).

Under a single-mode-dominant approximation, the response at that frequency is dominated by that mode, so the mode-shape ratio equals the amplitude ratio:

\[ \frac{\phi_j}{\phi_{\mathrm{ref}}} \approx \frac{A_j(\omega_r)}{A_{\mathrm{ref}}(\omega_r)}, \]

where \(A_j\) is the amplitude at sensor \(j\) at \(\omega_r\) (given by \(|X_j|\) or \(\sqrt{S_j}\) as below).

FFT vs power spectrum (PSD)

FFT (Fourier spectrum)
The ratio of \(\lvert X_j(\omega_r) \rvert\) across channels equals the ratio of amplitudes at that frequency (the modulus of \(X\) is proportional to amplitude, with the same scaling for all channels for a given window). So take \(A_j = \lvert X_j(\omega_r) \rvert\) to estimate the mode-shape ratio:

\[ \frac{\phi_j}{\phi_{\mathrm{ref}}} \approx \frac{\lvert X_j(\omega_r) \rvert}{\lvert X_{\mathrm{ref}}(\omega_r) \rvert}. \]

Power spectrum (PSD)
\(S\) has dimension [signal²/frequency] (e.g. (m/s²)²/Hz); \(S \propto \lvert X \rvert^2\), so amplitude \(\propto \sqrt{S}\)—only the ratio of \(\sqrt{S_j}\) across channels equals the amplitude ratio. With \(A_j = \sqrt{S_j(\omega_r)}\), use the ratio of square roots:

\[ \frac{\phi_j}{\phi_{\mathrm{ref}}} \approx \frac{\sqrt{S_j(\omega_r)}}{\sqrt{S_{\mathrm{ref}}(\omega_r)}}. \]

Using \(S_j/S_{\mathrm{ref}}\) instead gives \((\phi_j/\phi_{\mathrm{ref}})^2\)—wrong dimension and wrong physical meaning.

Procedure (outline)¶

FFT or power spectrum (e.g. Welch) for each channel.
Identify dominant peaks in the frequency range of interest; record their frequencies \(\omega_r\).
Choose a reference channel; at each \(\omega_r\), read the amplitude at every channel: use \(|X_j(\omega_r)|\) when using FFT, or \(\sqrt{S_j(\omega_r)}\) when using PSD (do not use \(S_j\) directly—wrong dimension).
Form the ratio of each channel’s amplitude to the reference; normalize the vector (e.g. by max or unit norm).
(Optional) If complex spectra are available, use the complex ratio to retain relative phase.

When it works and limitations¶

Use when: Modes are well separated, damping is low, and SNR is good; suited to quick checks or first estimates.
Limitations: No damping estimate; peak overlap or close modes are hard to separate; sensitive to noise and reference choice; mode shape is an amplitude-ratio approximation only.

Engineering practice: shortcomings and mitigations¶

In practice, PP’s weaknesses and the measures that can mitigate them are as follows; understanding them helps choose when PP is “good enough” versus when to add complexity.

Main shortcomings — Spectrum variance and noise (single-segment FFT has high variance, risk of false/missed detections); frequency resolution vs window length (\(\Delta f \approx 1/T\): too short blurs close modes, too long increases latency); peak-picking strategy (simple threshold mistakes harmonics/noise for peaks, discrete bins add scalloping); reference and mode-shape sign (amplitude ratio gives only \(|\phi_j/\phi_{\mathrm{ref}}|\), and reference near a node makes the ratio unstable); close modes (PP cannot disentangle when two frequencies are within \(\sim 1/T\) or damping bandwidth); no damping (PP does not yield damping for assessment).

Practical mitigations

Shortcoming	Mitigation	Notes
Spectrum variance, noise	Segment averaging (e.g. Welch), band-pass filter	Averaging stabilises peaks; band-pass limits band and suppresses out-of-band noise.
Peak reliability	Peak prominence, local fit (e.g. parabola)	Prominence rejects broad bumps; local fit gives sub-bin frequency.
Multiple peaks	Multi-window / multi-segment voting, prior band	Persistent peaks are more credible; search only in known band when available.
Reference and sign	Reference away from nodes, use complex ratio	Choose sensor with large amplitude; complex spectrum retains phase/sign.
Close modes	Use PP for screening only, upload suspect data	“Significant change yes/no”; use FDD/SSI for dense bands offline or in cloud.
Window length	Resolution–latency trade-off	\(\Delta f\) smaller than minimum mode spacing; shorten window when real-time matters.

Summary — PP’s strength is simplicity and low compute, suited to edge and online screening; its weaknesses come from “no model, just peaks”. Averaging, filtering, robust peak picking, and reference choice improve usability; close modes and damping are better handled downstream.

Edge and online¶

Why it fits — Compute and memory: only FFT (\(O(N\log N)\)) and peak search (\(O(N)\)), no matrix factorisations or iterations, memory linear in window length; suitable for MCUs and low-power SoCs. Streaming and latency: sliding or block FFT with fixed window gives predictable latency, “compute as you sample, alert on device”. Potential: when modes are well separated and excitation stable, PP at the edge can screen “has frequency or mode shape shifted noticeably?” and upload suspect segments for FDD/SSI, reducing bandwidth and cloud load as the first stage of a tiered SHM pipeline.

Challenges and trade-offs — Noise and robustness: edge SNR is often lower; trade off window length (resolution and noise rejection) vs latency; light improvements (smoothing, multi-window voting) help but add compute. Close modes and mode-shape quality: a single peak can mix two modes; PP is better for “significant change yes/no” than fine decomposition. No damping: if damage shows mainly in damping, combine with other features or uplink. Reference: choose a robust reference (away from nodes) from prior or offline analysis, or accept limited reliability for some modes.