How to Obtain a Stable FRF Using Welch Averaging

Problem

A single-shot FFT of excitation and response gives an FRF estimate whose variance does not decrease with record length — it is essentially a one-sample estimate at every frequency. The result looks noisy, and worse, coherence computed from a single segment is meaningless (it is identically 1). How should the record be processed to get a stable, assessable estimate?

Likely Causes

Noisy or unrepeatable FRFs usually trace back to one of:

• too few averages (variance),
• leakage from a poor window/segment-length choice (bias, fake damping),
• excitation energy missing at the frequencies of interest,
• drift or transients included in the averaged segments.

Measurement Method

The Welch H1 recipe:

$$\hat{G}_1(f) = \frac{\hat{S}_{yu}(f)}{\hat{S}_{uu}(f)}, \qquad \gamma^2(f) = \frac{|\hat{S}_{yu}(f)|^2}{\hat{S}_{uu}(f)\,\hat{S}_{yy}(f)}$$

1. Segment length sets frequency resolution ($\Delta f = f_s / N_{seg}$). Choose it so the sharpest expected resonance spans several bins; otherwise damping is overestimated.
2. Window: Hann is the default for broadband excitation. For periodic excitation with an integer number of periods per segment, use a rectangular window and no overlap — leakage is zero by construction.
3. Overlap: 50 % with Hann recovers most of the information lost to the window taper.
4. Averages: the coherence estimate needs enough segments to be meaningful; single-digit averages give optimistic coherence. Increase until the FRF stops changing between runs.
5. Discard segments containing transients (start-up, touch events) before averaging.

Engineering Interpretation

Welch averaging trades resolution against variance for a fixed record length. The practical discipline: decide the required $\Delta f$ from the expected modal density first, then buy variance reduction with a longer total record — not by shortening segments. Low coherence localizes where the estimate cannot be trusted (noise, nonlinearity, or missing excitation), so publish the coherence with every FRF, not just the pretty magnitude curve.

What to Test Next

• Repeat the measurement with doubled segment length: if resonance peaks sharpen, the original resolution was biasing the damping estimate.
• Compare H1 against H2 estimators where coherence dips — divergence between them indicates where noise enters (input vs output).
• For MIMO cases, check the input spectral matrix condition number, not only per-channel coherence.