FFT, Windowing and Damping: Where "Always Use Hann" Is Wrong

1. The closed loop: how verification is set up

The usual way to demonstrate that a calculation tool is correct is to compare it with the output of third-party software. In signal processing this is unnecessary, because we can produce the input ourselves.

On top of that, the mathematical properties of the transform itself are tested. The cleanest is Parseval's identity — the energy in the time and frequency domains must be equal:

2. Resolution is 1/T and nothing else

In SHM-01 we saw that the sampling rate sets the upper limit. What sets the frequency resolution is the record length:

The practical consequence is sharp: two modes closer together than 1/T cannot be separated. Closely spaced modes, coupled translational–torsional pairs and near-frequency pairs in symmetric structures all run into this limit. There is only one remedy — a longer record.

3. Zero padding is not resolution

An FFT wants a length that is a power of two, and the shortfall is filled with zeros. Because this increases the number of points in the spectrum, it is often read as "the resolution improved". It did not.

The real benefit of zero padding is that it samples the position and width of a single peak more finely. That is useful in the half-power measurement of section 9.

4. Leakage and the case for windowing

The FFT assumes the record repeats forever. If the start and the end of the record do not match, that assumption produces a discontinuity, and the discontinuity spreads through the spectrum as leakage: the energy of a component does not stay in its own bin but spills into neighbouring bins.

The most visible consequence is that the amplitude reads low. In the table below, the amplitude of a pure sinusoid that does not sit on a bin centre has been measured — the true amplitude is 1.0000.

Up to this point the table confirms the textbook advice, "window your data to reduce leakage". The next section shows where that advice breaks down.

5. Finding: on a transient, Hann halves the damping

Now apply the same chain to a free-decay record. Because we generated the signal ourselves, we know the true damping ratio: ζ = 2.00%. We then try to read it back with the half-power method.

This is not a negligible bias. Reporting damping as 1% instead of 2% means understating the structure's energy dissipation capacity by half — and damping enters base shear directly in seismic response calculations.

6. Why? Where the energy sits

The cause lies jointly in the shape of the window and the shape of the signal.

All of the energy of a free-decay response is at the beginning of the record. The signal is largest at t = 0 and decays exponentially. The Hann window, on the other hand, goes to exactly zero at both ends and rises to one in the middle. So when the two curves are multiplied, the window suppresses precisely the most informative part of the signal.

What is left is no longer an exponentially decaying signal: it is something that first rises and then falls, artificially symmetrised. Its spectral peak therefore differs from the true Lorentzian shape and its bandwidth narrows — hence a small ζ.

7. The right rule: look at the signal type

Ambient vibration records are stationary, so Hann is the right choice in operational modal analysis. That is why "always Hann" looks harmless in most SHM contexts — until someone tries to extract damping from a free-decay record.

8. Peak picking and parabolic interpolation

Modal frequencies are local maxima in the spectrum. But there is no reason for the true frequency to fall exactly on a bin centre; it usually falls between two. Fitting the peak together with its neighbours to a parabola locates it below bin resolution.

9. Half-power bandwidth damping

The width of a damped peak is directly related to the damping ratio. The band between the two frequencies where the amplitude falls to 1/√2 of the peak value:

The method has two conditions, and both must travel with the result:

• Light damping. The approximation degrades as ζ grows; above 10% the result should be read as an upper bound.
• Sufficient resolution. If the bandwidth spans only a few bins the measurement is meaningless. Our engine reports the number of bins in the band and warns below four.

The second condition has an unexpected consequence: the record length needed to measure damping is inversely related to the damping itself. Small ζ means a narrow peak, a narrow peak needs fine resolution, and fine resolution needs a long record. Measuring damping in a lightly damped structure takes a longer record than in a heavily damped one.

10. End-to-end verification

The whole chain is tested with a two-mode synthetic signal: known frequencies 1.25 and 3.75 Hz.

11. Common mistakes

1. Choosing the window without looking at the signal type. The whole subject of this article. On a transient record, Hann halves the damping.
2. Reading zero padding as improved resolution. The number of points grows; the ability to discriminate does not.
3. Expecting better resolution from a higher sampling rate. f_s sets the upper limit, T sets the resolution.
4. Reporting a half-power result without checking the bin count. If the bandwidth spans 2–3 bins, the result is noise.
5. Confusing parabolic interpolation with mode separation. Two peaks that are not visible do not become visible through interpolation.
6. Skipping the amplitude correction. A tapering window lowers the amplitude; read without dividing by the coherent gain, it is systematically low.

12. Test yourself

1. What is the frequency resolution of a 60-second record? Can two modes 0.01 Hz apart be separated with it?
2. You are extracting damping from the free-decay record of a hammer-excited beam. Which window do you choose, and why?
3. You recorded the same beam vibrating continuously in wind. Does your window choice change?
4. What is the bandwidth of a mode with ζ = 0.5% at f₀ = 2 Hz? What is the minimum record length needed to sample it with 8 bins?
5. Does satisfying Parseval's identity also show that the spectrum is being interpreted correctly? Justify your answer.