Issue 78

C. Vendittozzi et alii, Fracture and Structural Integrity, 78 (2026) 519-540; DOI: 10.3221/IGF-ESIS.78.30

where H is the hop size and k is the discrete frequency bin. The corresponding spectrogram, expressed as power, was obtained as:     2 H,j H,j , , S m k X m k  ∣ ∣ (6) Thomson multitaper [18, 19]. In the multitaper formulation, the same signal segment was multiplied by K orthogonal discrete prolate spheroidal sequence (DPSS) tapers r v [n], with time–bandwidth product (NW). For the (r)-th taper, the tapered spectrum was computed as:



L

1

    n X mk xn mHvne     , r j 0 , j r  



2 / 

i

kn L

 

r

K

,

1, ,

(7)

and the multitaper spectrogram was obtained by averaging the corresponding power spectra:

K

1

1 K    ∣ r







 2 ∣

MT,j S m k ,

, r j X m k ,

(8)

In the DPSS-based multitaper formulation, NW denotes the time–bandwidth product, which controls the spectral concentration of each taper and therefore the trade-off between leakage reduction and frequency smoothing. The parameter denotes the number of orthogonal DPSS tapers used for each frame; the corresponding tapered spectra are then averaged to obtain the final multitaper estimate. Relative to a single-taper estimate with the same frame length (L) and hop (H), the multitaper average reduces spectral-estimate variance and mitigates leakage, improving the stability of narrowband components that are subsequently used for ridge tracking. The conceptual difference between the single-taper Hann estimate and the Thomson multitaper estimate is illustrated schematically in Fig. 3.

Figure 3: Schematic comparison between single-taper and multitaper time-frequency estimation. (a) Representative analysis frame of length . (b) In the Hann STFT, the frame is multiplied by a single Hann window, producing one tapered segment and one spectral estimate. (c) In the Thomson multitaper approach, the same frame is multiplied by mutually orthogonal DPSS tapers, producing tapered segments and spectral estimates that are subsequently averaged. The DPSS time-bandwidth product controls taper concentration and the leakage-versus-smoothing trade-off, whereas denotes the number of tapers used per frame. The conceptual comparison is performed under matched frame length and hop , so that only the tapering strategy changes. In the present study, the main multitaper configuration used =2048, =512, =3, and =5. This schematic highlights that the comparison adopted in the present work preserves the same frame length L and hop H, while changing only the tapering strategy and, in the multitaper case, averaging the K tapered spectral estimates. In the current application, the following windows were applied:  Main Hann STFT. The primary configuration used a Hann-window STFT with a window length L=2048 samples (2.048 sec at fs 1.0 kHz) and hop H=512 samples (0.512 sec), corresponding to 75% overlap. This choice yields a frequency resolution Δ / s f f L   0.488 Hz and a feature update rate / upd s r f H   1.953 Hz.

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