Transient Spectral Centroid

The centroid is calculated as the weighted mean of the frequencies present in the transient's Short-Time Fourier Transform (STFT) spectrum, where the weights are the normalized magnitudes of each frequency bin. It is mathematically expressed as the sum of the product of frequency and spectral magnitude divided by the total spectral magnitude. This collapses a high-dimensional spectrum into a single scalar value representing the center of mass of the spectral energy distribution.

The centroid value is highly sensitive to the physical dynamics of the transmitter's components during the power-up sequence. A faster rise-time, often caused by a high-slew-rate power amplifier, generates sharper edges that inject energy into higher frequency bins, shifting the centroid upward. Conversely, a slow, damped response from a poorly regulated power supply biases energy toward lower frequencies, lowering the centroid value.

As a feature for device identification, the spectral centroid effectively separates transmitters with different frequency settling profiles. For instance, a device with an underdamped Phase-Locked Loop (PLL) will exhibit ringing at a specific high frequency, creating a distinct, repeatable upward shift in the centroid compared to a critically damped device of the same model. This makes it a robust metric for distinguishing between otherwise identical hardware units.

The centroid is computationally efficient and exhibits robustness against additive white Gaussian noise, as noise energy is uniformly distributed and does not systematically bias the weighted mean. However, it is vulnerable to narrowband interference. A strong, non-transient interferer, such as a continuous-wave tone from another transmitter, can completely dominate the spectral magnitude and pull the centroid toward the interferer's frequency, masking the true transient signature.

Rather than a single global value, the centroid is often computed over a sequence of overlapping short-time windows to create a centroid trajectory. This trajectory visualizes how the spectral balance evolves during the transient. A typical trajectory might start high due to initial spectral splatter, dip as the PLL settles, and then stabilize at the steady-state modulation centroid. The shape of this trajectory provides a richer fingerprint than a static value.

Extraction involves three core steps: 1) Burst Onset Detection to isolate the transient from the noise floor. 2) Computation of the STFT with a window size optimized for the transient duration (typically microseconds). 3) Application of the centroid formula to each time slice. The resulting feature vector is often normalized against the carrier frequency to make the fingerprint invariant to the absolute operating channel, allowing a model trained at one frequency to recognize a device at another.

A companion spectral shape descriptor often used alongside the centroid. The roll-off point is the frequency below which a specified percentage (typically 85% or 95%) of the total spectral energy is concentrated. While the centroid indicates the balancing point of the spectrum, the roll-off defines its bandwidth. A transient with a high centroid but a low roll-off suggests energy is concentrated in a narrow high-frequency band, whereas a high centroid with a high roll-off indicates broadband high-frequency energy, such as that from a fast-switching DAC glitch.

A measure of how quickly the power spectrum of a signal changes, calculated as the squared difference between the normalized magnitudes of successive STFT frames. During a turn-on transient, spectral flux is exceptionally high as the signal rapidly evolves from noise to a stable modulated carrier. The Transient Spectral Centroid's trajectory over time, when combined with spectral flux, can pinpoint the exact moments of maximum spectral instability, such as the PLL overshoot peak or the onset of ringing artifacts.

A higher-order statistical measure applied to the STFT magnitude spectrum to detect non-Gaussian, impulsive events in the frequency domain. While the centroid provides the mean location of energy, spectral kurtosis indicates the peakedness of the spectral distribution. A transient with high spectral kurtosis contains energy highly concentrated in a few frequency bins, which is characteristic of transient carrier feedthrough or a pure synthesizer glitch. This feature is particularly robust against Gaussian background noise.

An alternative to the STFT that provides a multi-resolution analysis of the transient, often yielding more robust features than the spectral centroid alone. Wavelets decompose the signal into components at different scales, capturing both the fine, high-frequency details of a leading edge and the slower, low-frequency settling time behavior. The energy distribution across wavelet scales serves as a complementary feature vector that is more resilient to the time-frequency resolution trade-off inherent in STFT-based centroid calculation.

Though traditionally used in speech processing, MFCCs can be adapted for transient analysis to provide a perceptually-weighted spectral representation. By warping the frequency axis to a Mel scale and applying a discrete cosine transform to the log-spectrum, MFCCs decorrelate spectral features. For RF transients, this technique can compactly represent the coarse spectral envelope, including the centroid's influence, while discarding fine harmonic structure. This is useful for creating low-dimensional device fingerprints from the transient's spectral shape.