Synchrosqueezing Transform (SST)

The core mechanism of SST is vertical reassignment. It calculates the instantaneous frequency of the signal at every time-frequency point. Instead of moving energy arbitrarily in the time-frequency plane, SST squeezes the coefficients only along the frequency direction, preserving the signal's causal timeline. This is distinct from the full reassignment method, which moves energy in both time and frequency.

SST typically operates on the output of the Continuous Wavelet Transform (CWT). It uses the phase information of the complex wavelet coefficients to derive a precise estimate of the instantaneous frequency. The transform then sums the CWT coefficients whose calculated instantaneous frequencies fall within a narrow bin around a central frequency, effectively 'squeezing' the diffuse scalogram into a concentrated spectrogram-like representation.

The primary visual result of SST is the elimination of spectral smearing. In a standard scalogram, a pure harmonic component appears as a thick band of energy. After synchrosqueezing, this energy is concentrated into a thin, highly localized time-frequency ridge. This dramatically improves the readability of multi-component signals, allowing for the clear separation of closely spaced frequency modulations.

A critical advantage of SST over other reassignment techniques is its invertibility. Because the squeezing operation is a simple summation of coefficients, the original signal's individual components can be reconstructed by integrating the concentrated energy around a specific ridge. This enables mode retrieval and signal denoising by isolating and extracting only the components of interest from the time-frequency plane.

SST exhibits strong noise robustness. While noise energy is also reassigned, it tends to spread out randomly across the time-frequency plane, whereas the coherent signal energy is concentrated into sharp ridges. This contrast makes it highly effective as a pre-processing step for feature extraction in low-SNR environments, such as identifying weak transient signals in electronic warfare or seismic analysis.

In Radio Frequency Fingerprinting, SST is used to extract precise, stable features from transient or steady-state signals. By concentrating the energy of subtle hardware impairment signatures—like I/Q imbalance or DAC clock jitter—into clear ridges, SST provides a highly discriminative input for deep learning classifiers. It transforms a diffuse, noisy waveform into a sparse, high-resolution image of the device's unique physical identity.

The precursor to SST. This classical technique sharpens a spectrogram by relocating each coefficient from its computed geometric coordinate (t, ω) to the local center of gravity (t̂, ω̂) of the signal's energy distribution. Unlike SST, the original reassignment method is not invertible, meaning the sharpened representation cannot be used to reconstruct the original signal. SST solves this by reassigning coefficients only along the frequency axis, preserving the signal's phase information and enabling mode reconstruction.

The foundational input for wavelet-based SST. CWT decomposes a signal into a two-dimensional time-scale representation using scaled and translated versions of a mother wavelet. The resulting scalogram often exhibits spectral smearing due to the wavelet's finite time-frequency support. SST operates directly on CWT coefficients, using the phase information to calculate the instantaneous frequency at each point and 'squeezing' the energy along the scale axis to produce a concentrated, sharpened time-frequency map.

The core mathematical parameter that drives the SST algorithm. Defined as the time derivative of the instantaneous phase of an analytic signal, it represents the dominant oscillatory frequency at a precise moment. SST estimates this value at every point in the CWT or STFT domain. Coefficients are then reassigned from their original geometric scale or frequency to this physically meaningful instantaneous frequency, collapsing diffuse energy into a sharp time-frequency ridge.

A competing adaptive decomposition method that breaks a signal into Intrinsic Mode Functions (IMFs) without a predefined basis. While EMD is fully data-driven, it suffers from mode mixing—where a single IMF contains disparate frequency components. SST addresses this limitation by providing a mathematically rigorous, invertible alternative that separates closely spaced components with high precision. The combination of EMD and SST is sometimes used for robust signal analysis.

A quadratic time-frequency distribution offering the highest possible joint resolution. However, it generates severe cross-term interference for multi-component signals, creating spurious oscillatory artifacts. SST provides a competing high-resolution representation that is cross-term free by design, as it operates on linear transforms (CWT/STFT) rather than quadratic ones. This makes SST far more interpretable for signals with multiple harmonic components.

The alternative input for Fourier-based SST. STFT computes the frequency content of windowed signal segments, but its fixed window imposes a rigid Heisenberg-Gabor uncertainty trade-off. SST can be applied to STFT results to dramatically sharpen the representation without violating the uncertainty principle. This variant, sometimes called the Fourier-based SST (FSST), is computationally efficient and well-suited for signals with slowly varying instantaneous frequencies.