Resolving Time-Frequency Ridge Crossings via Frequency-Rate Lifting

Ridge crossings, where two components share the same instantaneous frequency (IF) at a given time, are a fundamental bottleneck in multi-component time-frequency (TF) mode decomposition. We trace this difficulty to \emph{insufficient representational dimensionality} and resolve it by lifting signals into the 3-D time-frequency-frequency-rate (TFFR) space $(t,f,\dot{f})$, where crossing ridges separate because they typically differ in the instantaneous frequency rate (IFR). We formalize this via a separability proposition and confirm it empirically through Monte Carlo simulation (crossing rate: 78.6\% in 2-D $\to$ 0.9\% in 3-D). Based on this principle, we propose \textbf{TFFR-Net}, a query-based masked transformer that jointly predicts per-ridge masks and dense IFR maps. An IFR feedback loop injects the estimated frequency rate back into each query embedding, enabling the decoder to perform instance discrimination in TFFR space rather than the TF plane. On a synthetic benchmark with up to six chirp components under broadband noise (0--25\,dB SNR), TFFR-Net consistently reduces IF MAE by at least 5.8\% and Dice loss by at least 5.4\% relative to competing methods. Evaluation on real-world signals (bat echolocation, power-system oscillation, earthquake vibration) demonstrates effective generalization beyond synthetic training data. The source code is available at \url{https://anonymous.4open.science/r/TFFR-92D2/}.