Time-Frequency Reassignment Method

The core mechanism moves computed energy from its original geometric grid point (t, f) to a new coordinate (t̂, ω̂) that represents the local centroid of the distribution. This is not a new transform but a sharpening operation applied to existing representations like the spectrogram or scalogram.

• Corrects the smearing inherent in linear transforms
• Concentrates energy along the true instantaneous frequency ridges
• Preserves the total energy of the original representation

Reassignment vectors are derived from the phase information of the Short-Time Fourier Transform, which is typically discarded when taking the squared magnitude for a spectrogram. The local group delay and instantaneous frequency are computed from the phase's partial derivatives.

• Channelized Instantaneous Frequency: The frequency reassignment operator for STFT
• Local Group Delay: The time reassignment operator
• These operators point exactly to the center of gravity of the signal component's energy distribution

Synchrosqueezing is a special case of reassignment that operates only along the frequency axis, making the transform invertible. Unlike full reassignment, it allows for mode reconstruction and signal separation.

• Reassigns coefficients only in frequency, not time
• Enables extraction of individual Intrinsic Mode Functions
• Provides a mathematically rigorous foundation for empirical mode decomposition-style analysis
• Supports perfect signal reconstruction from the squeezed coefficients

When applied to quadratic distributions like the Wigner-Ville Distribution, reassignment significantly suppresses oscillatory cross-term interference. The reassignment process moves cross-term energy toward the geometric midpoint between interacting components, where it often destructively interferes.

• Reduces the readability barrier of high-resolution quadratic TFDs
• Cross-terms are relocated rather than filtered, preserving auto-term energy
• Enables the use of kernel-free distributions for multi-component signals
• Particularly effective for signals with non-linear frequency modulation

Efficient reassignment requires computing three STFTs: one with the standard window, one with a time-ramped window (t·h(t)), and one with the time-derivative window (dh/dt). These provide the partial derivatives needed for the reassignment operators.

• Three FFTs per frame instead of one
• Can be implemented recursively for real-time applications
• Modern GPU implementations enable high-resolution, real-time reassigned spectrograms
• Trade-off: increased computational load for dramatically improved readability

Reassignment sharpens the time-frequency representations of transmitter turn-on transients and steady-state waveforms, revealing subtle hardware impairment signatures that are smeared in standard spectrograms.

• Enhances visibility of DAC clock jitter artifacts in the time-frequency plane
• Reveals fine phase discontinuity structures during frequency hopping
• Improves feature extraction for deep learning classifiers by providing cleaner input representations
• Enables separation of closely spaced multi-component distortion products in power amplifier non-linearity analysis

The foundational input for reassignment. A spectrogram is the squared magnitude of the Short-Time Fourier Transform (STFT), visualizing signal energy on a 2D heatmap. Its resolution is fundamentally limited by the Heisenberg-Gabor uncertainty principle, causing energy to smear across adjacent time-frequency cells. Reassignment methods are applied directly to this smeared representation to refocus the energy onto the true instantaneous frequency ridges.

A quadratic time-frequency distribution offering the highest possible joint resolution by calculating the Fourier transform of the signal's instantaneous autocorrelation function. While it provides the ideal mathematical basis for reassignment, its practical use is complicated by severe cross-term interference for multi-component signals. Reassignment methods can be seen as a way to achieve WVD-like sharpness while suppressing these oscillatory artifacts.

A closely related, mathematically rigorous reassignment technique that sharpens a Continuous Wavelet Transform (CWT) or STFT. Unlike the general reassignment method, SST performs energy relocation strictly along the frequency axis only, based on an instantaneous frequency estimate. This restriction allows SST to support signal reconstruction, making it a powerful tool for mode decomposition and denoising in addition to visualization.

The core physical parameter that reassignment algorithms estimate. It is defined as the time derivative of the instantaneous phase of an analytic signal. In a smeared time-frequency representation, the true instantaneous frequency at a point corresponds to the local center of gravity of the energy distribution. Reassignment methods compute this center of gravity and relocate the energy to correct for the blurring induced by the analysis window.

A general family of quadratic time-frequency representations generated by applying a 2D kernel function to the ambiguity function. The reassignment method can be derived by applying a signal-dependent, adaptive kernel that shifts energy to the local centroid. Understanding Cohen's class provides the unified mathematical framework for comparing the WVD, Choi-Williams distribution, and spectrogram as different kernel selections.

Spurious oscillatory artifacts that appear midway between genuine signal components in quadratic distributions like the WVD. These artifacts arise from the bilinear nature of the transform interacting with multiple signal components. A primary motivation for the reassignment method is to sharpen the auto-terms of a spectrogram—which is inherently cross-term-free—rather than attempting to filter cross-terms from a WVD, thus avoiding the auto-term resolution trade-off.