Decomposition and Reconstruction in Wavelet Transform

Wavelet transform is a powerful time-frequency analysis tool that decomposes signals into components at different scales, revealing local signal characteristics. Traditional Discrete Wavelet Transform (DWT) achieves signal decomposition and reconstruction through multiresolution analysis, but its frequency resolution may be inadequate in high-frequency regions. The implementation typically involves filter banks with downsampling operations, where functions like wavedec() (in MATLAB) perform multi-level decomposition using quadrature mirror filters.

Synchrosqueezing Wavelet Transform (SWT) optimizes traditional methods by "squeezing" wavelet coefficients to enhance time-frequency representation clarity. Its core algorithm leverages instantaneous frequency information to reassign energy distribution of wavelet coefficients, concentrating energy more precisely on the time-frequency plane. This improvement not only increases frequency resolution but also strengthens weak signal detection capabilities. Code implementation generally involves three stages: continuous wavelet transform computation, instantaneous frequency estimation via phase derivative calculation, and energy reassignment through frequency bin redistribution.

During the decomposition phase, SWT generates initial time-frequency distribution through Continuous Wavelet Transform (CWT), then computes phase derivatives (instantaneous frequencies) at each point to concentrate energy along true frequency trajectories. The reconstruction phase employs inverse transforms or filtering techniques to recover the original signal with superior accuracy and stability compared to traditional methods. Key functions like icwt() or specialized synchrosqueezing inversion algorithms ensure faithful signal recovery while maintaining phase coherence.

Advantages of synchrosqueezing technology include: Reducing frequency-band ambiguity in wavelet transforms, particularly suitable for non-stationary signal analysis; Smaller reconstruction errors, ideal for high-fidelity applications; Effective for noise suppression and feature extraction in mechanical fault diagnosis or biomedical signal processing. The algorithm can be implemented using time-frequency toolbox functions like wsst() (Wavelet Synchrosqueezing) in MATLAB, which automatically handles energy reassignment and inverse transformation.

The enhanced method significantly improves time-frequency analysis usability while preserving signal details, providing more reliable data foundations for subsequent processing such as pattern recognition or compressed sensing. Practical implementation often involves parameter optimization for mother wavelets and scaling factors to balance resolution and computational efficiency.