Notes on Spectral Decomposition Approaches

Overview

The choice of spectral decomposition method has significant implications for the quality of features available to the downstream reconstruction algorithms. This note compares three approaches on a set of synthetic test signals with known spectral content.

Method 1: Fast Fourier Transform (FFT)

The FFT remains the workhorse of spectral analysis. Its strengths are computational efficiency (O(N log N)), well-understood mathematical properties, and broad tooling support.

Limitations: The FFT assumes stationarity within the analysis window. For signals with time-varying spectral content (e.g., a system whose properties change during measurement), the FFT smears transient features across the entire spectrum.

Method 2: Wavelet Transform

The continuous wavelet transform (CWT) provides time-frequency localization, trading frequency resolution for temporal resolution at different scales. Using a Morlet wavelet, we can track how spectral content evolves over time.

In our tests, the CWT correctly identifies frequency transitions in chirp signals and resolves closely spaced modes that the FFT conflates when using short windows. However, the computational cost is higher and the interpretation of the scalogram requires care.

Method 3: Empirical Mode Decomposition (EMD)

EMD is a data-driven method that decomposes a signal into intrinsic mode functions (IMFs) without requiring a pre-defined basis. It is particularly useful for non-linear and non-stationary signals.

The main drawback is mode mixing: when two spectral components are close in frequency, EMD may fail to separate them cleanly. Ensemble EMD (EEMD) mitigates this by averaging over noise-perturbed decompositions, at the cost of increased computation.

No single method is universally superior. The optimal choice depends on the signal characteristics and the requirements of the downstream reconstruction algorithm.

Recommendation

For the TEYA SOL LAB pipeline, we propose a hybrid approach:

1. Use FFT for initial broadband spectral characterization.
2. Apply CWT for time-frequency analysis of non-stationary segments.
3. Reserve EMD for signals with strong non-linear mode interactions.

This tiered strategy balances computational cost with analytical depth and avoids over-reliance on any single decomposition method.