Spectral Proper Orthogonal Decomposition with Variable Frequency Binning and Precise Tonal Mode Extraction

SPOD identifies coherent structures in statistically stationary flows. The method estimates the cross-spectral density matrix from an ensemble of temporal Fourier modes obtained by segmenting time series data into blocks. This statistical approach enables the extraction of coherent structures even in broadband turbulence. However, SPOD faces a tradeoff: better statistical convergence requires more blocks, which reduces frequency resolution. This variance–bias tradeoff necessitates long time series for accurate mode extraction.

In this talk we present a modified version of SPOD. Instead of block-segmenting the time series, the Fourier transform is computed using the full time history. SPOD modes are then obtained by averaging neighboring Fourier modes via Proper Orthogonal Decomposition. Importantly, information about the contribution of individual Fourier modes to each SPOD mode are stored in the expansion coefficients.

Compared to standard SPOD, the new approach offers two key advantages: (1) the number of Fourier modes used to estimate SPOD modes can be varied with frequency, enabling adaptive tradeoff between spectral variance and spectral bias; and (2) the contribution of individual frequencies to each SPOD mode can be directly quantified, allowing for more accurate extraction of tonal features without sacrificing statistical convergence. This makes the new algorithm particularly effective for analyzing flows that exhibit both broadband and tonal components.