Higher-Order Spectrum (Bispectrum) Analysis: Direct and Indirect Methods in MATLAB

This article presents MATLAB implementations of higher-order spectrum (bispectrum) analysis through both direct and indirect methods, with applications in time-difference estimation and advanced signal characterization. Higher-order spectral analysis constitutes a powerful signal processing technique for examining complex temporal and frequency domain characteristics beyond conventional power spectrum analysis. The direct method involves computing higher-order spectral matrices to derive time-difference estimates through multidimensional Fourier transforms of third-order cumulants. In MATLAB implementation, this typically utilizes the `bispecd` function or custom matrix operations to calculate the bispectral density directly from signal segments. The indirect method operates by computing phase differences between signal components, employing approaches like the Fourier transform of estimated cumulant sequences. MATLAB implementations often leverage functions such as `bispeci` which first estimates third-order cumulants before applying Fourier transformation to obtain bispectral estimates. These methodologies find extensive applications across multiple domains including telecommunications (for signal synchronization and interference analysis), acoustics (for source localization and echo detection), and biomedical engineering (for biological signal processing and feature extraction). The techniques enable more accurate time-difference estimation between correlated signals and extract additional phase coupling information not available in conventional spectral analysis. Key MATLAB functions for implementation include signal preprocessing tools, cumulant estimation routines, and Fourier transform operations, with proper windowing and segmentation techniques crucial for reducing variance in bispectral estimates. The direct method generally provides better resolution but requires more computational resources, while the indirect method offers improved statistical stability for practical applications. Through these advanced spectral analysis techniques, researchers can achieve enhanced signal characterization capabilities, particularly in scenarios involving non-Gaussian processes or nonlinear system identification where traditional second-order statistics prove insufficient.