Direct and Indirect Methods for Higher-Order Spectrum (Bispectrum) Analysis

Higher-order spectrum analysis (bispectrum analysis) serves as a crucial tool in signal processing for analyzing non-Gaussian and nonlinear signals. It enables extraction of phase information from signals while effectively suppressing Gaussian noise interference, making it particularly valuable in applications like time-delay estimation. In MATLAB implementation, two primary approaches exist for higher-order spectrum analysis: the direct method and the indirect method. Each technique possesses distinct characteristics suited for different analytical requirements. The direct method employs a Fourier-transform-based computational approach. Implementation typically involves segmenting the signal first, performing Fourier transformation on each segment, and then obtaining bispectrum estimates through multiplicative operations. This approach offers straightforward computation but requires longer data sequences to achieve satisfactory statistical properties. In MATLAB, this can be implemented using fft() functions combined with proper segmentation and averaging procedures. The indirect method initially estimates the signal's third-order cumulants before applying two-dimensional Fourier transformation to derive bispectrum estimates. While computationally more intensive, this method often provides superior estimation performance for shorter data sequences. A key advantage lies in the flexible selection of window functions for cumulant estimation, allowing trade-offs between bias and variance. MATLAB implementations might involve cumulant estimation functions followed by 2D-fft2() operations with appropriate windowing techniques. In time-delay estimation applications, higher-order spectrum analysis demonstrates particular strengths. By examining bispectral phase relationships between two signals, precise time-delay estimates can be obtained while maintaining robust performance even under Gaussian noise conditions. This capability makes it invaluable in fields such as acoustic source localization and seismic signal processing, where accurate delay estimation is critical. MATLAB implementations for these applications typically involve cross-bispectrum calculations and phase extraction algorithms.