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“Signal Processing VIA Higher-Order Statistics”
by Georgios B. Giannakis
July 1986
Although spectral theory is a well established approach to any signal processing tasks, its applicability depends heavily upon the assumptions of linearity and Gaussianity of the signal, and/or the minimum phase assumption of the underlying ARMA model.
Because 2nd-order statistics are "phase-blind," we introduce higher-order statistics that convey the complementary phase information required for realization and model reduction of non-minimum phase (NMP) systems. Identification of NMP stochastic systems, and signal phase reconstruction. The higher-order statistics that we use to replace or complement the auto-correlation information, are directly related to the ARMA parameters, and easy to handle mathematically.
With the introduction of cumulants and their multidimensional Fourier transforms known as polyspectra, new insight is gained about the stationarity, whiteness and non-Gaussianity conditions required for correct phase realization of finite-dimensional parametric models. For high resolution polyspectrum estimation, a parametric approach is developed as a byproduct of our NMP identification algorithm. We also derive a maximum entropy polyspectral estimator which turns out to have an AR structure.
The state-space and input-output methods that we propose for realization and model reduction of NMP models exploit higher-order output statistics to estimate the AR coefficients of the minimum phased part and higher-order statistics of the innovations to realize the all-pass part. By employing algorithms, we guarantee good numerical performance and a quantitative way of measuring the error of our approximate realization.
Because successful identification depends on correct model order we also develop two methods for ARMA order determination using higher-order statistics. Statistical analysis is performed and confidence intervals are provided to display the AR and MA orders with high probability, and suggest the most reliable statistic for efficient parameter estimation.
Our methods are robust with respect to additive, non-skewed, white output noise. Stability and consistency of our identification techniques are assured under mild conditions. Simulations verify our theoretical developments, and indicate that higher-order statistics will emerge as a powerful tool in various signal processing applications.