Novel recursive algorithms for frequency and spectrum estimation and their applications

Abstract

Frequency estimation from noisy observation is a long standing problem with myriad of applications. While the maximum likelihood estimator (MLE) is efficient asymptotically, good initial guess and reliable nonlinear optimization procedure are needed for good performance. Suboptimal solutions such as autoregressive (AR) modeling, subspace methods, and time-frequency distributions are simpler to implement and approach the Cramer Rao bound (CRB) asymptotically at high signal-to-noise ratios (SNR), their performances remain limited at low SNR. Conventional least-squares estimator for noisy AR models is biased and the Signal Parameters using Rotational Invariance Techniques (ESPRIT), a classical subspace method, also exhibits performance degradation at low SNRs.

This thesis aims to develop novel batch and recursive algorithms for estimating the parameters of multiple sinusoids under single/multiple channel AR noises. The ML principle is adopted due to its asymptotic efficiency. Specifically, improved AR estimator is explored to provide effective initial guess to the MLE and estimate the noise parameters. The first proposed approach integrates the extended instrumentation variable (EIV) to a novel Two-Stage Adaptive-Fading Variational Bayesian Kalman Filter (TS-AF-VBKF) to improve its noise immunity for tracking time-varying AR (TVAR) processes. Its smoother extension, EIV-TS-AF-VBKF with Iterative Smoothing (EIV-TS-AF-VBKS), can further reduce the estimation variance. By tracking the dominant poles of the estimated TVAR model, significant frequency components are identified and refined using a Gauss-Newton (GN)-based MLE for parameter estimation. A recursive version is also proposed for online applications. Experimental results on synthetic and bat sounds show that the proposed algorithm gives improved performance over other conventional algorithms tested.

Another contribution is the development of new batch and recursive MLE for AR additive noises. The sinusoidal parameters and the noise parameters are estimated alternately by the GN and the EIV/bias compensation (EBC) AR estimation methods respectively. The initial guess of the frequency parameters is also estimated by the EBC method. A recursive implementation is also proposed by employing the recursive EBC algorithm for obtaining the initial guess to the recursive GN-based MLE and tracking the noise parameters. The associated CRB is derived, which serves as the performance benchmark. Simulation results show that the proposed algorithm exhibits lower mean square error than conventional algorithm and approaches the CRB at median/high SNRs. Comparison with state-of-the-art methods on real dolphin signals also illustrates its improved performance.

The above approach is further extended to multiple channels where the sinusoidal and the AR noise parameters are estimated again alternately using the GN and the multichannel EBC methods with the initial frequency estimates provided by ESPRIT. A recursive GN method is also devised for online applications using the PAST-based ESPRIT initial estimates. The associated CRB is derived and simulation results show that the proposed algorithm offers lower MSE than other conventional algorithms, which approaches the CRB at medium to high SNRs. The algorithm is applied to the detection and localization of low frequency power system oscillations. The Direction Transfer Function is applied to locate the oscillations from the buses with detected oscillations. Experimental results show that the proposed algorithm significantly outperform existing algorithms.