| Spectral estimation is important in many fields including astronomy, meteorology, seismology, communications, economics, speech analysis, medical imaging, radar, sonar, and underwater acoustics. Most existing spectral estimation algorithms are devised for uniformly sampled complete-data sequences. However, the spectral estimation for data sequences with missing samples is also important in a number of applications ranging from astronomical time-series analysis to synthetic aperture radar imaging with angular diversity.; The classical approaches to spectral analysis include the discrete Fourier transform (DFT) or fast Fourier transform (FFT) based methods. These methods have been widely used for spectral estimation tasks due to their robustness and high computational efficiency. However, they suffer from low resolution and poor accuracy problems. Many advanced spectral estimation methods have also been proposed, including parametric and nonparametric adaptive filtering based approaches. In general, the nonparametric approaches are less sensitive to data mismodelling than their parametric counterparts. Moreover, the adaptive filter-bank based nonparametric spectral estimators can provide high resolution, low sidelobes, and accurate spectral estimates while retaining the robust nature of the nonparametric methods.; This research is aimed to investigate adaptive filter-bank based nonparametric complex spectral estimation of data sequences (one-dimensional case) or matrices (two-dimensional case) with missing samples occurring in arbitrary patterns. The main results of this work can be summarized in three parts.; First, we consider the one-dimensional (1-D) case. We develop two missing-data amplitude and phase estimation (MAPES) algorithms by using a "maximum likelihood (ML) fitting" criterion. Then we use the well-known expectation maximization (EM) method to solve the so-obtained estimation problem iteratively. Through numerical simulations, we demonstrate the excellent performance of the MAPES algorithms for missing-data spectral estimation and missing data restoration.; Next, we present two-dimensional (2-D) extensions of the MAPES-EM algorithms introduced in the 1-D case. Then we develop a new MAPES algorithm, referred to as MAPES-CM, by solving an ML problem iteratively via cyclic maximization (CM). We have compared MAPES-EM with MAPES-CM and have shown that MAPES-CM allows significant computational savings compared with MAPES-EM, which is especially desirable for 2-D applications. Numerical examples have been provided to demonstrate the performance of the 2-D MAPES algorithms.; Finally, by observing that all the missing-data algorithms developed previously estimate the missing data samples, we propose to achieve better spectral estimation performance, for example, higher resolution than APES, based on the (complete) data interpolated via MAPES. It is known that the Capon spectral estimator has better resolution (narrower mainlobes) compared with APES. In this part of the work, we develop a rank-deficient robust Capon filter-bank spectral estimator, which can achieve much higher resolution than the existing nonparametric spectral estimators. |
