Nonstationary signals are widely observed in many real-world applications, e.g., radar, sonar, radio astronomy, communication, acoustics, and vibration applications. Joint time-frequency (TF) domain representations provide a time-varying spectrum for their analyses, discrimination, and classifications. Nonstationary signals commonly exhibit sparse occupancy in the TF domain. In this dissertation, we incorporate such sparsity to enable robust TF analysis in impaired observing environments. In practice, missing data samples frequently occur during signal reception due to various reasons, e.g., propagation fading, measurement obstruction, removal of impulsive noise or narrowband interference, and intentional undersampling. Missing data samples in the time domain lend themselves to be missing entries in the instantaneous autocorrelation function (IAF) and induce artifacts in the TF representation (TFR). Compared to random missing samples, a more realistic and more challenging problem is the existence of burst missing data samples. Unlike the effects of random missing samples, which cause the artifacts to be uniformly spread over the entire TF domain, the artifacts due to burst missing samples are highly localized around the true instantaneous frequencies, rendering extremely challenging TF analyses for which many existing methods become ineffective. In this dissertation, our objective is to develop novel signal processing techniques that offer effective TF analysis capability in the presence of burst missing samples. We propose two mutually related methods that recover missing entries in the IAF and reconstruct high-fidelity TFRs, which approach full-data results with negligible performance loss. In the first method, an IAF slice corresponding to the time or lag is converted to a Hankel matrix, and its missing entries are recovered via atomic norm minimization. The second method generalizes this approach to reduce the effects of TF crossterms. It considers an IAF patch, which is reformulated as a low-rank block Hankel matrix, and the annihilating filter-based approach is used to interpolate the IAF and recover the missing entries. Both methods are insensitive to signal magnitude differences. Furthermore, we develop a novel machine learning-based approach that offers crossterm-free TFRs with effective autoterm preservation. The superiority and usefulness of the proposed methods are demonstrated using simulated and real-world signals. / Electrical and Computer Engineering

Non-stationary signals, particularly frequency modulated (FM) signals which arecharacterized by their time-varying instantaneous frequencies (IFs), are fundamental to radar, sonar, radio astronomy, biomedical applications, image processing, speech processing, and wireless communications. Time-frequency (TF) analyses of such signals provide two-dimensional mapping of time-domain signals, and thus are regarded as the most preferred technique for detection, parameter estimation, analysis and utilization of such signals. In practice, these signals are often received with compressed measurements as a result of either missing samples, irregular samplings, or intentional under-sampling of the signals. These compressed measurements induce undesired noise-like artifacts in the TF representations (TFRs) of such signals. Compared to random missing data, burst missing samples present a more realistic, yet a more challenging, scenario for signal detection and parameter estimation through robust TFRs. In this dissertation, we investigated the effects of burst missing samples on different joint-variable domain representations in detail. Conventional TFRs are not designed to deal with such compressed observations. On the other hand, sparsity of such non-stationary signals in the TF domain facilitates utilization of sparse reconstruction-based methods. The limitations of conventional TF approaches and the sparsity of non-stationary signals in TF domain motivated us to develop effective TF analysis techniques that enable improved IF estimation of such signals with high resolution, mitigate undesired effects of cross terms and artifacts and achieve highly concentrated robust TFRs, which is the goal of this dissertation. In this dissertation, we developed several TF analysis techniques that achieved the aforementioned objectives. The developed methods are mainly classified into two three broad categories: iterative missing data recovery, adaptive local filtering based TF approach, and signal stationarization-based approaches. In the first category, we recovered the missing data in the instantaneous auto-correlation function (IAF) domain in conjunction with signal-adaptive TF kernels that are adopted to mitigate undesired cross-terms and preserve desired auto-terms. In these approaches, we took advantage of the fact that such non-stationary signals become stationary in the IAF domain at each time instant. In the second category, we developed a novel adaptive local filtering-based TF approach that involves local peak detection and filtering of TFRs within a window of a specified length at each time instant. The threshold for each local TF segment is adapted based on the local maximum values of the signal within that segment. This approach offers low-complexity, and is particularly useful for multi-component signals with distinct amplitude levels. Finally, we developed knowledge-based TFRs based on signal stationarization and demonstrated the effectiveness of the proposed TF techniques in high-resolution Doppler analysis of multipath over-the-horizon radar (OTHR) signals. This is an effective technique that enables improved target parameter estimation in OTHR operations. However, due to high proximity of these Doppler signatures in TF domain, their separation poses a challenging problem. By utilizing signal self-stationarization and ensuring IF continuity, the developed approaches show excellent performance to handle multiple signal components with variations in their amplitude levels. / Electrical and Computer Engineering