The rapid progress of wireless communication technologies that has taken place in recent years has significantly improved the quality of everyday life. However with this expansion of wireless communication systems come significant security threats and significant technological challenges, both of which are due to the fact that the communication medium is shared. The ubiquity of open wireless Internet access networks creates a new avenue for cyber-criminals to impersonate and act in an unauthorized way. The increasing number of deployed wide-band wireless communication systems entails technological challenges for effective utilization of the shared medium, which implies the need for advanced interference rejection methods. Wireless security and interference rejection in wide-band wireless communications are therefore often considered as the two main challenges in wireless network's design and research. Important aspects of these challenges are illuminated and addressed in this dissertation. This dissertation considers signal processing approaches for exploiting or mitigating the effects of non-ideal components in wireless communication systems. In the first part of the dissertation, we introduce and study a novel, model-based approach to wireless device identification that exploits imperfections in the transmitter caused by manufacturing process nonidealities. Previous approaches to device identification based on hardware imperfections vary from transient analysis to machine learning but have not provided verifiable accuracy. Here, we detail a model-based approach, that uses statistical models of RF transmitter components: digital-to-analog converter, power amplifier and RF oscillator, which are amenable for analysis. Our proposed approach examines the key device characteristics that cause anonymity loss, countermeasures that can be applied by the nodes to regain the anonymity, and ways of thwarting such countermeasures. We develop identification algorithms based on statistical signal processing methods and address the challenging scenario when the units that need to be distinguished from one another are of the same model and from the same manufacturer. Using simulations and measurements of components that are commonly used in commercial communications systems, we show that our anonymity breaking techniques are effective. In the second part of the dissertation, we consider innovative approaches for the acquisition of frequency-sparse signals with wide-band receivers when a weak signal of interest is received in the presence of a very strong interference, and the effects of the nonlinearities in the low-noise amplifier at the receiver must be mitigated. All samples with amplitude above a given threshold, dictated by the linear input range of the receiver, are discarded to avoid the distortion caused by saturation of the low noise amplifier. Such a sampling scheme, while avoiding nonlinear distortion that cannot be corrected in the digital domain, poses challenges for signal reconstruction techniques, as the samples are taken non-uniformly, but also non-randomly. The considered approaches fall into the field of compressive sensing (CS); however, what differentiates them from conventional CS is that a structure is forced upon the measurement scheme. Such a structure causes a violation of the core CS assumption of the measurements' randomness. We consider two different types of structured acquisition: signal independent and signal dependent structured acquisition. For the first case, we derive bounds on the number of samples needed for successful CS recovery when samples are drawn at random in predefined groups. For the second case, we consider enhancements of CS recovery methods when only small-amplitude samples of the signal that needs to be recovered are available for the recovery. Finally, we address a problem of spectral leakage due to the limited processing block size of block processing, wide-band receivers and propose an adaptive block size adjustment method, which leads to significant dynamic range improvements.