The digital world becomes dominated by electronic devices. The demand for higher speeds is growing every day which implies a higher demand for the signal acquisition and processing speeds and a wider signal bandwidth. The Shannon/Nyquist sampling theorem states, the ADC must sample the signal at a rate two times faster than the signal bandwidth in order to avoid data loss. However, the advances of ADC cannot always meet these demands. Considerable research has been done to find new approaches to acquiring the signal at a rate lower than the Nyquist rate (sub-Nyquist). In recent years, compressive sensing (CS) has come to light as a new signal processing paradigm. CS exploits signal sparsity characteristics to acquire the signal using a number of samples much lower than the Nyquist rate. Our focus is studying CS and its applications to radar. We study different successful CS practical implementation approaches. However, these approaches lead to an undesired high-latency in the computation and signal reconstruction.

This thesis has four main contributions that build on each other. First, we study the effect of prior information on the CS recovery performance and introduce a new formula to improve the utilization of the prior information in CS pulse radar. Second, we exploit prior information and multi-resolution analysis properties to introduce a low-latency sub-Nyquist adaptive sampling algorithm. The proposed algorithm is a modified version of CS sampling approach called Xampling but uses Haar wavelets. The Haar wavelet based approach is found to be useful to extract target range information for radar applications that requires only target range detection. Third, we modified our Haar approach by using S-transform bases instead of Haar bases. The S-transform based algorithm is capable of analyzing signal frequency and phase that gives the algorithm the ability to extract target range and speed information in a single reconstruction step. The S-transform approach provides a reasonable balance between the number of samples and the reconstruction processing time. Fourth, we study the recovery guarantees for the S-transform algorithm. We introduce a mathematical proof to derive an upper bound for the number of detectable targets.