Wireless communication is omnipresent today, but this development has led to frequency spectrum becoming a limited resource. Furthermore, wireless devices become more and more energy-limited, due to the demand for continual wireless communication of higher and higher amounts of information. The need for cheaper, smarter and more energy efficient wireless devices is greater now than ever. This thesis addresses this problem and concerns the application of the recently developed sampling theory of compressive sensing in communication systems. Compressive sensing is the merging of signal acquisition and compression. It allows for sampling a signal with a rate below the bound dictated by the celebrated Shannon-Nyquist sampling theorem. In some communication systems this necessary minimum sample rate, dictated by the Shannon-Nyquist sampling theorem, is so high it is at the limit of what the current technology can manage. Even if the sampling rate is within the bounds of what is currently possible, the electrical components for acquiring highly oscillating signals are very expensive and energy demanding. Compressive sensing may mitigate this challenge by lowering the bound on the sample rate. The compressive sensing research area is still in its infancy and has so far been mainly a theoretical field. However, hardware implementations and actual application examples in current communication technologies have begun to emerge. The approach in this thesis has been to attack some of the current challenges with using compressive sensing in communication systems. The main contribution of this thesis is two-fold: 1) a new compressive sensing hardware structure for spread spectrum signals, which is simpler than the current state-of-the-art, and 2) a range of algorithms for parameter estimation for the class of translation-invariant signals, which outperform the current state-of-the-art algorithms for frequency and time delay estimation. Though the proposed methods and algorithms in this work have been designed for use in communication systems, they are also relevant outside this area.