Compressive sensing (CS) provides a new paradigm in data acquisition and signal processing in radar, based on the assumptions of sparsity of an unknown radar scene, and the incoherence of the transmitted signal. The resolution in the conventional pulse-compression radar is foreseen to be improved by the implementation of CS. An unknown sparse radar scene can then be recovered through CS with a high probability, even in the case of an underdetermined linear system. However, the theoretical framework of CS radar has to be verified in an actual radar system, accounting for practical system aspects, such as the signal bandwidth, ease of generation and acquisition, system complexity, etc. In this thesis, we investigate linear frequency modulated (LFM), Alltop and Björck waveforms, which show theoretically favorable properties in a CS-radar system, in the basic radar problem of range-only estimation. The aforementioned waveforms were investigated through a model of a digital radar system - from signal generation in the transmitter, to sparse signal recovery in the receiver. The capabilities of the CS-radar versus the conventional pulse compression radar were demonstrated, and the Alltop and Björck sequences are proven to outperform the commonly used linear LFM waveform in typical CS-radar scenarios.