Compressive Sensing (CS) theory is a newly developed theory which combines the signal sampling and compression based on the sparsity characteristics of the signal. Applying CS theory in radar signal processing may lead to a reduction in sampling rate, complexity, power consumption, and cost. On the other hand, performance is a critical point to be considered. In the present paper, an important question of the worthy of applying CS in the signal processing of Linear Frequency Modulated Continuous Wave (LFMCW) radar is considered. Two approaches of CS are considered; Nyquist rate based approach, and pseudo random based approach. The detection performance of LFMCW radar signal processor using CS based approaches is compared to the traditional one which is based on Fast Fourier Transform (FFT) through Receiver Operating Characteristics (ROC) curves. Comparative analysis between CS approaches and the traditional one regarding the performance and complexity is presented.