Bearing estimation algorithms obtain only a small number of direction of arrivals (DOAs) within the entire angle domain, when the sources are spatially sparse. Hence, we propose a method to specifically exploit this spatial sparsity property. The method uses a very small number of measurements in the form of random projections of the sensor data along with one full waveform recording at one of the sensors. A basis pursuit strategy is used to formulate the problem by representing the measurements in an overcomplete dictionary. Sparsity is enforced by l(1)-norm minimization which leads to a convex optimization problem that can be efficiently solved with a linear program. This formulation is very effective for decreasing communication loads in multi sensor systems. The algorithm provides increased bearing resolution and is applicable for both narrowband and wideband signals. Sensors positions must be known, but the array shape can be arbitrary. Simulations and field data results are provided to demonstrate the performance and advantages of the proposed method.