Sensitivity analysis of beamforming applied to coherent imaging systems

The implementation of coherent imaging system often uses beamformer as a fundamental element. The beamformer´s ability to precisely steer the beam in given direction determines the resolution of the imaging system. In the conventional delay and sum type of beamformer implemented using complex weights, accuracy of the weights decide the sharpness of the beam. In an ideal scenario, the phase term of the complex weight would have infinite resolution and no deviation due to timing jitter. However, in practice there exists the phase error in the beamforming weights due to RF imperfections. We can define sensitivity of the imaging system in terms of shape of beampattern and analyze its behavior against the phase errors. In fact, this behavior has been extensively investigated with the empirical approach in the literature [1]. In this paper, we derive the analytical expression for the minimum number of antenna elements required to ensure imaging system is immune to the given phase errors. The analytical expression has been derived using two different approaches. First approach follows rigorous stochastic analysis to formulate the expression for the envelope of beamformer output while the second approach utilizes the known results to obtain the expected value of the beamformer output power.