Differential Beamforming from the Beampattern Factorization Perspective

Differential beamformers have demonstrated a great potential in forming frequency-invariant beampatterns and achieving high directivity factors. Most conventional approaches design differential beamformers in such a way that their beampatterns resemble a desired or target beampattern. In this paper, we show how to design differential beamformers by simply taking advantage of the fact that the beampattern is actually a particular form of an exponential polynomial. Thanks to this quite obvious formulation, a target beampattern is not really needed while the zeros of the exponential polynomial and/or its factorization are fully exploited. The advantage of this factorization is twofold. First, it gives the relation between the beamformer and the roots of the polynomial, so the former can be directly determined from the latter, which seems natural and convenient. Second, based on this factorization, we propose a new formulation of the beamforming filter, which decomposes the filter into shorter ones with the Kronecker product. This formulation is very general, and many well-known beamformers such as the differential and delay-and-sum (DS) ones can be derived from it. Furthermore, the new formulation allows one to combine different kinds of beamformers together, which gives a great flexibility in forming different beampatterns and achieving a compromise among the directivity factor (DF), white noise gain (WNG), and frequency invariance.