Generalization of the classical delay-and-sum technique by using nonlinear dirac-delta functions

We presented a generalization of the delay-and-sum beamforming based on the Dirac-Delta functions but with nonlinear argument. For this end, a closed-form expression of the beampattern $\mathcal{B}(r)=\sum\nolimits_{k,q}w(k,q,r)x(k,q,r)$ with r = r(θ), was derived. This expression is computationally simulated through an algorithm that includes integer-order Bessel input functions and random noise. The 4M+N model parameters provided by the Dirac-Delta method are extracted by using a Monte-Carlo-like step which selects the best values for B(r) minimizing the Monte-Carlo error for Δθ = 0.5% for the case of beam response of θ0=30 degrees. These results might sustain the fact that beamforming techniques can use Dirac-Delta functions for modeling arrival signal even in those cases where strong nonlinearity is involved.