Reduction of computational complexity in the butterfly search technique

In the butterfly search technique, echoes from repeated firings of a transducer are resampled along a set of predetermined trajectories of constant velocities, called ''butterfly lines,'' because of their intersection and crossing at a reference range. The slope of the trajectory on which the sampled signals satisfy a predetermined criterion appropriate for the type of signal in question, provides an estimate of the velocity of the target. The search for this trajectory is called ''butterfly search,'' which can be carried out efficiently in a parallel processing scheme. The estimator can be based on the radio frequency (RF) A-lines, the envelopes, or the quadrature components. The butterfly search on quadrature components has shown outstanding noise immunity, even with relatively few successive scan lines, and was found to outperform all the common time domain and Doppler techniques in simulations and experiments with strong noise, It can be simply implemented using elementary digital signal processing hardware. However, it is possible to further improve upon its computational complexity to make the technique even simpler to implement, without any complex multipliers in the parallel channels. In this paper, we present some modifications that significantly reduce the computational complexity of butterfly search on quadrature components.