Demonstration of a unified approach to beamforming

In this paper, we discuss a unification of several well-known frequency domain beamforming methods into one working principle. The methods under consideration include Functional Beamforming, Asymptotic Beamforming, Adaptive Beamforming and - as a natural limiting case - Standard Beamforming. Common to most of these methods is the underlying eigenvalue decomposition of the cross-spectral matrix. Introducing a weighted power mean (also called weighted Hölder mean) in terms of these eigenvalues for every map point, each of the above methods is represented by a certain power p. Because of the latter, this unified approach will be called Power Beamforming throughout this paper. Going from the limiting case p=1 of Standard Beamforming to lower power values results in the attenuation of side lobes and sharpening of the main lobes in the corresponding beamforming map. We demonstrate this effect using simulations and several real-world measurements.

The Case for Shifting the Renyi Entropy

We introduce a variant of the Rényi entropy definition that aligns it with the well-known Hölder mean: in the new formulation, the r-th order Rényi Entropy is the logarithm of the inverse of the r-th order Hölder mean. This brings about new insights into the relationship of the Rényi entropy to quantities close to it, like the information potential and the partition function of statistical mechanics. We also provide expressions that allow us to calculate the Rényi entropies from the Shannon cross-entropy and the escort probabilities. Finally, we discuss why shifting the Rényi entropy is fruitful in some applications.