We propose a sampling scheme that can perfectly reconstruct a collection of spikes on the sphere from samples of their lowpass-filtered observations. Central to our algorithm is a generalization of the annihilating filter method, a tool widely used i
n array signal processing and finite-rate-of-innovation (FRI) sampling. The proposed algorithm can reconstruct $K$ spikes from $(K+sqrt{K})^2$ spatial samples. This sampling requirement improves over previously known FRI sampling schemes on the sphere by a factor of four for large $K$. We showcase the versatility of the proposed algorithm by applying it to three different problems: 1) sampling diffusion processes induced by localized sources on the sphere, 2) shot noise removal, and 3) sound source localization (SSL) by a spherical microphone array. In particular, we show how SSL can be reformulated as a spherical sparse sampling problem.
Euclidean distance matrices (EDM) are matrices of squared distances between points. The definition is deceivingly simple: thanks to their many useful properties they have found applications in psychometrics, crystallography, machine learning, wireles
s sensor networks, acoustics, and more. Despite the usefulness of EDMs, they seem to be insufficiently known in the signal processing community. Our goal is to rectify this mishap in a concise tutorial. We review the fundamental properties of EDMs, such as rank or (non)definiteness. We show how various EDM properties can be used to design algorithms for completing and denoising distance data. Along the way, we demonstrate applications to microphone position calibration, ultrasound tomography, room reconstruction from echoes and phase retrieval. By spelling out the essential algorithms, we hope to fast-track the readers in applying EDMs to their own problems. Matlab code for all the described algorithms, and to generate the figures in the paper, is available online. Finally, we suggest directions for further research.
We present the concept of an acoustic rake receiver---a microphone beamformer that uses echoes to improve the noise and interference suppression. The rake idea is well-known in wireless communications; it involves constructively combining different m
ultipath components that arrive at the receiver antennas. Unlike spread-spectrum signals used in wireless communications, speech signals are not orthogonal to their shifts. Therefore, we focus on the spatial structure, rather than temporal. Instead of explicitly estimating the channel, we create correspondences between early echoes in time and image sources in space. These multiple sources of the desired and the interfering signal offer additional spatial diversity that we can exploit in the beamformer design. We present several intuitive and optimal formulations of acoustic rake receivers, and show theoretically and numerically that the rake formulation of the maximum signal-to-interference-and-noise beamformer offers significant performance boosts in terms of noise and interference suppression. Beyond signal-to-noise ratio, we observe gains in terms of the emph{perceptual evaluation of speech quality} (PESQ) metric for the speech quality. We accompany the paper by the complete simulation and processing chain written in Python. The code and the sound samples are available online at url{http://lcav.github.io/AcousticRakeReceiver/}.
