Raking the Cocktail Party

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 multipath 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/}.

A Fast Hadamard Transform for Signals with Sub-linear Sparsity in the Transform Domain

A new iterative low complexity algorithm has been presented for computing the Walsh-Hadamard transform (WHT) of an $N$ dimensional signal with a $K$-sparse WHT, where $N$ is a power of two and $K = O(N^alpha)$, scales sub-linearly in $N$ for some $0 < alpha < 1$. Assuming a random support model for the non-zero transform domain components, the algorithm reconstructs the WHT of the signal with a sample complexity $O(K log_2(frac{N}{K}))$, a computational complexity $O(Klog_2(K)log_2(frac{N}{K}))$ and with a very high probability asymptotically tending to 1. The approach is based on the subsampling (aliasing) property of the WHT, where by a carefully designed subsampling of the time domain signal, one can induce a suitable aliasing pattern in the transform domain. By treating the aliasing patterns as parity-check constraints and borrowing ideas from erasure correcting sparse-graph codes, the recovery of the non-zero spectral values has been formulated as a belief propagation (BP) algorithm (peeling decoding) over a sparse-graph code for the binary erasure channel (BEC). Tools from coding theory are used to analyze the asymptotic performance of the algorithm in the very sparse ($alphain(0,frac{1}{3}]$) and the less sparse ($alphain(frac{1}{3},1)$) regime.

Pruned Continuous Haar Transform of 2D Polygonal Patterns with Application to VLSI Layouts

We introduce an algorithm for the efficient computation of the continuous Haar transform of 2D patterns that can be described by polygons. These patterns are ubiquitous in VLSI processes where they are used to describe design and mask layouts. There, speed is of paramount importance due to the magnitude of the problems to be solved and hence very fast algorithms are needed. We show that by techniques borrowed from computational geometry we are not only able to compute the continuous Haar transform directly, but also to do it quickly. This is achieved by massively pruning the transform tree and thus dramatically decreasing the computational load when the number of vertices is small, as is the case for VLSI layouts. We call this new algorithm the pruned continuous Haar transform. We implement this algorithm and show that for patterns found in VLSI layouts the proposed algorithm was in the worst case as fast as its discrete counterpart and up to 12 times faster.

Fast Continuous Haar and Fourier Transforms of Rectilinear Polygons from VLSI Layouts

We develop the pruned continuous Haar transform and the fast continuous Fourier series, two fast and efficient algorithms for rectilinear polygons. Rectilinear polygons are used in VLSI processes to describe design and mask layouts of integrated circuits. The Fourier representation is at the heart of many of these processes and the Haar transform is expected to play a major role in techniques envisioned to speed up VLSI design. To ensure correct printing of the constantly shrinking transistors and simultaneously handle their increasingly large number, ever more computationally intensive techniques are needed. Therefore, efficient algorithms for the Haar and Fourier transforms are vital. We derive the complexity of both algorithms and compare it to that of discrete transforms traditionally used in VLSI. We find a significant reduction in complexity when the number of vertices of the polygons is small, as is the case in VLSI layouts. This analysis is completed by an implementation and a benchmark of the continuous algorithms and their discrete counterpart. We show that on tested VLSI layouts the pruned continuous Haar transform is 5 to 25 times faster, while the fast continuous Fourier series is 1.5 to 3 times faster.