Aliasing-Free Wideband Beamforming Using Sparse Signal Representation

Abstract

Abstract-Sparse signal representation (SSR) is considered to be an appealing alternative to classical beamforming for direction-of-arrival (DOA) estimation. For wideband signals, the SSR-based approach constructs steering matrices, referred to as dictionaries in this paper, corresponding to different frequency components of the target signal. However, the SSR-based approach is subject to ambiguity resulting from not only spatial aliasing, just like in classical beamforming, but also from the over-completeness of the dictionary, which is typical to SSR. We show that the ambiguity caused by the over-completeness of the dictionary can be alleviated by using multiple measurement vectors. In addition, by considering the uniform linear array (ULA) structure, we argue that if the target signal contains at least two frequencies, whose absolute difference phrased in wavelengths is larger than twice the array spacing, the spatial aliasing corresponding to these frequencies will be completely distinct. These properties enable us to adapt the existing algorithms to extract the target DOAs without ambiguity.Index Terms-Direction-of-arrival estimation, multiple-dictionary, orthogonal matching pursuit, sparse signal representation, spatial aliasing, uniform linear array, wideband beamforming.

Sound source localization and speech enhancement with sparse Bayesian learning beamforming

“…Exploiting the underlying sparsity, sparse signal reconstruction improves significantly the resolution in DOA estimation. [9][10][11][12] While ' pnorm regularized maximum likelihood methods, with p 1, have been proposed to promote sparsity in DOA estimation [9][10][11]13 and wavefield reconstruction, 14,15 the accuracy of the resulting sparse estimate is determined by the ad hoc choice of the regularization parameter. 12,16 Sparse Bayesian learning (SBL) is a probabilistic parameter estimation approach which is based on a hierarchical Bayesian method for learning sparse models from possibly overcomplete representations resulting in robust maximum likelihood estimates.…”

Wireless Sensor Array Network DoA Estimation from Compressed Array Data via Joint Sparse Representation

“…In this case, the corresponding steering vector is given by: $\begin{array}{c}trueright\begin{array}{cc}{\mathrm{bolda}}_{\ell }\left(k\right)\hfill & ={[e^{\frac{-j{\omega }_k{u}_1\sin \left({\theta }_{\ell }\right)}{c}},e^{\frac{-j{\omega }_k{u}_2\sin \left({\theta }_{\ell }\right)}{c}},\cdots ,e^{\frac{-j{u}_H{\omega }_k\sin \left({\theta }_{\ell }\right)}{c}}]}^T\hfill \\ & =\chi {[1,e^{\frac{-j{\omega }_{k}d\sin \left({\theta }_{\ell }\right)}{c}},\cdots ,e^{\frac{-j{u}_H{\omega }_k(H-1)d\sin \left({\theta }_{\ell }\right)}{c}}]}^T\hfill \end{array}\end{array}$ where $\chi =e^{\frac{-j{\omega }_k{u}_1\sin \left({\theta }_{\ell }\right)}{c}}$ is a constant. Recall Equation (A2); the block sub-matrix is given by:$\begin{array}{c}trueright\mathrm{bold-sans-serif{\rm Y}}\left[k\right]={\mathbf{A}}_L\left(k\right)\otimes {\mathbf{\varphi }}_{e,k}\end{array}$The sparse representation matrix in Equation (8) that compactly expresses the spectrum of the H received signals is:$\begin{array}{c}trueright{\mathbf{A}}_L\left(k\right)=[{\mathrm{bolda}}_1\left(k\right),{\mathrm{bolda}}_2\left(k\right),\cdots ,{\mathrm{bolda}}_L\left(k\right)]\end{array}$ when ${\omega }_{k}d/c>\pi$ ($d>\lambda /2$) [37,50]; there are at least two different angles ${\theta }_1$ and …”

“…The sparse representation matrix in Equation (8) that compactly expresses the spectrum of the H received signals is:$\begin{array}{c}trueright{\mathbf{A}}_L\left(k\right)=[{\mathrm{bolda}}_1\left(k\right),{\mathrm{bolda}}_2\left(k\right),\cdots ,{\mathrm{bolda}}_L\left(k\right)]\end{array}$ when ${\omega }_{k}d/c>\pi$ ($d>\lambda /2$) [37,50]; there are at least two different angles ${\theta }_1$ and ${\theta }_2$ that satisfy:$\begin{array}{c}trueright{e}^{\frac{-j{\omega }_{k}d\sin \left({\theta }_1\right)}{c}}=e^{\frac{-j{\omega }_{k}d\sin \left({\theta }_2\right)}{c}}\end{array}$ in which ${\theta }_1\ne {\theta }_2\in (-\pi /2,\pi /2]$. This implies that there will be multiple identical columns in ${\mathrm{boldA}}_L\left(k\right)$, and hence, the coherence of ${\mathrm{boldA}}_L\left(k\right)$ will be one.…”

Underwater Localization via Wideband Direction-of-Arrival Estimation Using Acoustic Arrays of Arbitrary Shape

“…For example, the side-scan sonar of an autonomous underwater vehicle (AUV) is usually a 2D matrix of acoustic elements [4,5] designed to cover a given aperture with a given resolution, expressed in terms of the beamwidth of the main lobe of the array's beam patterns. Other relevant examples include linear arrays for DoA estimation [6] and three-dimensional general-purpose scanners such as tetrahedral and pyramidal arrays [7].…”