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1. Non-Uniform Frequency Spacing for Regularization-Free Gridless DOA

   https://doi.org/10.1109/ICASSP48485.2024.10446655  

   Wu, Yifan; Wakin, Michael B; Gerstoft, Peter; Park, Yongsung (April 2024, IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   Gridless direction-of-arrival (DOA) estimation with multiple frequencies can be applied to acoustic source localization. We formulate this as an atomic norm minimization (ANM) problem and derive a regularization-free semi-definite program (SDP) avoiding regularization bias. We also propose a fast SDP program to deal with non-uniform frequency spacing. The DOA is retrieved via irregular Vandermonde decomposition (IVD), and we theoretically guarantee the existence of the IVD. We extend ANM to the multiple measurement vector setting and derive its equivalent regularization-free SDP. For a uniform linear array using multiple frequencies, we can resolve more sources than the sensors. The effectiveness of the proposed framework is demonstrated via numerical experiments. 

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2. Convolutional Beamspace and Sparse Signal Recovery for Linear Arrays

   https://doi.org/10.1109/IEEECONF51394.2020.9443522  

   Chen, Po-Chih; Vaidyanathan, P. P. (November 2020, Proc. Asil. Conf. Sig., Sys., and Comp)

   null (Ed.)

   The convolutional beamspace (CBS) method for DOA estimation using dictionary-based sparse signal recovery is introduced. Beamspace methods enjoy lower computational complexity, increased parallelism of subband processing, and improved DOA resolution. But unlike classical beamspace methods, CBS allows root-MUSIC and ESPRIT to be performed directly for ULAs without additional preparation since the Vandermonde structure for ULAs are preserved in the CBS output. Due to the same reason, it is shown in this paper that sparse signal representation problems can also be directly formulated on the CBS output. Significant reduction in computational complexity and higher probability of resolution are obtained by using CBS. It is also shown how the regularization parameter involved in the method should be chosen 

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3. Convolutional beamspace for linear arrays

   https://doi.org/DOI: 10.1109/TSP.2020.3021670  

   Chen, P.C; Vaidyanathan, P.P (September 2020, IEEE transactions on signal processing)

   null (Ed.)

   Anew beamspace method for array processing, called convolutional beamspace (CBS), is proposed. It enjoys the advantages of classical beamspace such as lower computational complexity, increased parallelism of subband processing, and improved resolution threshold for DOA estimation. But unlike classical beamspace methods, it allows root-MUSIC and ESPRIT to be performed directly for ULAs without additional preparation since the Vandermonde structure and the shift-invariance are preserved under the CBS transformation. The method produces more accurate DOA estimates than classical beamspace, and for correlated sources, better estimates than element-space. The method also generalizes to sparse arrays by effective use of the difference coarray. For this, the autocorrelation evaluated on theULAportion of the coarray is filtered appropriately to produce the coarray CBS. It is also shown how CBS can be used in the context of sparse signal representation with dictionaries, where the dictionaries have columns that resemble steering vectors at a dense grid of frequencies. Again CBS processing with dictionaries offers better resolution, accuracy, and lower computational complexity. As only the filter responses at discrete frequencies on the dictionary grid are relevant, the problem of designing discrete-frequency FIR filters is also addressed. 

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4. Convolutional Beamspace for Array Signal Processing

   https://doi.org/10.1109/ICASSP40776.2020.9054051  

   Vaidyanathan, P. P.; Chen, Po-Chih (May 2020, Proc. IEEE Int. Conf. Acoust. Speech, and Signal Proc)

   null (Ed.)

   A new type of beamspace for array processing is introduced called convolutional beamspace. It enjoys the advantages of traditional beamspace such as lower computational complexity, increased parallelism of subband processing, and improved resolution threshold for DOA estimation. But unlike traditional beamspace methods, it allows root-MUSIC and ESPRIT to be performed directly for ULAs without any overhead of preparation, as the Vandermonde structure and the shift-invariance are preserved under the transformation. The method produces more accurate DOA estimates than traditional beamspace methods, and for correlated sources it produces better estimates than element-space methods. 

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5. Lebesgue Decomposition of Non-Commutative Measures

   https://doi.org/10.1093/imrn/rnaa231  

   Jury, Michael T; Martin, Robert T (September 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $$C^{\ast }-$$algebra, the $$C^{\ast }-$$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $$C^{\ast }-$$algebras; any *−representation of the Cuntz–Toeplitz $$C^{\ast }-$$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory. 

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