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Keywords = two-dimensional unitary transformation

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15 pages, 441 KB  
Article
Efficient Nyström-Based Unitary Single-Tone 2D DOA Estimation for URA Signals
by Liping Yuan, Ke Wang and Fengkai Luan
Mathematics 2025, 13(15), 2335; https://doi.org/10.3390/math13152335 - 22 Jul 2025
Viewed by 280
Abstract
We propose an efficient Nyström-based unitary subspace method for low-complexity two-dimensional (2D) direction-of-arrival (DOA) estimation in uniform rectangular array (URA) signal processing systems. The conventional high-resolution DOA estimation methods often suffer from excessive computational complexity, particularly when dealing with large-scale antenna arrays. The [...] Read more.
We propose an efficient Nyström-based unitary subspace method for low-complexity two-dimensional (2D) direction-of-arrival (DOA) estimation in uniform rectangular array (URA) signal processing systems. The conventional high-resolution DOA estimation methods often suffer from excessive computational complexity, particularly when dealing with large-scale antenna arrays. The proposed method addresses this challenge by combining the Nyström approximation with a unitary transformation to reduce the computational burden while maintaining estimation accuracy. The signal subspace is approximated using a partitioned covariance matrix, and a real-valued transformation is applied to further simplify the eigenvalue decomposition (EVD) process. Furthermore, the linear prediction coefficients are estimated via a weighted least squares (WLS) approach, enabling robust extraction of the angular parameters. The 2D DOA estimates are then derived from these coefficients through a closed-form solution, eliminating the need for exhaustive spectral searches. Numerical simulations demonstrate that the proposed method achieves comparable estimation performance to state-of-the-art techniques while significantly reducing computational complexity. For a fixed array size of M=N=20, the proposed method demonstrates significant computational efficiency, requiring less than 50% of the running time compared to conventional ESPRIT, and only 6% of the time required by ML methods, while maintaining similar performance. This makes it particularly suitable for real-time applications where computational efficiency is critical. The novelty lies in the integration of Nyström approximation and unitary subspace techniques, which jointly enable efficient and accurate 2D DOA estimation without sacrificing robustness against noise. The method is applicable to a wide range of array processing scenarios, including radar, sonar, and wireless communications. Full article
(This article belongs to the Section E2: Control Theory and Mechanics)
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137 pages, 3333 KB  
Review
Monte Carlo Based Techniques for Quantum Magnets with Long-Range Interactions
by Patrick Adelhardt, Jan A. Koziol, Anja Langheld and Kai P. Schmidt
Entropy 2024, 26(5), 401; https://doi.org/10.3390/e26050401 - 1 May 2024
Cited by 13 | Viewed by 3396
Abstract
Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum–optical platforms promises to gain deep insights into quantum-critical properties induced by the long-range nature of interactions. From a theoretical [...] Read more.
Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum–optical platforms promises to gain deep insights into quantum-critical properties induced by the long-range nature of interactions. From a theoretical perspective, long-range interactions are notoriously complicated to treat. Here, we give an overview of recent advancements to investigate quantum magnets with long-range interactions focusing on two techniques based on Monte Carlo integration. First, the method of perturbative continuous unitary transformations where classical Monte Carlo integration is applied within the embedding scheme of white graphs. This linked-cluster expansion allows extracting high-order series expansions of energies and observables in the thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo integration enables calculations on large finite systems. Finite-size scaling can then be used to determine the physical properties of the infinite system. In recent years, both techniques have been applied successfully to one- and two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg interactions on various bipartite and non-bipartite lattices. Here, we summarise the obtained quantum-critical properties including critical exponents for all these systems in a coherent way. Further, we review how long-range interactions are used to study quantum phase transitions above the upper critical dimension and the scaling techniques to extract these quantum critical properties from the numerical calculations. Full article
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57 pages, 592 KB  
Article
Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group
by Christopher S. Jackson and Carlton M. Caves
Entropy 2023, 25(8), 1221; https://doi.org/10.3390/e25081221 - 16 Aug 2023
Cited by 3 | Viewed by 2207
Abstract
The canonical commutation relation, [Q,P]=i, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the [...] Read more.
The canonical commutation relation, [Q,P]=i, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
19 pages, 609 KB  
Article
Target Parameter Estimation Algorithm Based on Real-Valued HOSVD for Bistatic FDA-MIMO Radar
by Yuehao Guo, Xianpeng Wang, Jinmei Shi, Lu Sun and Xiang Lan
Remote Sens. 2023, 15(5), 1192; https://doi.org/10.3390/rs15051192 - 21 Feb 2023
Cited by 9 | Viewed by 2105
Abstract
Since there is a frequency offset between each adjacent antenna of FDA radar, there exists angle-range two-dimensional dependence in the transmitter. For bistatic FDA-multiple input multiple output (MIMO) radar, range-direction of departure (DOD)-direction of arrival (DOA) information is coupled in transmitting the steering [...] Read more.
Since there is a frequency offset between each adjacent antenna of FDA radar, there exists angle-range two-dimensional dependence in the transmitter. For bistatic FDA-multiple input multiple output (MIMO) radar, range-direction of departure (DOD)-direction of arrival (DOA) information is coupled in transmitting the steering vector. How to decouple the three information has become the focus of research. Aiming at the issue of target parameter estimation of bistatic FDA-MIMO radar, a real-valued parameter estimation algorithm based on high-order-singular value decomposition (HOSVD) is developed. Firstly, for decoupling DOD and range in transmitter, it is necessary to divide the transmitter into subarrays. Then, the forward–backward averaging and unitary transformation techniques are utilized to convert complex-valued data into real-valued data. The signal subspace is obtained by HOSVD, and the two-dimensional spatial spectral function is constructed. Secondly, the dimension of spatial spectrum is reduced by the Lagrange algorithm, so that it is only related to DOA, and the DOA estimation is obtained. Then the frequency increment between subarrays is used to decouple the DOD and range information, and eliminate the phase ambiguity at the same time. Finally, the DOD and range estimation automatically matched with DOA estimation are obtained. The proposed algorithm uses the multidimensional structure of high-dimensional data to promote performance. Meanwhile, the proposed real-valued tensor-based method can effectively cut down the computing time. Simulation results verify the high efficiency of the developed method. Full article
(This article belongs to the Special Issue Theory and Applications of MIMO Radar)
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9 pages, 24178 KB  
Article
Programmable SCOW Mesh Silicon Photonic Processor for Linear Unitary Operator
by Liangjun Lu, Linjie Zhou and Jianping Chen
Micromachines 2019, 10(10), 646; https://doi.org/10.3390/mi10100646 - 26 Sep 2019
Cited by 9 | Viewed by 4890
Abstract
Universal unitary multiport interferometers (UMIs) can perform any arbitrary unitary transformation to a vector of input optical modes, which are essential for a wide range of applications. Most UMIs are realized by fixed photonic circuits with a triangular or a rectangular architecture. Here, [...] Read more.
Universal unitary multiport interferometers (UMIs) can perform any arbitrary unitary transformation to a vector of input optical modes, which are essential for a wide range of applications. Most UMIs are realized by fixed photonic circuits with a triangular or a rectangular architecture. Here, we present the implementation of an N × N rectangular UMI with a programmable photonic processor based on two-dimensional meshes of self-coupled optical waveguide (SCOW) resonant structures. Our architecture shows a high tolerance to the unbalanced loss upon interference. This work enriches the functionality of the SCOW mesh photonic processors, which are promising for field-programmable photonic arrays. Full article
(This article belongs to the Special Issue Silicon Photonics Bloom)
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17 pages, 237 KB  
Article
2-D Unitary ESPRIT-Like Direction-of-Arrival (DOA) Estimation for Coherent Signals with a Uniform Rectangular Array
by Shiwei Ren, Xiaochuan Ma, Shefeng Yan and Chengpeng Hao
Sensors 2013, 13(4), 4272-4288; https://doi.org/10.3390/s130404272 - 28 Mar 2013
Cited by 39 | Viewed by 8781
Abstract
A unitary transformation-based algorithm is proposed for two-dimensional (2-D) direction-of-arrival (DOA) estimation of coherent signals. The problem is solved by reorganizing the covariance matrix into a block Hankel one for decorrelation first and then reconstructing a new matrix to facilitate the unitary transformation. [...] Read more.
A unitary transformation-based algorithm is proposed for two-dimensional (2-D) direction-of-arrival (DOA) estimation of coherent signals. The problem is solved by reorganizing the covariance matrix into a block Hankel one for decorrelation first and then reconstructing a new matrix to facilitate the unitary transformation. By multiplying unitary matrices, eigenvalue decomposition and singular value decomposition are both transformed into real-valued, so that the computational complexity can be reduced significantly. In addition, a fast and computationally attractive realization of the 2-D unitary transformation is given by making a Kronecker product of the 1-D matrices. Compared with the existing 2-D algorithms, our scheme is more efficient in computation and less restrictive on the array geometry. The processing of the received data matrix before unitary transformation combines the estimation of signal parameters via rotational invariance techniques (ESPRIT)-Like method and the forward-backward averaging, which can decorrelate the impinging signalsmore thoroughly. Simulation results and computational order analysis are presented to verify the validity and effectiveness of the proposed algorithm. Full article
(This article belongs to the Section Physical Sensors)
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