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In this article, we consider the problem of planar graph-based simultaneous localization and mapping (SLAM) that involves both poses of the autonomous agent and positions of observed landmarks. We present complex (CPL)-SLAM, an efficient and certifiably correct algorithm to solve planar graph-based SLAM using the complex number representation. We formulate and simplify planar graph-based SLAM as the maximum likelihood estimation on the product of unit complex numbers, and relax this nonconvex quadratic complex optimization problem to convex complex semidefinite programming (SDP). Furthermore, we simplify the corresponding complex SDP to Riemannian staircase optimization (RSO) on the complex oblique manifold that can be solved with the Riemannian trust region method. In addition, we prove that the SDP relaxation and RSO simplification are tight as long as the noise magnitude is below a certain threshold. The efficacy of this work is validated through applications of CPL-SLAM and comparisons with existing state-of-the-art methods on planar graph-based SLAM, which indicates that our proposed algorithm is capable of solving planar graph-based SLAM certifiably, and is more efficient in numerical computation and more robust to measurement noise than existing state-of-the-art methods. The C++ code for CPL-SLAM is available at https://github.com/MurpheyLab/CPL-SLAM. [ABSTRACT FROM AUTHOR]

We provide a rigorous account of the equivalence between the complex-valued widely linear estimation method and the quaternion involution widely linear estimation method with their vector-valued real linear estimation counterparts. This is achieved by an account of degrees of freedom and by providing matrix mappings between a complex variable and an isomorphic bivariate real vector, and a quaternion variable versus a quadri-variate real vector. Furthermore, we show that the parameters in the complex-valued linear estimation method, the complex-valued widely linear estimation method, the quaternion linear estimation method, the quaternion semi-widely linear estimation method, and the quaternion involution widely linear estimation method include distinct geometric structures imposed on complex numbers and quaternions, respectively, whereas the real-valued linear estimation methods do not exhibit any structure. This key difference explains, both in theoretical and practical terms, the advantage of estimation in division algebras (complex, quaternion) over their multivariate real vector counterparts. In addition, we discuss the computational complexities of the estimators of the hypercomplex widely linear estimation methods. [ABSTRACT FROM AUTHOR]

Reduced biquaternion is a four dimensional hyper–complex number which is commutative algebra and an extension of complex. Due to this property, the corresponding fast algorithm is designed in time frequency analysis which can better fit the convolution theorem than the non-commutative quaternion. In addition, the reduced biquaternion can be interpreted as a single point in 2-dimensional hyperbolic space with its algebraic structure which has more capacity and variable ability than the Euclidean space. However, the algebra properties of the reduced biquaternion have not yet been well explored in the deep convolutional neural network. In this paper, we develop a new deep network structure, namely reduced biquaternion valued convolutional neural network (RQV-CNN). The proposed RQV-CNN includes basic modules of reduced biquaternion convolution layer and fully connection layer. Extensive experiments on color image classification and color image denoising are conducted to evaluate the promising performance of the RQV-CNN framework. The results show that RQV-CNN outperforms the real-valued CNN (RV-CNN), complex-valued CNN (CV-CNN), and quaternion-valued CNN (QV-CNN) with same structures. The source code can be found at https://github.com/tasteimage/RQVCNN. [ABSTRACT FROM AUTHOR]

The determination of complex elastic, piezoelectric, and dielectric coefficients of piezoelectric ceramics is important for precision engineering devices. Here, a novel method for determining the optimal material coefficients is presented. This method minimizes the average relative error in the values of conductance, susceptance, resistance, and reactance obtained from the 1-D model in the IEEE Standard (ANSI/IEEE Std 176-1987) and the experimental measurements of the first and second radial modes. Poisson’s ratio is assumed to be a complex number in addition to the elastic, piezoelectric, and dielectric coefficients in the present method. The global minimum of the average relative error is found by searching the minimum among all local minima of the average relative error, which are obtained with the Levenberg–Marquardt modification of Newton’s method from randomly chosen initial conditions. The optimal material coefficients of an APC 850 disk and an APC 855 disk are calculated with this method. The uncertainties in the optimal material coefficients are evaluated by calculating the minimum average relative error when the real or imaginary part of each coefficient is prescribed. [ABSTRACT FROM AUTHOR]

It has long been known that the ellipse normally used to model the phase space extension of a beam in linear dynamics may be represented by a complex number which can be interpreted similar to a complex impedance in electrical circuits, so that classical electrical methods might be used for the design of such beam transport lines. However, this method has never been fully developed, and only the transport transformation of single particular elements, like drift spaces or quadrupoles, has been presented in the past. In this article, we complete the complex formalism of linear beam dynamics by obtaining a general differential equation and solving it, to show that the general transformation of a linear beam line is a complex Moebius transformation. This result opens the possibility of studying the effect of the beam line on complete regions of the complex plane and not only on a single point. Taking advantage of this capability of the formalism, we also obtain an important result in the theory of the transport through a periodic line, proving that the invariant points of the transformation are only a special case of a more general structure of the solution, which are the invariant circles of the one-period transformation. Among other advantages, this provides a new description of the betatron functions beating in case of a mismatched injection in a circular accelerator. [ABSTRACT FROM AUTHOR]

We propose an impedance-matching technique for metasurfaces generating evanescent fields with complex free-space impedance for subwavelength focusing. The proposed technique is mandatory for the subwavelength focusing far beyond the diffraction limit. First, we theoretically show the fact that the wave impedance becomes a complex number due to the generated evanescent fields and give an explicit form of the wave impedance as a ratio of local electric and magnetic fields. Then, we discuss the impedance-matching technique for the distributed complex wave impedance by a metasurface with a position-dependent impedance matrix. To demonstrate the proposed technique, we design a subwavelength focusing 1-D metasurface with the null-to-null beamwidth of one-twentieth of the free-space wavelength at 2.45 GHz, and its performance is verified by wave propagation simulations based on the equivalent circuit model. In order to further verify the validity of the proposed technique, a metasurface is implemented with the trilayer configuration, and the subwavelength focusing is experimentally demonstrated. [ABSTRACT FROM AUTHOR]

3-D registration has always been performed invoking singular value decomposition (SVD) or eigenvalue decomposition (EIG) in real engineering practices. However, these numerical algorithms suffer from uncertainty of convergence in many cases. A novel fast symbolic solution is proposed in this article by following our recent publication in this journal. The equivalence analysis shows that our previous solver can be converted to deal with the 3-D registration problem. Rather, the computation procedure is studied for further simplification of computing without complex-number support. Experimental results show that the proposed solver does not loose accuracy and robustness but improves the execution speed to a large extent by almost 50%–80%, on both a personal computer (PC) and an embedded processor. Note to Practitioners—3-D registration usually has a large computational burden in engineering tasks. The proposed symbolic solution can directly solve the eigenvalue and its associated eigenvector. A lot of computation resources can then be saved for better overall system performance. The deterministic behavior of the proposed solver also ensures long-endurance stability and can help an engineer better design thread timing logic. [ABSTRACT FROM AUTHOR]

In this paper, simple methodologies of deep learning application to conventional multiple-input multiple-output (MIMO) communication systems are presented. The deep learning technologies with deep neural network (DNN) structure, emerging technologies in various engineering areas, have been actively investigated in the field of communication engineering as well. In the physical layer of conventional communication systems, there are practical challenges of application of DNN: calculating complex number in DNN and designing proper DNN structure for a specific communication system model. This paper proposes and verifies simple solutions for the difficulty. First, we apply a basic DNN structure for signal detection of one-tap MIMO channel. Second, convolutional neural network (CNN) and recurrent neural network (RNN) structures are presented for MIMO system with multipath fading channel. Our DNN structure for one-tap MIMO channel can achieve the optimal maximum likelihood detection performance, and furthermore, our CNN and RNN structures for multipath fading channel can detect the transmitted signal properly. [ABSTRACT FROM AUTHOR]

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if it has an ideal periodic auto-correlation function. This paper proposes a method for constructing sparse perfect Gaussian integer sequences (SPGISs) in which most of the sequence elements are zero. The proposed SPGISs are obtained by linearly combining four base sequences or their cyclic-shift equivalents using nonzero Gaussian integer coefficients of equal magnitudes. Each base sequence contains four nonzero elements belonging to the set \\pm 1, \pm j\. The number of nonzero elements of the constructed SPGISs depends on the choice of complex coefficients and cyclic shifts. However, each SPGIS has at most 16 nonzero elements, irrespective of the sequence length. A systematic investigation is performed into the properties of the SPGISs and their Fourier dual equivalents. Finally, a general expression is derived for a perfect Gaussian integer sequence (PGIS) of length $4n$, where $n$ is any positive integer and most of the sequence elements are nonzero. [ABSTRACT FROM AUTHOR]

Complex division is commonly used in various applications in signal processing and control theory including astronomy and nonlinear RF measurements. Nevertheless, unless reliability and assurance are embedded into the architectures of such structures, the sub-optimal (and thus erroneous) results could undermine the objectives of such applications. As such, in this paper, we present schemes to provide complex number division architectures based on Sweeney, Robertson, and Tocher-division with error detection mechanisms. Different error detection architectures are proposed in this paper which can be tailored based on the eventual objectives of the designs in terms of area and time requirements, among which we pinpoint carefully the schemes based on recomputing with shifted operands to be able to detect faults based on recomputations for different operands in addition to the unified parity (simplified detecting code) and hardware redundancy approach. The design also implements a minimized look up table approach which favors in error detection based designs and provides high fault coverage with relatively-low overhead. Additionally, to benchmark the effectiveness of the proposed schemes, extensive error detection assessments are performed for the proposed designs through fault simulations and field-programmable gate array (FPGA) implementations; the design is implemented on Xilinx Spartan-6 and Xilinx Virtex-6 FPGA families. [ABSTRACT FROM PUBLISHER]

To solve practical challenges for implementing deep learning-based algorithm in MIMO signal detector such as handling complex number or designing proper neural network for a specific communication system, (Baek et al., 2019) has proposed candidate implementation methodologies with simple verification experiments. According to (Baek et al., 2019), it was shown that the proposed algorithms can achieve the optimal ML performance. However, due to the lack of explanation on the experimental environment, it is difficult for readers to reproduce the presented experiments and obtain the same results. This document precisely explains on the experimental environments of (Baek et al., 2019) including the exact channel profiles. [ABSTRACT FROM AUTHOR]