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We propose an end-to-end learned bit-wise autoen-coder neural network (NN) for open loop multiple-input multiple-output (MIMO) antenna based communication systems. The optimized transmit vector constellations are learned based on the number of transmit and receive antennas, the number of bits conveyed per channel use and the signal-to-noise ratio. By training through an i.i.d. complex Gaussian (i.e., Rayleigh ergodic) matrix channel, the system implicitly picks up constellation shaping gains “along the way”. We evaluate and analyze these gains in comparison to the single-input single-output (SISO) results shown in [1] for different symmetric and asymmetric MIMO configurations. We first show that the NN-based receiver is able to compete with the a posteriori probability (APP) receiver performance up to certain configuration settings. Then we perform an end-to-end optimization of the transmit vector constellations using the conventional APP receiver as the decoder part. Thereby, we also investigate an edge case where the number of bits per vector symbol is not a multiple of the number of transmit antennas. Finally, we examine the performance of the system if a non-optimal zero forcing (ZF) receiver is used for inference, while the vector constellation was optimized for the APP receiver during training. We show that constellations optimized for the APP receiver are not necessarily optimal for such sub-optimal receivers that operate with reduced complexity. To fix this, we propose a detector-aware training scheme to learn constellations that have been optimized for a specific sub-optimal receiver and, thus, achieve higher performance in inference.

This paper proposes the use of systematic LDPC codes for hard decision Message Passing (hMP), which is a predecoder to Vector Symbol Decoding (VSD) for non-binary codes. VSD is a verification-based decoder whose outputs are decoded code words. LDPC are randomly generated and by nature, they are non-systematic codes. Using systematic instead of non-systematic codes allows the decoder to obtain the decoded data directly without a troublesome mapping step. In the simulations, 15 regular parity check matrices H are randomly generated and converted to their systematic forms. Results in a Gilbert-Elliot 2-state fading channel model with 6 different channel conditions show that hMP prefers codes with lower- weight H. For column weight of 9 bits and row weight of 18 bits, systematic H have lower weight after row operation and are better than the non-systematic ones for all channel conditions.

To have the burst error correcting feature of nonbinary codes with low complexity decoding, Vector Symbol Decoder (VSD) uses non-binary codes with the structure of binary $G$ and $H$ matrix. VSD can always correct if the number of error symbols is at most two fewer than the minimum distance $(d)$ . For more errors, weight distribution plays an important role in the ability to correct. This paper aims to analytically compare the performance of different codes with the same $d$ using their weight distributions. The goal is to select the nonbinary codes with better VSD decoding performance without simulations. For example, the (27,14,7) code has better correction capability than the (23,12,7) code from the proposed formula and confirmed by the simulation results. Additional simulation results in Rician fading channel with Doppler Effect also confirm that the first code is superior.

This paper proposes a new decoding technique called “MP-VSD” for LDPC codes with large nonbinary symbols. It combines Hard Decision Message Passing (HDMP) with Vector Symbol Decoding (VSD). VSD usually uses very short block codes to limit the number of error symbols, which limits the size of matrix inversion required in the VSD decoding step. MP-VSD can correct more than 60% of the correctable error patterns of VSD for an (60, 30) LDPC code with no matrix inversions in a 2-state fading channel model. Thus, longer block codes may be used for nonbinary symbols.

Vector projection and vector rejection are highly common and useful operations in mathematics, information theory, and signal processing. In this paper, we find the distribution of the norm of projection and rejection vectors when the original vectors are standard complex normally distributed.

This letter shows how to extend packet/vector symbol burst error correction capability beyond prior known limits with the aid of soft information interaction with the inner code of a concatenated code. If the errors prove not to be in a burst, further correction still is possible with the aid of soft inner code information. Also, the soft information provides strong additional error detection.

In this communication, simple methods are shown for correcting bursts of large size and bursts combined with random errors using vector symbols and primarily vector XOR and feedback shift register operations. One result is that any (n, k) cyclic code with minimum distance > 2 can correct all full vector symbol error bursts of length n-k-1 or less if the error vectors are linearly independent. If the bursts are not full but contain some error-free components, the capability of correcting bursts up to n-k or less is code dependent. Also, vector symbol decoding with Reed-Solomon component codes can correct, very simply, with probability ges 1- n(n - k)2-r, all cases of e les n - k - 1 r-bit random errors in any cyclic span of length les n - k. The techniques often work when there is linear dependence. In cases where most errors are in a burst but a small number of errors are outside, the solution, given error-correcting capability, can be broken down into a simple solution for the small number of outside errors, followed by a simple subtraction to reveal all the error values in the burst part.

In this paper, we propose a generalized expression of multi-target probability density in the framework of finite-set statistics (FISST). It differs from the existing expression in that it is completely defined by a set of parameters. The parameterized expression offers three important advantages. Firstly, the statistics of any subset of an random finite set (RFS) can be conveniently extracted by directly computing the marginal distribution of the proposed parameterized probability density. Note that the theoretically sound way of computing the marginal distribution of an RFS probability density is not well defined in current FISST. Secondly, it inherits advantages of the vector notation based density in that the identities of target states are implicitly embedded with no need to augment target state with an extra label. Thirdly, it is a generalized expression capable to describe the statistics of almost all kinds of RFSs and doesn't need to assume a specified measurement model for the development of target tracking filter. As a preliminary study, this paper provides both the derivations of the proposed expression and the resultant tracking filter for a two-target scenario. The aforementioned advantages are clearly highlighted by the numerical results.

We have attempted to express the frequency response functions of a linear and a quadratic non-linear system in terms of spectral vectors. These vector notations convey the system characteristics in physically realisable measures. One of the valuable tools to verify the non-linear system features is the expression of a coherence function using vector notation. The visualisation of a coherence function demonstrates the dependency of an output signal on the linear and quadratic non-linear terms in the Volterra model. Multiple coherence functions are also introduced to validate the modelling errors.

An analysis method based on vector triangle and vector locus is introduced in this paper. By using vector triangles, a set of quantities generally used in feedback theory can be intuitively visualized, which is impossible by using other theories. The vector locus records the evolvement of the vector triangle versus frequency in polar coordinates, and combines the magnitude and phase plot together in one track. Examples are given to show that the proposed method gives intuitive insights while still retains a general method to characterize feedback systems.

It is well known that vector–tensor notation is a compact and natural language for the mathematical formulation of continuum mechanics problems. Here we describe the application of vector technique to numerical simulation starting with a mathematical formulation. We provide an efficient numerical scheme and furnish an implementation as a computer program. As a result (in comparison with traditional “component” form), the two last steps are significantly simplified, especially for multidimensional problems with various boundary conditions in irregular geo- metries where nonorthogonal meshes are applied. Therefore, more attention can be focused on the physical nature of problems. Apart from the cleaner syntax, vector notation conserves the structure of traditional numerical algorithms and solves multidimensional problems with minimum additional programming effort. Complex practical applications of this technique are described as well.

A method is presented to calibrate transfer function-noise (TFN) models, operating at the same frequency as the input (auxiliary) variables, to sparsely or irregularly observed time series of the output (target) variable. Once calibrated, the TFN models can be used to predict or simulate the output variable at the same frequency as the input variable. Consequently, the method provides a useful tool for filling in gaps of irregularly or sparsely observed hydrological time series. Although generic and suitable for any type of time series, the method is described through the modeling of a time series of groundwater head data with precipitation surplus (precipitation minus potential evapotranspiration) as input variable. First, the TFN model is written in vector notation, yielding the state equation of a linear discrete stochastic system. Subsequently, the state equation is embedded in a Kalman filter algorithm. The Kalman filter is then combined with a maximum likelihood criterion to obtain estimates of the parameters of the TFN model for small time steps (e.g., 1 day) while using sparsely (e.g., two times a month) or even irregularly observed time series of groundwater head data. The method is illustrated using (subsets of) time series of groundwater head data with varying regular and irregular observation intervals.

This book employs homogeneous coordinate notation to compute the first- and second-order derivative matrices of various optical quantities. It will be one of the important mathematical tools for automatic optical design. The traditional geometrical optics is based on raytracing only. It is very difficult, if possible, to compute the first- and second-order derivatives of a ray and optical path length with respect to system variables, since they are recursive functions. Consequently, current commercial software packages use a finite difference approximation methodology to estimate these derivatives for use in optical design and analysis. Furthermore, previous publications of geometrical optics use vector notation, which is comparatively awkward for computations for non-axially symmetrical systems.

In previous work the author has presented a methodology for obtaining closed form equations of motion for general non-holonomic hybrid parameter multiple body systems (HPMBS). It is well known that generating closed form symbolic equations of motion for HPMBS of few degrees of freedom rapidly degenerates to an algebraic soup unless computer tools are utilized. Methods dependent on Lagrange multipliers are virtually unable to provide the constant free (minimal) symbolic equations for systems of moderate to high order due to the complication of eliminating the multipliers. In this paper, the intent is to provide the details, and example use of, a tool, based on commercial symbolic manipulation software, developed to assist the symbolic processing of the equations of motion as previously minimally formulated. The tool presented herein is novel in the way the vector notation of the formulation scheme is retained and the way that frame information is carried in the calculations. The symbolic tool provided in this paper allows the modelling methodology to be a practical tool for generating closed form equations of motion for extremely complicated HPMBS.

An embedded image coder based on wavelet transform coding and multistage vector quantization (VQ) is proposed in this research. We have examined several critical issues to make the proposed algorithm practically applicable. They include the complexity of embedded VQ, design of the successive vector quantizer, significance evaluation of a vector symbol, and integration of wavelet transform coding and vector quantization. It is shown in experiments that the new method achieves a superior rate-distortion performance.

It has been shown that a vector transform is a good tool for digital filtering and image coding. This paper addresses some of the issues related to vector transforms. First, a fundamental difference between vector transforms and conventional transforms is examined. Second, two approaches to constructing vector transforms are discussed. Finally, an application of vector transformation to an efficient implementation of 2-D nonseparable digital filters is proposed. >

The primitive logic functions NOT, AND, OR, NAND, NOR, XOR, and XNOR can be connected together to build slightly more complex functions. A single signal carrying one bit of binary data is known as a scalar entity. A set of signals carrying similar data can be gathered together into a group known as a vector. A key advantage of vector notation is that it allows all of the signals comprising the vector to be easily referenced in a single statement. A multiplexer uses a binary value, or address, to select between a number of inputs and to convey the data from the selected input to the output. A decoder uses a binary value, or address, to select between a number of outputs and to assert the selected output by placing it in its active state. There is a special category of gates called tri-state functions whose outputs can adopt three states: 0, 1, and Z. The tri-state buffer’s symbol is based on a standard buffer with an additional control input known as the enable. A more sophisticated function called a D-type latch can be constructed by attaching two AND gates and a NOT gate to the front of an RS latch. In the case of a D-type flip-flop, the data appears to be loaded when a transition, or edge, occurs on the clock input, which is therefore said to be edge-sensitive. Counter functions are also commonly used in digital systems. The number of states that the counter will sequence through before returning to its original value is called the modulus of the counter.

A vector notation is developed for describing complementary codes defined at finite apertures of particular relevance to measuring and imaging problems. The original complementary series of Golay defined at one-dimensional linear apertures, are extended for more general apertures, notably in two dimensions. Two generating rules of the codes are formulated, namely the doubling rule familiar from the one-dimensional Golay series, and the point symmetry rule for adding just one vector element to a code of a strict rotational symmetry. Both on-off-type and polarity-type modulation are included and decoding algorithms are presented based on the ideal delta-function-type autocorrelation function of the codes without any side structure. Statistical analysis of the application of the complementary vector codes to a modulated Poisson process is given and implementation of the codes to ultrasonics, powder diffraction and X-ray astronomy discussed.

We consider the communication system that transmits a sequence of binary vector symbols over a vector intersymbol interference (ISI) channels subject to additive white Gaussian noise. Conventionally, maximum likelihood (ML) sequence is computed using the Viterbi algorithm (VA), whose complexity scales exponentially in both the symbol vector length and the number of ISI channel taps. We show that, as the signal to noise ratio (SNR) goes to infinity, the ML sequence can be obtained with an asymptotic complexity scaling linearly in the number of channel taps and quadratically in the symbol vector length.

The performance of ML detection for the given channel is analyzed in spatially multiplexed MIMO system. In order to obtain the vector symbol error rate, we define error vectors which represent the geometrical relation between lattice points. The properties of error vectors are analyzed to show that all lattice points in an infinite lattice almost surely have four nearest neighbors after random channel transformation. Using this information and the minimum distance obtained by the modified sphere decoding algorithm, we formulate the analytical performance of vector symbol error rate(VSER) over the given channel. To verify the results, we simulate ML performance over the various random channels which are classified into three categories: unitary channel, dense channel, and sparse channel. From the simulation results, it is verified that the derived analytical result gives a good approximation for the performance of ML detector over all random MIMO channels.

Recently, the 3-D CAP system was introduced as a modification to the conventional 2-D CAP. This paper makes two contributions in the context of the 3-D CAP system. First, the QAM equivalent for the three-dimensional case is introduced by extending the 2-D complex notation into a 3-D vector notation. The definition is a modification of the well known quaternions. Second, the 3-D QAM filtering is redefined using algorithmic strength reduction to reduce the overall complexity by about one third with marginal degradation. The 3-D QAM system using 9 filters performs almost exactly as the 3-D CAP. Further complexity reduction is achieved by exploiting filtering reduction at the expense of only one dB of receiver SNR degradation.

Vector symbol decoding is an outer code decoding technique for a concatenated code which works with a (n-k)*r syndrome matrix S of n-k linear combinations of r-bit inner code symbol vectors. A Gauss-Jordan reduction provides an error location vector. The zeros in the error location vector are the apparent error positions, and the number of zeros should match the rank of S. Then the error values can be solved. Decoding success is related to the linear independence of error vectors. Data scrambling techniques for inner code symbols can make linear independence likely for moderately large r. Any parity check code structure can be used for the combination rules. The new result is that, if the outer code decoder has available a (possibly ordered) list of two or more alternative decisions for some or all inner symbols, a slightly modified vector symbol decoder automatically reveals most correct alternatives, allowing more powerful and often simpler correction. Moreover, the ability to recognize these alternatives does not require error vector linear independence. The main idea is to store differences between alternative choices and the first choice as additional rows below the syndrome matrix S.

The goal of this subcontract was to modify the FOSPACK code, developed by John Ruge, to call the BoomerAMG solver developed at LLNL through the HYPRE interface. FOSPACK is a package developed for the automatic discretization and solution of First-Order System Least-Squares (FOSLS) formulations of 2D partial differential equations (c.f [3-9]). FOSPACK takes a user-specified mesh (which can be an unstructured combination of triangular and quadrilateral elements) and specification of the first-order system, and produces the discretizations needed for solution. Generally, all specifications are contained in data files, so no re-compilation is necessary when changing domains, mesh sizes, problems, etc. Much of the work in FOSPACK has gone into an interpreter that allows for simple, intuitive specification of the equations. The interpreter reads the equations, processes them, and stores them as instruction lists needed to apply the operators involved to finite element basis functions, allowing assembly of the discrete system. Quite complex equations may be specified, including variable coefficients, user defined functions, and vector notation. The first-order systems may be nonlinear, with linearizations either performed automatically, or specified in a convenient way by the user. The program also includes global/local refinement capability. FOSLS formulations are very well suited for solutionmore » by algebraic multigrid (AMG) (c.f. [10-13]). The original version uses a version of algebraic multigrid written by John Ruge in FORTRAN 77, and modified somewhat for use with FOSPACK. BoomerAMG, a version of AMG developed at CASC, has a number of advantages over the FORTRAN version, including dynamic memory allocation and parallel capability. This project was to benefit both FRSC and CASC, giving FOSPACK the advantages of BoomerAMG, while giving CASC a tool for testing FOSLS as a discretization method for problems of interest there. The major parts of this work were implementation and testing of the HYPRE package on our computers, writing a C wrapper/driver for the FOSPACK code, and modifying the wrapper to call BoomerAMG through the HYPRE interface.« less

The input-output equation for an open loop linkage with six revolute joints is derived from a set of 20 equations in the 28 parameters that describe the equivalent closed loop seven link mechanism. Computer software is used to manipulate the resultant multinomials (potentially with an enormous number of terms). The resultant set of equations are easily given in a simple vector notation, with no necessity, on the part of the user, to invest in the underlying computations. This also provides further justification for the recent derivation of a 16th degree polynomial I/O equation, with simplification for further analysis. †

The mathematical solution for the true positions of features in space from the position of their images on pairs of radiographs has been discussed in Ballistic Research Laboratory reports by Grabarek and Herr (BRL TN 1634, September 1966, AD 807619), and by Henry C. Dubin (BRL MR 2470, April 1975, AD B003797L, now available for unlimited distribution). A truly general solution should not require, or be limited to, orthogonality of the image pairs. Dubin provides such a solution. Because of errors in setup and measurement, the two lines presumed to connect the respective source and image points through the common object point do not necessarily intersect. Using vector notation and partial derivatives, he obtains the line of minimum length between these two vectors. The midpoint of this line is the best estimate for the object position, and the length is a measure of the error of the estimate. The redundant image coordinate in Dubin's method contributes to increasing the accuracy of the estimate of the position of features. The solution of Grabarek and Herr uses analytic geometry and the assumption that the two lines between the respective sources and images intersect, and requires an orthogonal radiographic setup. This approach forgoes generality and some available accuracy. Driven by the need to provide as simple an approach as possible, this paper presents two similar derivations. The author uses analytic geometry to rederive the equations of Grabarek and Herr from a simpler perspective. The form of the solution provides a conceptual bridge to a more direct derivation by the author, using trigonometry. Both derivations are for orthogonal radiographic setups. The trigonometric approach does not require complicated computation of magnification factors, is more easily understood in terms of the geometry of the setup, and is easily implemented in computer or calculator programs to reduce orthogonal radiographs.