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In this study, a classification algorithm based on complex number feature is proposed. Specifically, the SVM framework is reformulated, so each example would be classified in the unitary space. The cost function is redefined by considering the maximum margin of real and imaginary units of the complex number feature at the same time. The cost function is based on the expectation of the hinge loss, and its derivatives can be calculated in closed forms. Using a stochastic gradient descent (SGD) algorithm, this method allows for efficient implementation. For complex number feature, the example uncertainty is modeled by a sample preprocessing method based on within-class Euclidean distance Gaussian distribution sample (DGS). In addition, a complex number feature selection method based on improved hybrid discrimination analysis (HDA) is proposed by considering the correlation between real and imaginary units of complex number feature. The proposed classification algorithm is tested on synthetic data and three publicly available and popular datasets, namely, MNIST, WDBC, and Voc2012. Experimental results verify the effectiveness of the proposed method. The codes are available: https://github.com/luckysomebody/paper-code.

As the existing complex number Nth root computation methods are relatively discrete, we propose a general method and architecture based on coordinate rotation digital computer (CORDIC) to compute arbitrary complex number Nth root for the first time. Our method performs the tasks of computing complex modulus, complex phase angle, real Nth root, sine function and cosine function, which can be implemented by circular CORDIC, linear CORDIC and hyperbolic CORDIC. Based on these CORDICs, our proposed architecture can not only improve the hardware efficiency just through shift-add operations, but also flexibly adjust the precision and the input range of complex number Nth root. To prove its feasibility, we conduct a software simulation and implement an example circuit in hardware. Under the TSMC 28nm CMOS technology, we synthesize it and get the report that it has the area of 6561μm2 and the power of 3.95mW at the frequency of 1.5GHz.

Google PageRank is widely used by search engines and known for its high calculation cost. This study proposes the usage of complex numbers in node ranking calculations for a network. The node scores are calculated in the graph using the Hermitian adjacency matrix and the complex plane through an algorithm, called HermitianStatus. The result of the proposed method is then compared to that of Google PageRank. In conclusion, HermitianStatus has advantages over PageRank.

Determination of complex number modulus and argument is commonly encountered task in digital signal processing. According to definition, this task requires evaluation of square root and inverse tangent functions. When computing hardware resources are limited, e.g. in real-time applications, an approximation by basic arithmetic and logical operations are of interest. We propose method for modulus approximation that uses addition, scaling and comparison, and provides also argument information. At each additional level of our algorithm, modulus approximation error decreases by factor of 4 and argument error by factor of 8.

In this paper, we present CPL-Sync, a certifiably correct algorithm to solve planar pose graph optimization (PGO) using the complex number representation. We formulate planar PGO as the maximum likelihood estimation (MLE) on the product of unit complex numbers, and relax this nonconvex quadratic complex optimization problem to complex semidefinite programming (SDP). Furthermore, we simplify the corresponding semidefinite programming to Riemannian staircase optimization (RSO) on complex oblique manifolds that can be solved with the Riemannian trust region (RTR) 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 comparisons with existing methods as well as applications on planar PGO in simultaneous localization and mapping (SLAM), which indicates that the proposed algorithm is capable of solving planar PGO certifiably, and is more efficient in numerical computation and more robust to measurement noises than existing state-of-the-art methods. The C++ code for CPL-Sync is available at https://github.com/fantaosha/CPL-Sync.

Network security is a general idea to ensure information transmission over PC and portable systems. Elliptic curve cryptosystems are nowadays widely used in public communication channels for network security. Their security relies upon the complexity of clarifying the elliptic curve discrete alogarithm issue. But, there are several general attacks in them. Elliptic bend number juggling is actualized over complex fields to enhance the security of elliptic curve cryptosystems. This paper starts with the qualities of elliptic curve cryptosystems and their security administrations. At that point we talk about limited field number-crunching and its properties, prime field number-crunching, twofold field math and complex number-crunching, and elliptic bend number-crunching over prime field and parallel field. This paper proposes how to execute the unpredictable number of math under prime field and double field utilizing java BigInteger class. also, we actualize elliptic bend math and elliptic bend cryptosystems utilizing complex numbers over prime field and double field and talk about our trials that got from the usage.

Arithmetic operations involving complex numbers are widely used in the signal processing functions in the physical layer of modern wireless and wireline communication systems, electronic instrumentation and control systems. With the ever increasing throughput requirements of such systems, the power consumption of the hardware realization is increasing beyond the allowed budget. Arithmetic circuits based on binary numeral system that have been optimized rigorously over the past few decades are currently being used for the computation involving complex numbers. In this paper, we present the potential of arithmetic circuits for complex number computations based on the Quater-imaginary (QI) base numeral system to reduce power consumption. We show that for a simple multiplier implementation in the QI base, the savings in power and area consumption could be up to 40% when synthesized in 28nm TSMC standard cell technology node.

DNA Computing, since its inception in 1994, has caught the eyes of researchers due to its massive parallelism and extremely high data density. These powers have given DNA Computer the ability to solve computationally "hard" problems using search over large search space, as well as a powerful data storage technique. This would be much more powerful and general purpose when its ability is increased to solve general-purpose problems too, which can be solved easily on a conventional computer. As an example, DNA Computing falls short in mathematical computations, such as performing simple arithmetic as well as complex number arithmetic etc. In this work, the major purpose is to increase the power of DNA Computing to solve such general purpose problems. This work attempts to perform complex number arithmetic, such as addition, subtraction, multiplication, and division of two complex numbers. True to the best of authors knowledge, no work has been performed to perform representation and arithmetic of complex numbers using DNA molecules. This work has several advantages, such as bio-molecular operations being used here are easy to implement, and operations can be carried out on arbitrarily large numbers.

This work presents a simple study of stochastic arithmetic complex number operators for addition and multiplication. Their usage is demonstrated by design of a sum of product circuit As the stochastic complex number operators need more control random streams than stochastic rational number operators, we optimized the number of random generators used in the real circuit. In the end our sum of product circuit contains two LFSRs thus we analyzed the impact of the choice of the seeds for the LFSRs on the quality of the calculated results. By using exhaustive search over the LFSR state space we were able to reduce the output RMSE by 34% in comparison to choice of the equally spaced seeds over the LFSR state space.

The paper examines mathematical representations of electrical magnitudes in a.c. Circuit theory. It gives an historical and technical perspective of the development of the power concept and its geometrical and algebraic interpretations. The paper criticises the existing mathematical model of electrical power for being an entanglement of two mutually inconsistent, algebras: 1) standard vector algebra (Gibbs-Heaviside) and 2) complex algebra. The paper examines the ubiquitous expressions for power: S = P + jQ a#x1E60; = a#x1E7E;I* The paper analyzes Steinmetz's symbolic method and exposes its inconsistencies. The paper proves that Steinmetz hypothesis, of a new and noncommutative algebra for power theory, represents a rediscovery of Grassmann-Clifford Algebra. The paper proposes a new didactic of power theory that should include Geometric Algebra.

Today, the use of computers is very common and unavoidable almost in every field. The enhancement of the circuit integration process from micrometer scale to nanometer scale escalates the great demand for less costly and reliable embedded solutions. Computer systems, embedded systems, or electronic systems are very generalized in many working sectors like automotive, biomedical, communication, aviation, and transport industries. Thus to perform their work reliably, it is important to have sophisticated arithmetic and logical unit to perform its task intelligently. Many efforts have been made from the beginning to implement addition, subtraction, multiplication, and division type basic arithmetic operations electronically to provide a high computation quality. The division is critical and vital for a sophisticated and reliable computational system among all basic arithmetic operations, but the implementation of the divider is difficult due to its inherent properties. We have to derive division operation in terms of successive subtraction or multiplication formation. Electronic implementation of divider takes multiple clock cycles. A complex number is represented as a vector equivalent of a real number and an imaginary number; thus, we have to perform the same operation twice with more working conditions in complex number arithmetics. This criticality of complex division implementations leads to the use of software-based complex division whenever required, but the demand for complex numbers in various essential engineering application such as earth fault distance protection, acoustics pulse reflectometry, astronomy, non-linear radio frequency measurements, digital signal processing demands for a formulation of sophisticated and efficient operational and implementational algorithms for the complex number representation and computation in the central processing unit of microprocessors or embedded, computer, electronic systems. In this article, we explained the basic stages of complex division by Baudhayan Triplet algorithm using novel state of the art USP-Awadhoot divider. In this article, we explained the workflow of the Baudhayan Triplet algorithm for implementing complex division operations efficiently. The Baudhayan Triplet algorithm provides an area-efficient way of implementing a complex divider.

In this paper, datasets using complex numbers are applied to the Feature Selective validation (FSV) algorithm directly making full use of the original algorithm. The Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) in the standard FSV algorithm support complex number by nature although the currently available FSV tools do not support it. The distance in complex number plane is more reasonable to measure the error between two complex numbers than the weighted errors by real and imaginary parts, or magnitude and phase parts, separately. A complete flow chart is designed and implemented to deal with the standard S-parameter data file in Touchstone format. To serve for most Signal Integrity/Power Integrity (SI/PI) design applications, the FSV is applied element by element on the whole S-matrix. The FSV results include Point-by-Point Mean Value Amplitude Difference Measure (ADM), Feature Difference Measure (FDM), Global Difference Measure (GDM), and Grade and Spread can be displayed in 3D Bar format on every element of the whole S-matrix. The Confidence Histograms of each S-Matrix element for ADM, FDM, and GDM can also be displayed. Several practical application examples are displayed including 12-port printed circuit board (PCB) via S-parameter models derived from electromagnetic simulation tool. The application examples show easy usage of the method to compare two Touchstone S-parameter models in practical SI/PI design.

Stochastic computing (SC) has been applied on the implementations of complex arithmetic functions. Complicated polynomial-based approximations lead to large hardware complexity of previous SC circuits for arithmetic functions. In this paper, a novel piecewise approximation method based on Taylor series expansion is proposed for complex arithmetic functions. Efficient implementations based on unipolar stochastic logic are presented for the monotonic functions. Furthermore, detailed optimization schemes are provided for the non-monotonic functions. Using NAND and AND gates as main computing elements, the optimized hardware architectures have extremely low complexity. The experimental results show that a broad range of arithmetic functions can be implemented with the proposed SC circuits, and the mean absolute errors can achieve the order of 1 × 10-3. Compared with the state-of-the-art works, the approximation precision for some typical functions can be increased by more than 8× with our method. In addition, the proposed circuits outperform the previous methods in hardware complexity and critical path significantly.

The paper presents algorithms for the generation of Residue Number System (RNS) triples with $SQ=2^{k}-1$ and quadruples with $SQ=2^{k}$ for some k. Triples and quadruples allow us to design efficient hardware implementations of non-modular operations in RNS such as division, sign detection, comparison of numbers, reverse conversion with using of a diagonal function from requiring division with the remainder by the diagonal module SQ. Division with a remainder in the general case is the most complex arithmetic operation in computer technology. However, the consideration of special cases can significantly simplify this operation and increase the efficiency of hardware implementation. We show that there are 5573 good RNS triples (2301 even and 2372 odd) with elements less than 10 000, as the values of SQ vary from $2^{5}-1$ to $2^{27}-1$ . In contrast, RNS quadruples with $SQ=2^{k}$ seem to be quite rare. Restricting our search to sums of the elements in a quadruple less than 4000 we find that exactly 31 such quadruples exist. Their values of SQ vary between 220 and 230 with always even exponent. We suggest the measure of RNS balance and find perfectly balanced RNS among triples according to this measure. We demonstrate the advantages of more balanced quadruples by means of hardware implementation.

This paper studies the integration of modular arithmetic, matrix algebra and complex number arithmetic. It describes implementation strategies of matrix transformations for residue matrices over complex plane that can be applied in the computer science fields which deal with cryptographic applications for more secure. Their mathematical properties over complex plane are analyzed to apply in cryptographic matrix transformation techniques. A new implementation approach to Hill cipher is proposed using the residue matrices over complex plane.

Using time-harmonic analysis, a method to identify load-dependent parameters of the induction motor (IM) equivalent circuit considering iron-loss is presented. Time-harmonic analysis is generally used for IM analysis due to its fast speed. However, iron-loss caused by harmonics cannot be calculated, since it regards that all parameters are sinusoidal. Thus, time-transient analysis is used to consider harmonics in general. A complete period of rotor flux density is needed for iron-loss calculation, and is inversely proportional to slip. Since IM is operated under very low slip condition, it takes immoderate time. Accordingly, novel analysis method for flux density harmonics for calculation of iron-loss. Based on the nature that result of time-harmonic analysis is in the form of complex number, method, which enables harmonics analysis with very short analysis range, is presented. Finally, the impact of considering iron-loss to equivalent circuit is investigated to demonstrate the accuracy improvement.

Recently, models of neural networks in the real domain have been extended into the high dimensional domain such as the complex number and quaternion domain, and several high-dimensional models have been proposed. These extensions are generalized by introducing Clifford algebra (geometric algebra). In this paper we extend conventional real-valued Hopfield-type neural networks into the octonion domain and discuss their dynamics. The octonions represent a particular extension of the quaternions which also represent a particular extension of the complex numbers and have 7 imaginary parts. They are non-commutative and non-associative on multiplication and do not belong to Clifford algebra due to the latter fact. With this in mind we propose four models of octonion Hopfield-type neural networks. We derive existence conditions of an energy function and construct energy function for each model.

Stochastic computing enables computationally complex arithmetic using binary numbers converted to stochastic bitstreams. A large number of applications have used stochastic computing due to its fault-tolerant nature. However, stochastic circuit synthesis presents a larger solution space when compared to classical logic synthesis. Previous methods synthesize stochastic circuits using a heuristic method. In this paper, a novel exact synthesis method using Boolean satisfiability (SAT) is proposed to obtain an optimal stochastic circuit represented by majority-inverter graphs (MIGs). The experimental results suggest that the proposed approach can achieve 21% area reduction, 4% delay improvement, with a 3% mean absolute error trade-off.

the use of homomorphic encryption to ensure computational security imposes additional overhead associated with encryption and decryption of data. One of the mechanisms that can be used to increase the data coding rate is the quadratic residue number system. The quadratic residue number system reduces the number of decimal multiplications by half when multiplying two complex numbers. However, the use of a quadratic residue number system imposes a number of restrictions related to the representation of a number. In this article, we propose a mechanism for choosing the parameters of a quadratic residue number system and develop an efficient data encoding algorithm

We derive the Kullback-Leibler average of von Mises distributions to fuse circular informations in multi-agent systems. Similar to the covariance intersection for Gaussian distributions, the derived fusion protocol does not require the independence among incoming distributions but maintains the estimation consistency, if those von Mises distributions represent estimates. Therefore, this fusion protocol is especially useful in the distributed estimation problem to avoid the overconfidence problem. For example, together with von Mises filters, the derived fusion protocol can estimate the dynamic circular term in a distributed manner. In addition, we apply this fusion protocol to determine the consensus of von Mises distributions over a network. Since the fusion protocol can be easily achieved by calculating the weighted average of the associated complex numbers, the corresponding convergent conditions of the consensus algorithm are then elegantly determined.

In this paper, we focus on the extensively utilized algorithm for split radix FFT. It proposes two the 6mpoint split radix fast Fourier transform (SRFFT), where the complex numbers are represented in a special basis (1, μ) and μ is the complex cube root of unity. Two SRFFTs, termed radix-2/6 and radix-3/6, are proposed and both algorithms are based on radix 2 and radix 3 FFT. Furthermore, we utilize them to design appropriate algorithm structure for length 6m• In addition, fast multiplication in (1, μ) is also proposed. Compared with prior results, the proposed SRFFT requires fewer real multiplications. To our knowledge, this is the first SRFFTs over the basis (1, μ) and this work achieves better specifications for area use and delay. Meanwhile, the occupied resources are approximately same. Moreover, the performance of different FFT length is analyzed.

InP heterojunction bipolar transistor (HBT) small signal modeling technique based on artificial neural network(ANN) is proposed in this paper. Two ANN models with different outputs form are given and compared. In the frequency range of 2-110 GHz, good agreements between the measured and model-calculated data can be achieved to demonstrate that the ANN model outputs with complex numbers form is more accurate than amplitude-phase form.

We develop a novel deep learning architecture for naturally complex valued data, which are often subject to complex scaling ambiguity. We treat each sample as a field in the space of complex numbers. With the polar form of a complex number, the general group that acts on this space is the product of planar rotation and non-zero scaling. This perspective allows us to develop not only a novel convoluation operator using weighted Frechet mean (wFM) on a Riemannian manifold, but also to a novel fully connected layer operator using the distance to the wFM, with natural equivariant properties to non-zero scaling and planar rotations for the former and invariance properites for the latter. We demonstrate our method on widely used SAR dataset MSTAR and RadioML dataset. On MSTAR data, without any preprocessing, our network can achieve 98% classification accuracy on this highly imbalanced dataset using only ~44,000 parameters, as opposed to 94% accuracy with more than 500,000 parameters with a baseline real-valued network on the two-channel real representation of the complex valued data. On RadioML data, we got comparable classification accuracy with the baseline with only using 10% of the parameters as the baseline model.

In Land Mobile Satellite Communication system, MIMO (Multiple Input Multiple Output) Polarization Diversity along with spatial and temporal diversity is studied and analyzed. In general diversity, schemes offer better link reliability. Therefore, to achieve this, a combination of space, time and polarization diversity schemes is proposed. In this work, Orthogonal Space- Time Polarization Block Codes is generalized using quaternions which is a generalization of an orthogonal complex number. To decode OSTPBC, Sphere Decoding algorithm is used. Sphere Decoding is an effective technique to find the solution of Maximum Likelihood Decoding. But computational complexity is still high in the traditional Sphere Decoding. To further reduce the complexity of SD, Contraction Factor scheme is introduced which will speed up the contraction in each layer of the sphere radius and reduce the decoding complexity.

this paper presents a novel power flow method based on the Modified Backward-Forward Sweep algorithm of radial Ship's Shore Power Supplying Networks without involving node number and special rules. This method makes full use of the structure characteristics of the radiation distribution network; through the establishment of loop chain and a directional node table, by using breadth-first to search node sequence of generated power flow calculation, using the linear trend equation iteration to get each node voltage, it does not require complex number of nodes and branch, does not need to form the node admittance matrix and triangular matrix, with simple input data format, rapid computing speed, less memory and good convergence. Finally, the model of power distribution system should be set up on MATLAB platform and realized by fast plural matrix operation function, a practical calculation example proved the effectiveness of the proposed algorithm.

In this paper we extend the successive concave sparsity approximation (SCSA) algorithm to complex numbers in order to make it available for applications benefiting of it e.g. radar applications. SCSA is an attractive reconstruction algorithm for compressive sensing (CS) problems due to its high reconstruction performance (superior to $\ell_{1}$ algorithms) and in particular since it does not require any “hard” parameters, unlike all hard thresholding (HT) algorithms, which may be unknown in radar applications. We call this extended version complex successive concave sparsity approximation (CSCSA) and evaluate its performance by use of phase transition plots for random and discrete Fourier transform (DFT) sensing operators and further compare it with the two well known algorithms normalized iterative hard thresholding (NIHT) and fast iterative shrinkage-thresholding algorithm (FISTA), where we especially show how the reconstruction performance of NIHT declines in case of wrongly assumed parameters.

Stochastic logic performs computation on data represented by random bit-streams. The representation allows complex arithmetic to be performed with very simple logic, but it suffers from high latency and poor precision. Furthermore, the results are always somewhat inaccurate due to random fluctuations. In this paper, we show that randomness is not a requirement for this computational paradigm. If properly structured, the same arithmetical constructs can operate on deterministic bit-streams, with the data represented uniformly by the fraction of 1’s versus 0’s. This paper presents three approaches for the computation: relatively prime stream lengths, rotation, and clock division. Unlike stochastic methods, all three of our deterministic methods produce completely accurate results. The cost of generating the deterministic streams is a small fraction of the cost of generating streams from random/pseudorandom sources. Most importantly, the latency is reduced by a factor of $({1}/{2^{n}})$ , where $n$ is the equivalent number of bits of precision. When computing in unary, the bit-stream length increases with each level of logic. This is an inevitable consequence of the representation, but it can result in unmanageable bit-stream lengths. We discuss two methods for maintaining constant bit-streams lengths via approximations, based on low-discrepancy sequences. These methods provide the best accuracy and area $\times $ delay product. They are fast-converging and therefore offer progressive precision.

In stochastic computing, values are represented as sequences of random bits and arithmetic computations are computed on the bit streams. Since bit-wise operations are performed on random bit streams, stochastic computing offers low-cost error-tolerant architectures for its hardware implementations. In stochastic computing, complex arithmetic operations can be computed using linear finite state machines (FSMs). However, the synthesis of a linear FSM for a given target function is nontrivial. In this paper, we exploit linear regression and demonstrate a general approach to synthesize linear FSMs for stochastic computations. We show that our approach outperforms traditional numerical synthesis methods in terms of mean-squared error. We also demonstrate that fault-tolerance of FSMs synthesized using linear regression can be improved by injecting noise during the synthesis phase, allowing the synthesized functions to tolerate up to 35% of random bit flips.

This paper aims to provide a useful framework for extreme learning machines (ELMs) on Cayley-Dickson algebras. Cayley-Dickson algebras, which include complex numbers, quaternions, and octonions as particular instances, are hyper-complex algebras defined using a recursive procedure. Firstly, we review some basic concepts on Cayley-Dickson algebras and formulate Cayley-Dickson matrix product using real-valued linear algebra. Then, we propose the Cayley-Dickson ELMs and derive their learning using Cayley-Dickson least squares problem. Lastly, we compare the performance of real-valued and four-dimensional Cayley-Dickson ELM models, including quaternion-valued ELM, in an experiment on color image auto-encoding using the well-known CIFAR dataset.

The techniques used in deep learning for channel estimation are generally model-centric. These models have changed significantly over the years with each iteration yielding a better estimator than the last. Fundamentally, channel estimation works by exploiting correlations in an array of complex numbers, in particular the channel gains for a fading channel. In this paper, we study the effects of the spatial arrangement of channel response and input data, on channel estimation. With the right spatial arrangement, we improved the performance of our convolutional neural network that was used for estimation. Additionally, we optimized the training procedure simultaneously. We experimentally validate the importance of spatial arrangement of data in obtaining an accurate deep learning model for the channel.

A major hurdle in machine learning is scalability to massive datasets. One approach to overcoming this is to distribute the computational tasks among several workers. \textit{Gradient coding} has been recently proposed in distributed optimization to compute the gradient of an objective function using multiple, possibly unreliable, worker nodes. By designing distributed coded schemes, gradient coded computations can be made resilient to \textit{stragglers}, nodes with longer response time comparing to other nodes in a distributed network. Most such schemes rely on operations over the real or complex numbers and are inherently numerically unstable. We present a binary scheme which avoids such operations, thereby enabling numerically stable distributed computation of the gradient. Also, some restricting assumptions in prior work are dropped, and a more efficient decoding is given., Conference

The Half-Unit-Biased (HUB) format has interesting advantages for implementing floating-point arithmetic which has been proved for the four basic arithmetic operations as well as square root. Nevertheless, although Floating-point Fused Multiply-add (FMA) operation (AxB + C) is one of the most important and complex arithmetic instructions in modern processors, FMA operation for HUB numbers has not been confronted yet. In this paper, we present a design to deal with this operation under HUB format. The key points to turn the conventional FMA architecture into a HUB unit are explained. Comparing the ASIC implementation of a HUB FMA unit with the conventional one, the former reduces the required area and power up to 38% and 35%, respectively, for single-precision. For BFloat16, the HUB FMA increases the speed a 15%, and even then, reduces the area and power by 26% and 12%, respectively.