Search

This investigation concerns the finding of complex eigenvalues of axisymmetric oscillations in acoustical and electromagnetic open spherical cavities. The openings are circular holes or slits. The characteristic equations for the complex eigenvalues are obtained by the Method of Regularization. This approach reduces the problem to finding the complex roots of the determinants of the truncated systems of the algebraic equations. The fast convergence of solution of the truncated system to the exact solution of the infinite system provides any desired accuracy of computations. As examples, calculations of the complex eigenvalues of the sound-hard spherical shell with two symmetric circular holes (spherical "barrel") and of the complex eigenvalues of the TE-modes of the perfectly electrically conducting spherical shell with a single circular aperture are presented.

Quadratic Residue Number System (QRNS) arithmetic became very popular in the early 1980’s. QRNS allows one to compute the multiplication of two complex numbers with only two real multiplies as compared to the four multiplies required in normal complex multiplication. This is particularly useful in the computation of Fast Fourier Transforms (FFT), Complex Number Theoretic Transforms (CNTT) and digital filters with complex coefficients. However, QRNS requires the RNS moduli to be prime numbers of the form 4k + 1 or composite numbers with prime factors of that form. This restriction eliminates some of the most desirable moduli for RNS number systems. In 1986, Graham Jullien and William Miller introduced what they called Modified Quadratic Number Systems (MQRNS) which allowed any set of valid RNS moduli to be used in an MQRNS system in which two complex numbers could be multiplied with three real multiplications: one more that QRNS, but one less than normal complex multiplication. This MQRNS system lent itself well to applications in FFT, CNTT and digital filters with complex coefficients including complex heterodyne tunable filters.

An alternative stabilization procedure based on a Lyapunov function is presented for PVTOL aircraft. The rotational dynamics of the aircraft is represented with complex numbers, and the controller design procedure is given based on this model. Almost global asymptotic stability of the desired position for the aircraft is shown with a Lyapunov function and by the utilization of LaSalle's invariant set principle. The performance of the developed controller is tested via numerical simulations and the results are presented.

In this paper, we present a frequency domain neural network for image super-resolution. The network employs the convolution theorem so as to cast convolutions in the spatial domain as products in the frequency domain. Moreover, the non-linearity in deep nets, often achieved by a rectifier unit, is here cast as a convolution in the frequency domain. This not only yields a network which is very computationally efficient at testing but also one whose parameters can all be learnt accordingly. The network can be trained using back propagation and is devoid of complex numbers due to the use of the Hartley transform as an alternative to the Fourier transform. Moreover, the network is potentially applicable to other problems elsewhere in computer vision and image processing which are often cast in the frequency domain. We show results on super-resolution and compare against alternatives elsewhere in the literature. In our experiments, our network is one to two orders of magnitude faster than the alternatives with an imperceptible loss of performance.

Global Navigation Satellite System Reflectometry can be used to derive information about the composition or the properties of ground surfaces, by analyzing GPS signals reflected by the ground. The received power of the signal is proportional to the modulus of the perpendicular and parallel polarization Fresnel coefficients. These coefficients depend on the incidence angle $\theta$ , and on the ground's dielectric constant $\varepsilon$ , which provides information on the composition and properties of the ground. Thus, one has to solve the inverse problem, consisting of finding the value of $\varepsilon$ from the known value of $\theta$ and the measured values of the Fresnel reflection coefficients. In general, $\varepsilon$ is a complex number; in some cases (e.g., for non-dispersive soils), the imaginary part of $\varepsilon$ can be neglected, and a real value of $\varepsilon$ is sought. We discuss the mathematical solvability of a particular type of inverse Fresnel problem for real unknown $\varepsilon$ .

Chaotic secure communication has several advantages over conventional crypto-based communication due to its simplicity, diversity and robustness. In this paper, we first present difference function projective synchronization (DFPS) for complex chaotic systems. It extends the category of synchronization and enables the development of applications based on complex chaotic signals instead of real signals, thereby increasing both the security and robustness for chaotic communications. We derive the control law using the active control theory based on Lyapunov stability analysis. A novel communication scheme is further designed based on DFPS. Its main idea is chaotic masking in essence, but the transmitted signal is the derivative of the sum of the information signal and chaotic signal. As the difference functions are complex and arbitrary, they are more unpredictable than real scaling functions, and the operations of complex numbers are complicated, thus greatly reducing the possibility that an interceptor extracts the information from the transmitted signal. The communication system can transmit both analog signal and digital symbols with fast transmission speed and high security, especially with low bit-error rate and strong robustness to noise for digital symbols. The main ideas of the proposed method are veri?ed by the example of coupled complex Lorenz system used as both transmitter and receiver.

The paper describes an application that allows to visualize four-dimensional graphs of a complex variable function. Three coordinates of the graph are Cartesian, and fourth is the parametric coordinate. Graphs of complex polynomials are considered in detail. We demonstrates the well-known basic theorem of algebra about the number of polynomial roots. The study of graphs of complex polynomials made it possible to construct an exact geometric algorithm for finding the real and complex roots of a polynomial on the same plane. The algorithm assumes the construction of a graph of the main polynomial and graphs of two auxiliary functions. The application of this method is considered in detail for a cubic polynomial. In this case the method has exceptional features in comparison with polynomials of other degrees. Taking into account the well-known expressions for the roots of polynomials of order 3 and 4, the auxiliary graphs of the method have exact formulas for polynomials with order from 3 to 10.

Theory of complex waves (see e.g. [1] and references therein) accumulated a big volume of results. However, its biggest drawback remains which brakes significantly the development of the theory: absence of rigorous proofs of the existence of complex waves. In this respect, studies of complex waves in open metal-dielectric waveguides has been essentially enhanced when two new families of symmetric (angle-independent) complex waves have been identified [2], [3] in a dielectric rod (DR) and Goubau line (GL) with homogeneous dielectric cover: surface complex waves occurring as perturbations of real surface waves caused by the presence of lossy dielectric, and pure complex waves which have no counterpart in the set of real waves. Existence of the latter family of complex waves has been shown using a special technique and the proof [3] of the existence of an infinite set of complex roots of the dispersion equation (DE) describing waves in DR and GL.

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.

Space-time processing of signals allows improving a reception characteristics and extraction of information from signals in the presence of interference. When synthesizing and realizing the algorithms of space-time processing, the description of signals is in the form of complex numbers with further use of complex arithmetic. However, the practical implementation of complex arithmetic requires significant computational costs. So, for example, when realizing multiplication of two complex numbers in hardware, three multiplications of real numbers are required. Therefore, in order to reduce computational costs when implementing algorithms of space-time processing, it is desirable to obtain algorithms in real numbers. This desire is justified also because the real signal is valid, and nothing indicates that for its processing it is necessary to move on to some other description different from the description with real numbers. If for the received real signal to use the best methods of synthesis of processing algorithms, the synthesized algorithm will operate only with real numbers and give the best characteristics for the chosen criterion of optimality. This article is devoted to the solution of the described problem, namely: the synthesis of the algorithm for estimating signal parameters with space-time processing based on antenna array, and with calculations in real numbers.

With increased interests of neural networks, hardware implementations of neural networks have been investigated. Researchers pursue low hardware cost by using different technologies such as stochastic computing and quantization. For example, the quantization is able to reduce total number of trained weights and results in low hardware cost. Stochastic computing aims to lower hardware costs substantially by using simple gates instead of complex arithmetic operations. In this paper, we propose a new stochastic multiplier with shifted unary code adders (SUC-Adder) for quantized neural networks. The new design uses the characteristic of quantized weights and tremendously reduces the hardware cost of neural networks. Experimental results indicate that our stochastic design achieves about 10x energy reduction compared to its counterpart binary implementation while maintaining slightly higher recognition error rates than the binary implementation.

This paper proposes a new approach of representing decimal number system using Vinculum number representation. Vinculum number system consists of numerals 0,1,2,3,4,5 same as decimal number system and 6,7,8 and 9 are represented using negative numbers less than or equal to 5. Therefore Vinculum number system consists a set of numbers as {0, 1, 2, 3, 4, 5, −4, −3, −2, −1}. Hence complexity of higher order numerals like 6,7,8 and 9 are converted into less complex numbers. Vinculum is the Vedic method of representing decimal number system. Decimal numbers are representing in Binary Coded Decimal numbers for getting compatibility with Computer systems. Similarly we have used 2’s complement number system for representing Vinculum numbers. This helped in representing signed Vinculum numbers. A unique set of tuples are represented in Vinculum number system which are suitable for any decimal arithmetic operation.

In this work, we propose using Complex Nonseparable Oversampled Lapped Transform (CNSOLT) as a sparse representation method for denoising of complex radar image. CNSOLT is an image transform which is an extended version of the existing Nonseparable Oversampled Lapped Transform to the complex number field, and simultaneously satisfies non-separabe, redundant, linear-phase, and compact-support property. The lattice structure is able to constitute a Parseval tight frame with redundancy of rational numbers, and it is possible to design a dictionary with directional atomic images. In this work, we propose an image restoration method combining CNSOLT and an iterative shrinkage/thresholding algorithm (ISTA) modified for complex-valued signals. The performance evaluation is conducted through denoising of complex radar images. We also compare the performance with that of the complex K-SVD as a reference method.

In this paper, we propose an improved image encryption algorithm based on double random phase encoding (DRPE). Our contribution is the design of a new algorithm which uses Discrete Cosine Transform (DCT) to replace Discrete Fourier Transform (DFT) so as to avoid operations on complex numbers. We use a logistic map to generate random matrices instead of random phase masks in the traditional DRPE so as to decrease the number of secret keys. We tested the algorithm using five types of attacks. The results indicate that the improved algorithm overcomes weaknesses of traditional scrambling methods and it has shown strong resilience against statistical attacks. An advantage of this algorithm is that it is convenient to verify whether the encrypted images have been tampered with.

This paper proposes a reconfigurable high performance multiplier (RHPM) based on multi-granularity design and parallel acceleration. Capable of supporting multiple precisions for different processing requirements, the RHPM can perform one 32×32, two 16×16, or four 8×8 bit unsigned/signed multiplication, or one 16×16, or two 8×8 bit complex number multiplication. The structures of the partial product generator and the partial product accumulator are improved in the paper, so as to reuse most of the hardware resources. Compression can be completed automatically by means of recording the validity of every bit in the partial product array which accelerates the computation dramatically. The RHPM is implemented with TSMC 28nm technology, exhibiting a 0.68s of the critical path delay, while consuming only 0.6281mW in power. Results show its significant superiority in terms of performance and power efficiency compared with our previous work or other similar products.

Symbolic algebra is a promising approach to verify large and complex arithmetic circuits. Existing algebraic-based verification methods generate a remainder to indicate buggy implementation. The remainder is beneficial for debugging of the faulty implementation since it can be used for automated test generation, bug localization, and bug correction. However, existing equivalence checking approaches are not scalable and lead to explosion in size of the remainder when the design is faulty. To make the matters worse, the location of the bug can also lead to the explosion in the number of remainder terms. In this paper, we propose an incremental equivalence checking method to address the scalability challenges by solving the verification problem in the increasing order of design's input complexity. Our proposed approach makes two important contributions. It is able to generate smaller and compact remainders for large designs. Our proposed incremental debugging is capable of localizing and correcting hard-to-detect bugs irrespective of their location in the design. Experimental results demonstrate that our approach can efficiently debug most difficult bugs in large arithmetic circuits when the state-of-the-art methods fail.

To have more effective fault ride through operation for the power conditioner of the photovoltaic system, the detection of the imbalanced condition is important factor. In this paper, new detection method by using the complex number algebra in real time control system is shown. Very fast detection of the imbalanced condition is verified by simulations and experiments.

Traditional method to estimate formant based on LPC calculates the roots from the LP polynomial coefficients, and then gets formant feature from conjugate complex roots. There is a problem of formant combination when calculating the formants. An improved formant extraction method based on LPC has been proposed in this paper. First pairs of complex-conjugated roots are calculated by conventional method, and then getting corresponding angle of poles by maxima method in LPC spectrum. In this part, we use the maxima method to search local maximum points and avoid getting all angle data, this process accelerates the algorithm speed. Finally an interaction factor is used to consider pole interacion effect and modified the formant bandwidths to make the spectral energy of formant polynomial to be equal to the spectral before deleting spurious roots. The results show that this method can effectively solve combination of formant, and this new formant extraction method gets a higher recoginition rate in classification of singer speech feature than traditional method.

The need for placing more number of transistors in the core area by shrinking of electronic devices has brought out the necessity of scaling down the Field Effect Transistor (FET) devices. According to the current equation of existing MOSFET device the parameters which can increase the device current is probably channel width and the mobility of electrons and holes. The parameter which can help in improving the device performance will be the mobility of electrons and holes. Hence the necessity of replacing the channel material has become a leading research work in nano electronics. The allotrope of carbon atom — Graphene has the most required mobility range which can help in scaling down the currently available MOSFET devices. Some of the parameters which can be improved due to the replacement of channel material is the device drain current, reduction in the amount of impurity added to obtain the required drain current. A scaled down 15nm Dual Gate Graphene Field Effect Transistor is modeled in this work. As the device is scaled down to a such a narrow range the necessity of investigating the device parameters in a nano scale has been carried out using a novel approach of segmenting the channel. An additional focus in this paper is to employ a computationally efficient method for including the Quantum Capacitance effect. The most common methodology for obtaining the values for quantum capacitance is through solving the equations simultaneously as proposed in [1],[2]. The parameters of interest like quantum capacitance and channel potential are assumed initially and the equations are solved until the assumed parameter value and the obtained value are equal. This not only increases the simulation time but also ends up in approximate result values which are complex. The complex solutions cannot be separated into a real and imaginary value in verilog-a as well as in verilog-ams language as given in [3],[4], to enable circuit compatibility. Moreover the approximate results obtained through the existing methodologies are of huge margin for a 15nm scaled down device. This makes it difficult to implement dual gate GFET in cadence — virtuoso environment. Hence the only way to implement the equations in a circuit level of simulation is to bring out the results as real value and not a complex number. In order to avoid complex solutions the only way is to convert the equations into a polynomial. Hence a novel method which uses a fourth order polynomial known as quartic equation in which the quantum capacitance is derived from the channel potential equation is proposed. The methodology adopted in this work is quartic equation whose results are real valued solutions and hence overcomes the complex number solutions obtained so far. This makes it possible for the model to be developed in verilog-a and incorporated as circuit model. The proposed fourth order polynomial equation can be used to calculate quantum capacitance for any device which obeys the ballistic transport. Hence the device structure proposed in this work is limited to ballistic structure. Ballistic transport of electrons can be obtained when the length and width of the channel is lesser than the mean free path. A N-type impurity of about 2.63∗1011cm−2 (calculated, not presented here) is added in order to achieve a mobility of 2497 cm/Vs. Impurities can be further added but it will lead to increased scattering of electrons which will affect the ballistic structure. Hence further addition of impurity to enhance the mobility is stopped. In this model the carriers' Fermi velocity of 106 m/s, tox of 1.5nm (top and bottom), length (L) of 15nm, a width (W) of 150 nm, Cox(bottom gate) of 23.024×10-3 F/m are used and the determined value of mobility is found to be better as reported in Table 1.

The paper deals with signal analysis in frequency domain oriented on improving its effectiveness. New approach to the problem is proposed. This leads to obtaining better results than the ones being well known from standard techniques based on DFT and FFT transforms. Our method uses adaptive synthesis aided by evolutionary algorithm. Main attention is focused on signal analysis with short duration time. Just under such conditions our approach is especially attractive and exhibits superiority over the other methods. Complex-valued studies have been considered. Using complex numbers, the obtained results turned out to be much more effective than in case of real-valued ones. The conditions necessary to obtain right results of the analysis have been discussed in [13-16]. From results of many studies performed recently it arises, how important in our analysis is satisfying these conditions and how to choose the analysis parameters. Even though the obtained study outcomes not always are as good as expected, which concerns mainly the analysis precision, diagnostic values of the realized analysis can be valuable. In this paper we show, how to modify the algorithm parameters in order to improve precision and how to decrease the used number of samples.

Existence of complex TM and TE waves in a dielectric waveguide of circular cross section and a Goubau line is proved by analyzing functional properties of the dispersion equations (DEs) using the theory of functions of several complex variables and validating the existence of complex roots of DE. The method proceeds from the analysis of general setting involving hybrid nonsymmetric azimuthally dependent real and complex waves.

The method for finding inverse element by the module is based on the stepwise addition of residue is developed in this paper. This method allows avoiding performing the complex arithmetic operations and implementing calculations on numbers much lower bit compared to the classical approach based on the Euclid's algorithm and its consequences. Analytical expressions of time complexity characteristics are obtained and their graphic depending is built. Software and hardware modules of specified methods on the basis of environment of the development AldecActive-HDL 9.1 are implemented and studies of their temporal characteristics that point to the advantages of the proposed method are conducted.

Implementing complex arithmetic routines with Single Instruction Multiple Data (SIMD) instructions requires the use of instructions that are usually not found in their real arithmetic counter-parts. These instructions, such as shuffles and addsub, are often bottlenecks for many complex arithmetic kernels as modern architectures usually can perform more real arithmetic operations than execute instructions for complex arithmetic. In this work, we focus on using a variety of data layouts (mixed format) for storing complex numbers at different stages of the computation so as to limit the use of these instructions. Using complex matrix multiplication and Fast Fourier Transforms (FFTs) as our examples, we demonstrate that performance improvements of up to 2× can be attained with mixed format within the computational routines. We also described how existing algorithms can be easily modified to implement the mixed format complex layout.

The phase correction algorithm of dual channel mono-pulse radar is studied in this paper. The angle error voltage of the tracking receiver is represented by a complex number. The expression of the relation between the corrected phase and the corrected slope and the error voltage is derived, and a fast phase detection method is proposed. phase detection rate of this method is significantly faster than the automatic phase calibration of the equipment itself for the case of without the boresight tower. Moreover, the method presented here has higher accuracy.

This paper presents a new approach to approximate various complex arithmetic functions for applications in wideband RF transceivers by using stochastic computing based on the piecewise linear (PWL) approximation. The optimized design parameters for PWL approximation are chosen by an optimization algorithm targeting the best tradeoff between hardware complexity and approximation accuracy. Then, this approximation is implemented by stochastic logic circuits. The proposed approach outperforms previous works based on Maclaurin expansions, Bernstein polynomial and finite-state-machine implementations while achieve similar computation accuracy.

Parametrisation of the shape of deformable objects is of paramount importance in many computer vision applications. Many state-of-the-art statistical deformable models perform landmark localisation via optimising an objective function over a certain parametrisation of the object's shape. Arguably, the most popular way is by employing statistical techniques. The points of shape samples of an object lie in a 2D lattice and they are normally represented by concatenating the 2D coordinates into a vector. As the 2D coordinates can be naturally represented as a complex number, in this paper we study statistical complex number representations of an object's shape. In particular, we show that the real representation provides a similar statistical prior as the widely linear complex model, while the circular complex representation results in a much more condensed encoding.

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.

In this work a general algorithm is presented for the efficient computation of the complex roots of determinantal equations arising in microwave applications. The method is based on a two-dimensional minimization problem, which is efficiently solved by eliminating complex points which do not satisfy certain conditions. The method does not need initial points, and the complete spectrum of the system matrix is not computed on the selected grid. The efficiency and applicability of the algorithm is demonstrated to a variety of microwave applications, including the calculation of complex propagation constants in multilayer waveguides, complex propagation constants of surface waves in microstrip antennas, or complex eigenfrequencies in spherical resonators.

We present the design and implementation of a parallel and fully algebraic preconditioner based on an approximate sparse factorization using low-rank matrix compression. The sparse factorization uses a multifrontal algorithm with fill-in occurring in dense frontal matrices. These frontal matrices are approximated as hierarchically semi-separable matrices, which are constructed using a randomized sampling technique. The resulting preconditioner has (close to) optimal complexity in terms of flops and memory usage for many discretized partial differential equations. We illustrate the robustness and performance of this new preconditioner for a number of unstructured grid problems. Initial results show that the rank-structured preconditioner could be a viable alternative to algebraic multigrid and incomplete LU, for instance. Our implementation uses MPI and OpenMP and supports real and complex arithmetic and 32 and 64 bit integers. We present a detailed performance analysis. The code is released as the STRUMPACK library with a BSD license, and a PETSc interface is available to allow for easy integration in existing applications.

The spherical-multipole expansion of an inhomogeneous electromagnetic plane wave is derived by extending the spherical source coordinates v to a complex number. The analytical results are successfully validated by comparing the numerical evaluation of the multipole expansion with the corresponding closed-form results. The paper also contains a first result for an inhomogeneous plane electromagnetic wave diffracted by a perfectly conducting semiinfinite circular cone.

It is important to know the power system condition by calculating power flows. As an alternative to the Newton-Raphson method which requires much calculation and complex program, this paper proposes a highly reliable complex power flow calculation method that requires not Jacobian matrix but the I=YE equation. To consider the proposed method, two approaches were tested. Simulations are carried out to determine which approach is suitable for the proposed method and a reliable and fast power flow calculation technique was developed. In addition, the proposed method is applied to circuit calculations for transient stability calculations, which use the I=YE equation. Simulations were run to verify the validity of the proposed method.

This paper presents a BCD Multiplier that operates on Vedic Mathematics called Vedic sutras. It uses a method called Vinculum which converts higher complex numbers into its simplest form. In Decimal number system the numerials 6,7,8,9 are called high complex numbers and numbers consisting of 6,7,8,9 are converted into 4,3,2,1 there by total number or any number is between 1 to 5 digits only for any arithmetic operation. This feature reduces Carry generations and Carry propagations there by performance parameter like delay reduces especially in adders and multiplier structures. We choosen an arithmetic operation multiplication and it is compared with Conventional Multiplier [1] [2] [3] and Vedic Multiplier[5] and it has been observed that improvement in speed is 83.5% in case of conventional multiplier and 47.8% in case of vedic multiplier which is suitable for High Speed Applications. The Architecture is implemented using Xilinx Vertex 4 FPGA and the same is done using Cadence Digital Encounter Tools of TSMC180nm Technology. The results indicate that the proposed BCD multipliers is very efficient in terms of speed when compared to decimal multipliers implemented with direct manipulation of BCD numbers.

Quaternions, a four dimensional extension of complex numbers, are increasingly being used in various signal processing applications. At the same time, there are few probabilistic representations for capturing such random variables. In this work we present a new copula-based distribution over quaternions. The distribution extends the classical Gaussian and independent distributions, while maintaining the circular symmetry conditions required by such applications, namely properness. Importantly, the new distribution allows for flexible control of both the marginals and the correlations. We develop methods for estimating the parameters of the new distribution, and demonstrate its potential advantages via numerical experiments.

How to improve consensus conditions for double-integrator multi-agent systems is investigated. First, a consensus algorithm is presented and the corresponding convergence condition is described. However, the given convergence condition is conservative such that consensus is still obtained when the parameters of the consensus algorithm violate the given convergence condition. Second, to cope with this issue, a less conservative convergence condition is proposed by using complex number and matrix properties. Finally, simulation results illustrate the effectiveness of the convergence condition through two examples.

A numerical approach to calculate multivariate transcendental equations in complex number domain that can be applied to solve dispersion equation is presented here. The mathematical derivations of the convergence of the moduli of the equations around null point are presented strictly, when the transcendental equation is univariate. Similar process also applies to the case when the equations are multivariate. For multivariate equations, the forms of scanning elements are chosen according to the numbers of the variables and the dimensions of the solution. To validate the proposed approach, we calculate the dispersion cures of wave propagation in an infinite piezoelectric plate. As a result, the three-dimensional dispersion curves of complex wave numbers and real frequencies are obtained correctly.

In Dsp applications complex arithmetic operations are performed. In order to improve the performances of Dsp applications we focus on optomizing the design of fused add multiply operator. The proposed method implements a newly designed recoding technique. This technique implements recoding of the multiplier in its Modified Booth sum (S-MB) form. The proposed technique results in reductions in terms of critical delay, hardware complexity and power consumption of the AM unit.

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.

In order to reduce data dimensions, autoencoders with neural networks have been proposed by Hinton et al. Autoencoders are composed of input, one hidden, and output layers, which tune weights and biases by a back propagation to minimize an error between inputs and outputs. The learned weights have input features, and can be applied to pretrainings of deep neural networks. However, these autoencoders have been developed for real-valued neural networks. In this study, we propose complex and quaternion autoencoders for complex and quaternion neural networks, respectively. In the complex-valued autoencoder, inputs, weights, biases and outputs of the real-valued autoencoder are extended to complex numbers. In the quaternion autoencoder, these parameters are extended to quaternion numbers. We show the learning abilities of the proposed methods using handwritten digit images. The results show that the proposed methods can recognize the images as the real-valued methods.

A Gaussian integer is a complex number whose real and imaginary parts are both integers. This paper proposed a unified construction of perfect Gaussian integer sequences based on cyclic difference sets. It turns out that this construction produces an abundance of perfect Gaussian integer sequences. The proposed construction includes all the sequences recently given by Lee et. al as special cases, and many new families of Gaussian integer sequences. To illustrate, two classes of perfect Gaussian integer sequences defined from Kasami-Welch functions and Helleseth-Gong functions are given.

A positional numeration system is given by a base and by a set of digits. The base is a real or complex number aamp;#x03B2; such that |aamp;#x03B2;| > 1, and the digit set A is a finite set of real or complex digits (including 0). In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that aamp;#x03B2; is any real or complex number, and digits are real or complex. We show that if (aamp;#x03B2;, A) satisfies the so-called (OL) Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base aamp;#x03B2; and alphabet A of contiguous integers, the system (aamp;#x03B2;, A) has the (OL) Property if #A > |aamp;#x03B2;|. Provided that addition and subtraction are realizable in parallel in the system (aamp;#x03B2;, A), our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base aamp;#x03B2; = 3+aamp;#x221A;5/2 with alphabet A = {-1, 0, 1}, base aamp;#x03B2; = 2i with alphabet A = {-2, -1, 0, 1, 2} (redundant Knuth numeration system), and base aamp;#x03B2; = -3/2 + aamp;#x0269; aamp;#x221A;3/2 = -1 + aamp;#x03C9;, where aamp;#x03C9; = exp 2iaamp;#x03C0;/3, with alphabet A = {0, ±1, ±aamp;#x03C9;, ±aamp;#x03C9;2} (redundant Eisenstein numeration system).

On the basis of the coherent concept a decomposition of the polarimetric scattering matrix of a radar remote sensing object is performed. With the usage of the complex numbers mathematics a scattering operator that compactly describes transformation of the backscattered signal polarization is received. The expressions for invariant polarization transfer functions that separately describe transformations of ellipticity and orientation of the sensing signal at its backscattering by a remote sensing object are derived. On the basis of experimental data it is shown that the suggested polarization transfer functions may be used for classification and recognition of the remote sensing objects.

PFC is known to be effective and simple for a large number of industrial scenarios. However, there are some cases where the default PFC algorithm is less effective and needs significant insight or modification. This paper proposes a modification to the conventional algorithm to improve the ease of tuning for systems with significant under-damped modes. The proposed approach is almost identical to the pole placement PFC for processes with real poles, but exploits complex number algebra and linear combinations in order to deliver guarantees of stability and performance. The potential advantages are demonstrated with numerical examples.

Complex arithmetic operations dominate Digital Signal Processing (DSP) applications heavily degrading their performance. Targeting to accelerate the Residue Number Systems-based DSP applications, we optimize the design of the Add-Multiply (AM) operation with modulo 2n + 1 diminished-1 operands by incorporating the direct recoding of the sum of two numbers in its Modified Booth form. Compared to the conventional allocation of an adder and a subsequent multiplier, the proposed fused AM design yields delay, area and power gains.

Digital Signal Processing (DSP) applications are dominated by complex arithmetic operations, which heavily degrade their performance. Targeting to accelerate the execution of Residue Number Systems (RNS)-based DSP applications, in this work, we focus on optimizing the design of the modulo 2n + 1 Add-Multiply (AM) operation with weighted operands. We incorporate in our design a new direct recoding of the modulo 2n + 1 sum of two weighted operands in its Modified Booth form. Compared to the conventional design of first instantiating an adder and then, driving its output to a multiplier, the proposed fused AM design yields considerable delay, area and power gains.

This paper describes sound source localization (SSL) based on deep neural networks (DNNs) using discriminative training. A naive DNNs for SSL can be configured as follows. Input is the frequency-domain feature used in other SSL methods, and the structure of DNNs is a fully-connected network using real numbers. The training fails because its network structure loses two important properties, i.e., the orthogonality of sub-bands and the intensity- and time-information saved in complex numbers. We solved these two problems by 1) integrating directional information at each sub-band hierarchically, and 2) designing a directional activator that could treat the complex numbers at each sub-band. Our experiments indicated that our method outperformed the naive DNN-based SSL by 20 points in terms of the block-level accuracy.

The problem of approximating the complex-valued function modulus using a minimax criterion is of interest in many technical applications, such as standard process controlling systems with limiting the transient oscillations, low-side-lobe antenna arrays, or multiplexing devices having a deep channel isolation. The paper introduces approximate formulas to compute the absolute value of a complex number based on piecewise linear inequalities, thanks to which the approximation problem may be reduced to the minimax linear programming problem allowing the use of standard application packages. Computational experiments, the results of which are discussed, have proven the efficiency of the proposed computing algorithm combining high speed and good approximation accuracy.

To square large size complex numbers using n×n embedded multipliers as in using FPGAs, one needs to first partition the real and imaginary parts into segments of size less or equal to n and then to multiply the segments using these embedded multipliers. When the size of some segments is small, we show that three multiplications can be carried out using one n×n multiplier thus two n×n embedded multipliers can be saved. The idea is also applicable in carrying out two small squarers using one n×n embedded multiplier.

The accuracy of subpixel estimation of Discrete Cosine Transform (DCT) -sign phase correlation (-SPC) is presented using real images involved by actual movement. So far, the accuracy had been reported just using simulated images. Using real images and two types of simulated images, we evaluate the performance with subpixel accuracy of DCT-SPC with curve fitting compared to phase correlation with curve fitting. For real images captured by a commercial-off-the-shelf product camera, the error of estimated value with subpixel accuracy of DCT-SPC with curve fitting is lower than that of phase correlation with curve fitting, although DCT-SPC uses only the signs of DCT coefficients, while phase correlation uses the phase term with complex numbers.