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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.

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.

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.

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.

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

Many optimization problems in power transmission networks can be formulated as polynomial problems with complex variables. A polynomial optimization problem with complex variables consists in optimizing a real-valued polynomial whose variables and coefficients are complex numbers subject to some complex polynomial equality or inequality constraints. These problems are usually directly converted to real variables, either using the polar form or the rectangular form. In this work, we propose a Julia module allowing the representation of polynomial problems in their original complex formulation. This module is applied to power systems optimization and its generic design enables the description of several variants of power system problems. Results for the Optimal Power Flow in Alternating Current problem and for the Preventive-Security Constrained Optimal Power Flow problem are presented.

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.

International audience; Micro-Doppler analysis commonly makes use of the log-scaled, real-valued spectrogram, and recent work involving deep learning architectures for classification are no exception. Some works in neighboring fields of research directly exploit the raw temporal signal, but do not handle complex numbers, which are inherent to radar IQ signals. In this paper, we propose a complex-valued, fully temporal neural network which simultaneously exploits the raw signal and the spectrogram by introducing a Fourier-like layer suitable to deep architectures. We show improved results under certain conditions on synthetic radar data compared to a real-valued counterpart.

The native implementation of the N-point digital Fourier Transform involves calculating the scalar product of the sample buffer (treated as an N-dimensional vector) with N separate basis vectors. Since each scalar product involves N multiplications and N additions, the total time is proportional to $N^{2}$ , in other words, its an $O(N^{2})$ algorithm. However, it turns out that by cleverly re-arranging these operations, one can optimize the algorithm down to $O(Nlog_{2}(N))$ , which for large N makes a huge difference. The optimized version of the algorithm is called the Fast Fourier Transform, or the FFT. In this paper, we discuss about an efficient way to obtain Fast Fourier Transform algorithm (FFT). According to our study, we can eliminate some operations in calculating the FFT algorithm thanks to property of complex numbers and we can achieve the FFT in a better execution time due to a significant reduction of $N/8$ of the needed twiddle factors and to additional factorizations.

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.

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.

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.

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.

This paper advocates the use of complex variables to represent votes in the Hough transform for circle detection. Replacing the positive numbers classically used in the parameter space of the Hough transforms by complex numbers allows cancellation effects when adding up the votes. Cancellation and the computation of shape likelihood via a complex number's magnitude square lead to more robust solutions than the "classic" algorithms, as shown by computational experiments on synthetic and real datasets., The paper has been withdrawn since the authors concluded a more comprehensive study on the choice of parameters needs to be performed

The hypercomplex number quaternion is an extension to the complex number and is widely used in computer graphic, image processing and multiple dimensional linear time-invariant systems. In this paper, the quaternion algebra is applied to head-related impulse response (HRIR) modeling. Four HRIRs measured at different elevations or for different ears are used to construct a quaternion impulse response. A two dimensional quaternion common factor decomposition (QCFD) algorithm is developed to represent each quaternion impulse response as the convolution of two factor impulse responses. By using the proposed QCFD algorithm, quaternion impulse responses with the same elevation will share the same elevation factor while quaternion impulses with the same azimuth will share the same azimuth factor. Experimental results show that the QCFD algorithm has better performance as compared to the traditional two-dimension common factor decomposition (CFD) algorithm.

Recommendation can be reduced to a sub-problem of link prediction, with specific nodes (users and items) and links (similar relations among users/items, and interactions between users and items). However, the previous link prediction algorithms need to be modified to suit the recommendation cases since they do not consider the separation of these two fundamental relations: similar or dissimilar and like or dislike. In this paper, we propose a novel and unified way to solve this problem, which models the relation duality using complex number. Under this representation, the previous works can directly reuse. In experiments with the Movie Lens dataset and the Android software website AppChina.com, the presented approach achieves significant performance improvement comparing with other popular recommendation algorithms both in accuracy and coverage. Besides, our results revealed some new findings. First, it is observed that the performance is improved when the user and item popularities are taken into account. Second, the item popularity plays a more important role than the user popularity does in final recommendation. Since its notable performance, we are working to apply it in a commercial setting, AppChina.com website, for application recommendation.

Digital computation on stochastic bit streams is a non-conventional type of computation, which uses digital circuits to process probabilistic inputs. In this paradigm, digital circuits compute on the probability values. This leads to extremely simple digital implementations for complex arithmetic operations. The combinational logic-based designs can be implemented in parallel to realize fast computation by trading off silicon area with delay. However, it is impossible to implement the current sequential logic-based designs in parallel to reduce delay, since they need to compute for a large number of clock cycles to obtain the result. In this work, we propose a novel design methodology for synthesizing sequential circuits that compute on stochastic bit streams, which does not require a long computation delay. We further demonstrate a parallel implementation based on our design, which provides ultra-fast arithmetic computation.

In considering the reliability of numerical programs,it is normal to “limit our study to the semanticsdealing with numerical precision” (Martel, 2005). On the otherhand, there is a great deal of work on the reliability ofprograms that essentially ignores the numerics. The thesisof this paper is that there is a class of problems that fallbetween these two, which could be described as “does the lowlevelarithmetic implement the high-level mathematics”. Manyof these problems arise because mathematics, particularlythe mathematics of the complex numbers, is more difficultthan expected: for example the complex function log is notcontinuous, writing down a program to compute an inversefunction is more complicated than just solving an equation,and many algebraic simplification rules are not universallyvalid.The good news is that these problems are theoreticallycapable of being solved, and are practically close to being solved,but not yet solved, in several real-world examples. However,there is still a long way to go

Orthogonalization consumes much of the run time of many iterative methods for solving sparse linear systems and eigenvalue problems. Commonly used algorithms, such as variants of Gram-Schmidt or Householder QR, have performance dominated by communication. Here, "communication" includes both data movement between the CPU and memory, and messages between processors in parallel. Our Tall Skinny QR (TSQR) family of algorithms requires asymptotically fewer messages between processors and data movement between CPU and memory than typical orthogonalization methods, yet achieves the same accuracy as Householder QR factorization. Furthermore, in block orthogonalizations, TSQR is faster and more accurate than existing approaches for orthogonalizing the vectors within each block ("normalization"). TSQR's rank-revealing capability also makes it useful for detecting deflation in block iterative methods, for which existing approaches sacrifice performance, accuracy, or both. We have implemented a version of TSQR that exploits both distributed-memory and shared-memory parallelism, and supports real and complex arithmetic. Our implementation is optimized for the case of orthogonalizing a small number (5 -- 20) of very long vectors. The shared-memory parallel component uses Intel's Threading Building Blocks, though its modular design supports other shared-memory programming models as well, including computation on the GPU. Our implementation achieves speedups of 2 times or more over competing orthogonalizations. It is available now in the development branch of the Trilinos software package, and will be included in the 10.8 release.

We investigate the low-SNR sum rate performance of QPSK for symmetric interference channels. The QPSK performance is described by the minimum energy per bit and the wideband slope pertaining to the sum capacity. Comparing this with the minimum energy per bit and wideband slope of corresponding interference channels using optimal inputs, we find that QPSK achieves optimal performance in all of the cases where exact sum capacities are known. We also show that a simplified Han-Kobayashi scheme is suboptimal in the low-SNR regime when the input alphabet is the whole set of complex numbers.

We consider the problem of recovering a signal/image (x) with a k-sparse representation, from hybrid (complex and real), noiseless linear samples (y) using a mixture of complex-valued sparse and real-valued dense projections within a single matrix. The proposed Hybrid Compressed Sensing (HCS) employs the complex-sparse part of the projection matrix to divide the n-dimensional signal (x) into subsets. In turn, each subset of the signal (coefficients) is mapped onto a complex sample of the measurement vector (y). Under a worst-case scenario of such sparsity-induced mapping, when the number of complex sparse measurements is sufficiently large then this mapping leads to the isolation of a significant fraction of the k non-zero coefficients into different complex measurement samples from y. Using a simple property of complex numbers (namely complex phases) one can identify the isolated non-zeros of x. After reducing the effect of the identified non-zero coefficients from the compressive samples, we utilize the real-valued dense submatrix to form a full rank system of equations to recover the signal values in the remaining indices (that are not recovered by the sparse complex projection part). We show that the proposed hybrid approach can recover a k-sparse signal (with high probability) while requiring only m ≈ 3√n/2k real measurements (where each complex sample is counted as two real measurements). We also derive expressions for the optimal mix of complex-sparse and real-dense rows within an HCS projection matrix. Further, in a practical range of sparsity ratio (k/n) suitable for images, the hybrid approach outperforms even the most complex compressed sensing frameworks (namely basis pursuit with dense Gaussian matrices). The theoretical complexity of HCS is less than the complexity of solving a full-rank system of m linear equations. In practice, the complexity can be lower than this bound.

Dealing with various inexact pieces of information has become an intrinsically important issue in knowledge-based reasoning, because many problem domains involve imprecise, incomplete and uncertain information. Indeed, different approaches exist for reasoning with inexact knowledge and data. However, the common strategy they adopt is to integrate various types of inexact information into a global measure. This may destroy the underlying semantics associated with different information components. This paper presents an innovative notion of fuzzy complex numbers (FCNs), which extends real complex numbers to representing two-dimensional uncertainties conjunctively without necessarily integrating them. This new framework is applied to supporting Compositional Modelling (CM). In particular, calculus of FCNs over arithmetic and propositional relations is developed to entail scenario model synthesis from model fragments, and modulus of FCNs is introduced to constrain the scenario descriptions. The utility and usefulness of this work are illustrated by means of an example for constructing possible scenario descriptions from given evidence in the crime investigation domain.

In this paper, a new algorithm based on the Tschirnhaus transforms is developed to reduce the computation complexity of the 10-order line spectrum pairs (LSP) frequencies. The first step of the proposed algorithm is to derive a quartic equation from the 1st derivative of the given 5-degree LSP polynomial. Then the extremes of the 5-degree LSP polynomial can be found by applying the Tschirnhaus transform to the above quartic equation. By the use of these extremes as the initial approximations, one can easily solve the roots of the 5-degree LSP polynomial via the Newton's method and get the accurate LSP frequencies. One of the main advantages of the proposed algorithm is the rapid root determination of a quartic equation without complex number operations and resulting in considerable computational saving. Compared to other methods, the proposed algorithm can determine the precise LSP frequencies with the lowest computational complexity.

This letter proposes a simple but highly efficient technique to jointly utilize space, time, and polarization diversities. The technique is based on extending orthogonal space-time block codes into the quaternion domain and utilizing a description of the dual-polarized signal by means of quaternions, which are a generalization of complex numbers. In the given example, the achievable performance gain for two transmit and one receive antennas is approximately 6 dB at a bit error rate of 10-4 when compared with the Alamouti code.

Support vector regressors (SVRs) are a class of nonlinear regressor inspired by Vapnik's support vector (SV) method for pattern classification. The standard SVR has been successfully applied to real number regression problems such as financial prediction and weather forecasting. However in some applications the domain of the function to be estimated may be more naturally and efficiently expressed using complex numbers (eg. communications channels) or quaternions (eg. 3-dimensional geometrical problems). Since SVRs have previously been proven to be efficient and accurate regressors, the extension of this method to complex numbers and quaternions is of great interest. In the present paper the standard SVR method is extended to cover regression in complex numbers and quaternions. Our method differs from existing approaches in-so-far as the cost function applied in the output space is rotationally invariant, which is important as in most cases it is the magnitude of the error in the output which is important, not the angle. We demonstrate the practical usefulness of this new formulation by considering the problem of communications channel equalization.

An insight into the potential benefits of using complex valued models for real valued data is provided. The problem itself is not new; it is however timely and important to revisit this issue, due to a plethora of modern applications based on multidimensional and multichannel measurements which can be cast into an equivalent problem in the field of complex numbers C. The analysis and simulations highlight the duality between several classes of real domain problems and their complex valued representations. This is supported by case studies on image processing, modelling of point processes for brain prosthetics, and forecasting of vector fields.

We propose a digit recurrence algorithm for division in real and complex number domains using a variable radix. The objective of the approach is to simplify the prescaling of the operands by using a suitable low radix, and switch to higher radices in the remaining iterations to reduce their number. The prescaling is used to allow a simple quotient digit selection by rounding of the residual. We discuss the algorithm, its implementation, and estimate its time and cost characteristics with respect to fixed high radix division algorithms.

In this paper, we present an implementation for the Complex Householder Transform, using complex number on-line arithmetic, based on adopting a redundant complex number system (RCNS) to represent complex operands as a single number. We present comparisons with (i) a real number on-line arithmetic approach, and (ii) a real number arithmetic parallel approach, to demonstrate a significant improvement in cost.

Trace-orthogonality is an important property of linear space-time codes that has emerged relatively recently. In this work we propose a quite general procedure for the generation of such codes for any number of transmit antennas and uses of the channel. Simulations suggest that the use of rectangular encoding matrices, in place of square ones, could be beneficial to the coding gain. Moreover we are able to guarantee full-diversity of the proposed codes when information symbols are carved from the quadratic field Q(j). Since the components of any complex number can be approximated with arbitrary precision by rational numbers, our scheme could be used to guarantee full-diversity for any conceivable complex constellation.

In this paper, we present a novel implementation for an N-tap complex finite impulse response (FIR) filter, using complex number on-line arithmetic, based on adopting a redundant complex number system (RCNS) to represent complex operands as a single number. We present cost comparisons with (i) a real number on-line arithmetic approach and (ii) a real number parallel arithmetic approach, to demonstrate a significant improvement in cost.

This paper describes a CORDIC-based architecture to efficiently compute the phase difference between two complex numbers. The problem of fast phase difference computation is central in many signal processing algorithms. Our main focus has been posed on the phase correlation technique applied to motion estimation. A reduced complexity solution is proposed and specifically tailored to suit the application needs. The presented algorithm has been completely implemented in 0.25 /spl mu/m standard-cell CMOS technology. As far as the performance are concerned the designed core outperforms a recently designed solution by more than 50% under area and energy standpoints.

Two subspaces of a vector space are here called "nonintersecting" if they meet only in the zero vector. Motivated by the design of noncoherent multiple-antenna communications systems, we consider the following question. How many pairwise nonintersecting M/sub t/-dimensional subspaces of an m-dimensional vector space V over a field F can be found, if the generator matrices for the subspaces may contain only symbols from a given finite alphabet A/spl sube/F? The most important case is when F is the field of complex numbers C; then M/sub t/ is the number of antennas. If A=F=GF(q) it is shown that the number of nonintersecting subspaces is at most (q/sup m/-1)/(q/sup Mt/-1), and that this bound can be attained if and only if m is divisible by M/sub t/. Furthermore, these subspaces remain nonintersecting when "lifted" to the complex field. It follows that the finite field case is essentially completely solved. In the case when F=C only the case M/sub t/=2 is considered. It is shown that if A is a PSK-configuration, consisting of the 2/sup r/ complex roots of unity, the number of nonintersecting planes is at least 2/sup r(m-2)/ and at most 2/sup r(m-1)-1/ (the lower bound may in fact be the best that can be achieved).

This paper describes a cellular neural network (CNN) for estimating the time-to-contact from a one dimensional image. The CNN arrays used for this algorithm consist of cells with second order dynamics. The key feature of these arrays is that the spatial information in a region around each cell is represented by the phase of a complex number. The velocity is encoded as the temporal variation of that phase. By modelling this variation using adaptive temporal oscillators, the velocity can be estimated. Velocity information extracted over the entire array can be combined to estimate the time-to-contact.

Among various transform techniques for image compression, the discrete cosine transform (DCT) is the most popular and effective one in practical applications because it gives an almost optimal performance and can be implemented at an acceptable cost. We describe a novel 8/spl times/8 2-D DCT/IDCT architecture based on the direct 2-D approach and the rotation technique. The computational complexity is reduced by taking advantage of the special attribute of complex numbers. Unlike other direct approach, the proposed architecture is regular, hence, it is suitable for VLSI implementation.

Most data in digital signal processing belong to the complex number system. Some characteristics of complex number computations can be utilized to reduce hardware overhead for concurrent error detection (CED) scheme. This paper presents a novel design of CED technique based on realization of direct complex computation approach. It is a time-redundant method and the recomputation step is performed through exchanging strategy on both the real part and imaginary part circuits in a complex function. The hardware overhead then can be eliminated and the capability of error detection is still robust as duplicated module technique. Since pipelined processing can be introduced to kill the recomputation cycle, the fast Fourier transform (FFT) processor based on pipelined butterfly module with the proposed CED technique is able to achieve real-time fault diagnosis. This technique can be applied to other complex number computing systems, and is more attractive especially in high-speed and high-reliability system. >

A class of on-line algorithms for complex number arithmetic is presented These algorithms adopt a redundant complex number system (RCNS) to represent complex numbers as a single number. Such a scheme simplifies the specification of the design, and has the additional effect that single precision complex arithmetic can be easily reconfigured for double-precision real arithmetic. We present cost/spl times/delay comparisons with the more conventional approach to show a significant improvement, demonstrating that the presented algorithms are attractive for VLSI systems demanding complex number operations.