Output Cumulants Research Articles

Relation between fluctuations and efficiency at maximum power for small heat engines

We study the ratio between the variances of work output and heat input, $\eta^{(2)}$, for a class of four-stroke heat engines which covers various typical cycles. Recent studies on the upper and lower bounds of $\eta^{(2)}$ are based on the quasistatic limit and the linear response regime, respectively. We extend these relations to the finite-time regime within the endoreversible approximation. We consider the ratio $\eta_{\text{MP}}^{(2)}$ at maximum power and find that the square of the Curzon-Ahlborn efficiency, $\eta_{\text{CA}}^2$, gives a good estimate of $\eta_{\text{MP}}^{(2)}$ for the class of heat engines considered, i.e., $\eta_{\text{MP}}^{(2)} \simeq \eta_{\text{CA}}^2$. This resembles the situation where the Curzon-Ahlborn efficiency gives a good estimate of the efficiency at maximum power for various kinds of finite-time heat engines. Taking an overdamped Brownian particle in a harmonic potential as an example, we can realize such endoreversible small heat engines and give an expression of the cumulants of work output and heat input. The approximate relation $\eta_{\text{MP}}^{(2)} \simeq \eta_{\text{CA}}^2$ is verified by numerical simulations. This relation also suggests a trade-off between the efficiency and the stability of finite-time heat engines at maximum power.

Improved resolving capabilities of linear array using 2 q th order non‐circular statistics

Resolving more sources than sensors is always of interest to researchers. For this purpose, generation of the virtual array from the non-uniform linear array has recently gathered a lot of attention. The covariance/cumulant lags of the array output define virtual sensors and thereby virtual array. The aperture of the designed virtual array is much more than the aperture of physical array. This large aperture provides highest degrees of freedom to solve the underdetermined system. In the case of non-circular signals, pseudo-covariances/cumulants are significant, and this additional information can further be used to increase the virtual array aperture. Herein, a framework is proposed to extend the virtual array aperture by additionally using pseudo-/non-circular cumulants along with the circular cumulants of the array output for non-circular signals. The suggested framework not only increases the resolvability but also improves the DoA estimation accuracy. With fourth-order statistics, the virtual array aperture becomes almost double, and the increment is much more with further higher order statistics. Numerical simulations demonstrate the efficacy of the claims.

Fast Cumulant Method for Probabilistic Power Flow Considering the Nonlinear Relationship of Wind Power Generation

Currently, the increasing wind power penetration, with consequent randomness and variability, presents great challenges to power system planning and operation. Probabilistic power flow (PPF) has been developed to calculate the power flow under uncertain circumstances. However, the current wind power models are subject to specific probability distributions, limiting their accuracies in wider applications. Additionally, the cumulant method (CM)-based PPF, if nonlinear relationship is considered in, would face an impractically high computational complexity. To address these problems in modeling and cumulant calculation, this article proposes a novel generalized density/distribution fitting method (GDFM) combining with the Copula function to establish a joint probability model for wind power generation. A special impulse- mixed probability density (IMPD) integration method is also introduced to derive the input cumulants from the model. Finally, a fast cumulant method (FCM) is proposed to reduce the computational burden of output cumulant calculation while retaining a high accuracy in a nonlinear context. Case study on the IEEE-118 test system validates the effectiveness of the proposed methods, and a real application to a provincial power grid in China provides some useful power flow risk information for decision making. The whole FCM-based PPF scheme can be helpful for future power flow examination in power system planning and operation.

Probability distributions of continuous measurement results for two noncommuting variables and postselected quantum evolution

We address the statistics of a simultaneous continuous weak linear measurement of two noncommuting variables on a few-state quantum system subject to a postselected evolution. The results of both postselected quantum measurement and simultaneous monitoring of two noncommuting variables differ drastically from the results of either classical or quantum projective measurement. We explore the peculiarities arising from the combination of the two. We concentrate on the distribution function of two measurement outcomes integrated over a time interval. We formulate a proper formalism for the evaluation of such distribution, and further compute and discuss the resulting statistics for idealized and experimentally relevant setups. We demonstrate the visibility and manifestations of the interference between initial and final states in the statistics of measurement outcomes for both variables in various regimes. We analytically predict the peculiarities at the circle ${\mathcal{O}}_{1}^{2}+{\mathcal{O}}_{2}^{2}=1$ in the distribution of measurement outcomes ${\mathcal{O}}_{1,2}$ in the limit of short measurement times and confirm this by numerical calculation at longer measurement times. We demonstrate analytically the anomalously large values of the time-integrated output cumulants in the limit of short measurement times and zero overlap between initial and final states, and give the detailed distributions for this case. We term this situation sudden jump. We present the numerical evaluation of the probability distributions for experimentally relevant parameters in several regimes and demonstrate that interference effects in the postselected measurement can be accurately predicted even if they are small.

Extending HOC-based methods for identifying the diagonal parameters of quadratic systems

In this paper, methods developed for the linear case of identifying the diagonal parameters of quadratic systems are extended to nonlinear case. Firstly, nonlinear relationships between model kernels and output cumulants are presented. Secondly, the relationship linking output cumulants and the coefficients of systems presented in the linear case, is extended to the general case of nonlinear quadratic systems identification. According to this concept, two nonlinear approaches are developed, the first use the fourth-order cumulants, and the second combined the third- and fourth-order cumulants. The numerical simulation results, for various signal to noise ratio (SNR) and 200 Monte Carlo runs, show that the proposed approaches achieve better accuracy, as compared with the related algorithm in the literature. Furthermore, the second algorithm is more precise in high noise environment (smallest \(\mathrm{SNR}=0\) dB), but the first algorithm more efficient in the weak noise environment case (highest SNR \(\ge \) 8 dB) comparing to using others methods.

Sparse Cumulants Fitting for Direction-of-Arrival Estimation without Redundancy

A novel direction-of-arrival (DOA) estimation method is proposed based on the sparse cumulants fitting without redundancy. Firstly, we derive that some fourth order cumulants of the array output are redundant and therefore are removed to reduce computational complexity. Then, the left cumulants are sparsely represented on an overcomplete basis and the DOAs are resolved by using a software package. Despite introducing a high variance, the proposed method shows several advantages including the ability to detect more sources than sensors, high resolution, and robustness to all kinds of Gaussian noise. Besides, our method does not have to know, a priori, the number of sources. Simulation results are presented to illustrate the effectiveness and efficiency of the proposed method.

Nonlinear Blind Identification with Three‐Dimensional Tensor Analysis

This paper deals with the analysis of a third‐order tensor composed of a fourth‐order output cumulants used for blind identification of a second‐order Volterra‐Hammerstein series. It is demonstrated that this nonlinear identification problem can be converted in a multivariable system with multiequations having the form of Ax + By = c. The system may be solved using several methods. Simulation results with the Iterative Alternating Least Squares (IALS) algorithm provide good performances for different signal‐to‐noise ratio (SNR) levels. Convergence issues using the reversibility analysis of matrices A and B are addressed. Comparison results with other existing algorithms are carried out to show the efficiency of the proposed algorithm.

Blind direction-of-arrival estimation algorithm for conformal array antenna with respect to polarisation diversity

Owing to the curvature of the conformal carrier, the polarisation diversity of element patterns is one of the most distinct characteristics of conformal array manifolds. Consequently, direction-of-arrival (DOA) estimation with conformal array antennas always couples with the estimation of the polarisation. Based on the fourth-order cumulants of the array outputs and an elaborately designed array structure, an estimation of signal parameters via rotational invariance techniques-based blind DOA estimation algorithm with respect to polarisation diversity is proposed for conformal array antennas, in which the displacement vectors required by ESPRIT are acquired by utilising ‘virtual array elements’ derived from actual array elements by ‘virtual cross-correlation computation’. Owing to the fact that these distance vectors used in algorithm implementation are not affected by element patterns and source polarisation, the 2D DOA estimates are decoupled from polarisation and obtained with no need for exact knowledge of array element polarised patterns. The proposed method achieves high-resolution 2D DOA estimation and is applicable to cylindrical, conical and spherical conformal carriers. The array set-up examples of conical, cylindrical and spherical carriers are presented for demonstration and the Monte-Carlo simulation results of DOA estimation with cylindrical conformal array are also provided to illustrate the effectiveness of the proposed algorithm.

New approaches to finite impulse response systems identification using higher-order statistics

In this study, new approaches for the identification of finite impulse response (FIR) systems using higher-order statistics are proposed. The unknown model parameters are obtained using optimisation algorithms. In fact, the proposed method consists first in defining an optimisation problem and second in using an appropriate algorithm to resolve it. Moreover, a new method is developed for estimating the order of FIR models using only the output cumulants. The results presented in this study illustrate the performance of the proposed methods and compare them with a range of existing approaches.

Blind identification of Hammerstein channels using QAM, PSK, and OFDM inputs

This paper is concerned with the blind identification of passband and baseband Hammerstein channels using cumulants of the received sequence. The channel is excited by common communication signals such as QAM, PSK and OFDM; sparseness of the higher order cumulant lags is exploited. Exact expressions and algorithms involving the output cumulants are developed. Performance is assessed by simulations.

Blind Identification of Second Order Volterra Systems With Complex Random Inputs Using Higher Order Cumulants

In this correspondence, closed-form expressions for the blind identification of linear-quadratic Volterra systems are established. The system is excited by a complex valued random sequence and the output cumulants (of order up to 4) are employed. It is assumed that the memory of the linear part is greater than or equal to the memory of the quadratic part. Cumulant-based formulas are developed demonstrating that the system is uniquely identifiable. An SVD based variant with improved performance is also derived. Simulations and comparisons with existing techniques are presented.

Blind identification of MISO-FIR channels

In this paper, we address the problem of determining the order of MISO channels by means of a series of hypothesis tests based on scalar statistics. Using estimated 4th-order output cumulants, we exploit the sensitiveness of a Chi-square test statistic to the non-Gaussianity of a stochastic process. This property enables us to detect the order of a linear finite impulse response (FIR) channel. Our approach leads to a new channel order detection method and we provide a performance analysis along with a criterion to establish a decision threshold, according to a desired level of tolerance to false alarm. Afterwards, we introduce the concept of MISO channel nested detectors based on a deflation-type procedure using the 4th-order output cumulants. The nested detector is combined with an estimation algorithm to select the order and estimate the parameters associated with different transmitters composing the MISO channel. By treating successively shorter and shorter channels, it is also possible to determine the number of sources.

Parafac-based Blind Identification of Convolutive MIMO Linear Systems

In this paper, we introduce a new tensor model for the 4th-order space-time output cumulants in the context of convolutive MIMO linear systems. We also propose a single-step least-squares (SS-LS) algorithm for blind identification of convolutive MIMO linear systems. The proposed tensor-based identification algorithm generalizes our methods recently developed for SISO and memoryless MIMO linear systems. Uniqueness conditions are derived showing that a large range of system configurations can be considered. Simulation results illustrate the performance of the proposed algorithm.

Blind paraunitary equalization

In this paper a blind multiple-input/multiple-output (MIMO) space–time equalizer is described, dedicated to convolutive mixtures when observations have been pre-whitened. Filters preserving space–time whiteness are paraunitary; a parameterization of such filters with plane rotations is proposed. Theoretical developments then lead to a numerical algorithm that sweeps all pairs of delayed outputs. This algorithm involves the solution of a polynomial system in two unknowns, whose coefficients depend on the output cumulants. Simulations and performance of the numerical algorithm are reported.

Blind channel identification using higher-order statistics

This article addresses the problem of blind identification of a non-minimum phase system from only third- and fourth-order cumulants of the output noisy observations of the system. Nonlinear optimization algorithms, namely the gradient descent, the Gauss–Newton and the Newton–Raphson algorithms, are proposed for estimating the parameters of the moving average models. A relationship between third- and fourth-order cumulants of the noisy system output and the parameters of the model is exploited to build a set of non-linear equations that is solved by means of the three non-linear optimization algorithms cited above. Simulation results demonstrate the performance of the proposed algorithms.

Blind channel identification algorithms based on the Parafac decomposition of cumulant tensors: The single and multiuser cases

In this paper, we exploit the symmetry properties of 4th-order cumulants to develop new blind channel identification algorithms that utilize the parallel factor (Parafac) decomposition of cumulant tensors by solving a single-step (SS) least squares (LS) problem. We first consider the case of single-input single-output (SISO) finite impulse response (FIR) channels and then we extend the results to multiple-input multiple-output (MIMO) instantaneous mixtures. Our approach is based on 4th-order output cumulants only and it is shown to hold for certain underdetermined mixtures, i.e. systems with more sources than sensors. A simplified approach using a reduced-order tensor is also discussed. Computer simulations are provided to assess the performance of the proposed algorithms in both SISO and MIMO cases, comparing them to other existing solutions. Initialization and convergence issues are also addressed.

Blind identification of Volterra-Hammerstein systems

This paper is concerned with the blind identification of Volterra-Hammerstein systems. Two identification scenarios are covered. The first scenario assumes that, although the input is not available, the statistics of the input are a priori known. This case appears in communication applications where the input statistics of the transmitter are known to the receiver. The second scenario assumes that the input statistics are unknown. In the case of known input statistics, the input is stationary higher order white noise with arbitrary probability density function. Under the scenario of unknown input statistics, the input is restricted to Gaussian white process. New cumulant-based identification methods are described for the above scenarios. The problem is converted into a linear multivariable form and the output cumulants are calculated using Kronecker products. First, initial conditions are determined by a linear system of equations. These correspond to the boundary values of the Volterra kernels. The remaining kernel coefficients can be determined under both identification schemes from a possibly overdetermined system of linear equations.

Blind identification of bilinear systems

This paper is concerned with the blind identification of a class of bilinear systems excited by non-Gaussian higher order white noise. The matrix of coefficients of mixed input-output terms of the bilinear system model is assumed to be triangular in this work. Under the additional assumption that the system output is corrupted by Gaussian measurement noise, we derive an exact parameter estimation procedure based on the output cumulants of orders up to four. Results of the simulation experiments presented in the paper demonstrate the validity and usefulness of our approach.

Blind Neural Equalizer using Higher-Order Statistics

This paper discusses a blind equalization technique for FIR channel system, that might be minimum phase or not, in digital communication. The proposed techniques consist of two parts. One is to estimate the original channel coefficients based on fourth-order cumulants of the channel output, the other is to employ RBF neural network to model an inverse system fur the original channel. Here, the estimated channel is used as a reference system to train the RBF. The proposed RBF equalizer provides fast and easy teaming, due to the structural efficiency and excellent recognition-capability of R3F neural network. Throughout the simulation studies, it was found that the proposed blind RBF equalizer performed favorably better than the blind MLP equalizer, while requiring the relatively smaller computation steps in tranining.

Blind identification of second order Hammerstein series

Blind identification of second order Hammerstein series is considered in this paper. The output cumulants up to order 5 are used to determine the Volterra kernels and the system orders, when the input is a stationary zero mean Gaussian white stochastic process. Both infinite and finite extent kernels are considered. The algorithm is validated by simulations.