Optimal Nonlinearity Research Articles

Improved compensation of intrachannel four-wave mixing in dispersion-managed transmission links with mid-span optical phase conjugation

Recently, a novel approach to satisfy the nonlinearity compensation criteria by mid-span optical-phase-conjugation (OPC) was proposed and excellent transmission performance was achieved, there the OPC propagation symmetry was obtained through optimized dispersion management with inverse-dispersion fiber (IDF). In this work, the author numerically investigates the possibilities of using dispersion-compensating fiber (DCF) instead of IDF for construction of the link. Nonlinearity compensation effectiveness of the two links are compared with respected to the intrachannel four-wave mixing (IFWM) induced intensity fluctuation of the ‘1’ bits and the generation of the ghost pulse. It is found that, by launching different powers (or energy of a limited bits) into the spans before and after the OPC, nonlinearity mismatch between the spans of the DCF-managed link can be counterbalanced with significantly improved transmission performance. For a given input energy, there exists an optimum ratio of the energies into the spans before and after the OPC, at which optimum nonlinearity compensation can be achieved, and the compensation effectiveness is very close to that of the IDF-managed link. Further studies show that the optimum energy ratio increases with the input energy but is independent of the span number of the link.

A new method to construct Boolean functions with good cryptographic properties

To resist fast correlation attacks, Boolean functions used in stream ciphers should have high nonlinearity. n-variable bent functions have the maximum nonlinearity. However, they are not balanced and their algebraic degrees are at most n2. Therefore, they cannot be used directly as filter functions. In this paper, we give a new method to construct cryptographically significant Boolean functions. As an example, based on bent functions, we construct an infinite class of functions with good cryptographic properties: balancedness, optimum algebraic degree, almost optimum algebraic immunity and an almost optimum nonlinearity (higher than all other infinite classes of balanced functions with high algebraic immunity).

Optimum Nonlinear Response of Composite Panels Subjected to Aerodynamic and Thermal Loadings

A nonlinear finite element model is provided for the thermal post buckling and linear flutter behavior of composite panels. Panel subjects to combined aerodynamic and thermal loads. The governing equations are derived using the classical plate theory and the principle of virtual work. The effect of large deflection is included in the formulation through the von Karman nonlinear strain-displacement relations. To account for the temperature dependence on material properties, the thermal strain is stated as an integral quantity of the thermal expansion coefficient with respect to temperature. The aerodynamic pressure is modeled using the quasi-steady first order piston theory. The Newton–Raphson iteration method is employed to obtain the nonlinear aero-thermal post-buckling deflections, and a frequency-domain solution is presented to predict the critical dynamic pressure at different elevated temperatures. Finally, numerical results are provided to depict the optimum lamination scheme in order to maximize the aero-thermal stability of such panels. The optimum solution is obtained by Genetic Algorithms.

Construction of 1-resilient Boolean functions with optimum algebraic immunity

In this paper, for an integer n≥10, two classes of n-variable Boolean functions with optimum algebraic immunity (AI) are constructed, and their nonlinearities are also determined. Based on non-degenerate linear transforms to the proposed functions, we can obtain 1-resilient n-variable Boolean functions with optimum AI and high nonlinearity if n−1 is never equal to any power of 2.

Optimum and suboptimum memoryless nonlinearities for the detection of ultrashort light pulses in Gaussian noise

Detection of ultrashort light pulses with conventional bandlimited photodetectors can lead to substantial performance degradation in digital lightwave communication systems. All-optical nonlinear processing of the received field prior to the optical to electrical conversion can greatly overcome this problem. In this letter, through a decision-theoretic framework and using a basic invariance property of the photodetector, we obtain the optimal zero-memory nonlinear element and evaluate its performance. Possible suboptimum nonlinearities are also investigated and their performance is examined. The results in this letter strongly justify the intuitive use of power-law devices, such as those implemented via two photon absorption (TPA) or second harmonic generation (SHG), in the detection of ultrashort light pulses and show how by using them almost optimal performance can be achieved.

Optimal and perfect difference systems of sets

Difference systems of sets (DSS) were introduced in 1971 by Levenstein for the construction of codes for synchronization, and are closely related to cyclic difference families. In this paper, algebraic constructions of difference systems of sets using functions with optimum nonlinearity are presented. All the difference systems of sets constructed in this paper are perfect and optimal. One conjecture on difference systems of sets is also presented.

Impact of the Signal and Nonlinearity Extinction Ratios on the Design of Nonideal 2R All-Optical Regenerators

The impact of the signal and nonlinearity extinction ratios on the optimum parameters of a reamplification and reshaping (2R) all-optical regenerator is investigated for different nonlinearity characteristic shapes. The bit error ratio (BER) is numerically evaluated along the 2R all-optical regenerator chain using the rigorous distribution and the Gaussian distribution for the optical power. It is shown that, when the extinction ratio of the signal at the transmitter output decreases, the optimum nonlinearity threshold increases. The results also show that, independent of the nonlinearity characteristic shape, the requirements on the extinction ratio of the nonlinearity can be strongly relaxed when the extinction ratio of the nonlinearity is higher than the extinction ratio of the signal at the transmitter output. Numerical results show that, only for nonlinearities with a small degree of nonlinearity and small extinction ratio, the Gaussian distribution correctly estimates the optimum regenerator parameters. For a small extinction ratio of the nonlinearity or a small extinction ratio of the signal at the transmitter output, the Gaussian distribution only provides correct estimates of the minimum BER along the regeneration chain. Furthermore, for high and moderate extinction ratios and degrees of nonlinearity, and for high and moderate extinction ratios of the signal at the transmitter output, the Gaussian distribution fails to correctly estimate the optimum regenerator parameters and minimum BER.

Signal Sets From Functions With Optimum Nonlinearity

Signal sets with the best correlation property are desirable in code-division multiple-access (CDMA) systems. In this paper, the construction of Wootters and Fields for mutually unbiased bases is extended into a generic construction of signal sets using planar functions. Then, specific classes of planar functions and almost bent functions are employed to obtain (q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> +q,q) signal sets. The signal sets derived from planar functions are optimal with respect to the Levenstein bound, and those obtained from almost bent functions nearly meet the Levenstein bound. The signal sets constructed in this paper could have a very small alphabet size, and have applications in synchronous DS-CDMA systems, where the number of users is greater than the signal space dimension or the spreading factor

On the optimization of regenerator parameters in a chain of 2R all-optical regenerators

The dependence of the bit-error ratio (BER) along a 2R all-optical regenerator chain on the nonlinearity threshold and degree is investigated numerically using the rigorous distribution and the Gaussian distribution for the optical power. It is shown that, for high nonlinearity degree, the optimum nonlinearity threshold has a weak dependence on the number of regenerators and, for low nonlinearity degree, increases with the number of regenerators. Numerical results show that the Gaussian distribution may under-estimate or over-estimate the BER and fails to indicate the optimum regenerator parameters

Locally Optimum Nonlinearities for DCT Watermark Detection

The issue of copyright protection of digital multimedia data has attracted a lot of attention during the last decade. An efficient copyright protection method that has been gaining popularity is watermarking, i.e., the embedding of a signature in a digital document that can be detected only by its rightful owner. Watermarks are usually blindly detected using correlating structures, which would be optimal in the case of Gaussian data. However, in the case of DCT-domain image watermarking, the data is more heavy-tailed and the correlator is clearly suboptimal. Nonlinear receivers have been shown to be particularly well suited for the detection of weak signals in heavy-tailed noise, as they are locally optimal. This motivates the use of the Gaussian-tailed zero-memory nonlinearity, as well as the locally optimal Cauchy nonlinearity for the detection of watermarks in DCT transformed images. We analyze the performance of these schemes theoretically and compare it to that of the traditionally used Gaussian correlator, but also to the recently proposed generalized Gaussian detector, which outperforms the correlator. The theoretical analysis and the actual performance of these systems is assessed through experiments, which verify the theoretical analysis and also justify the use of nonlinear structures for watermark detection. The performance of the correlator and the nonlinear detectors in the presence of quantization is also analyzed, using results from dither theory, and also verified experimentally.

Memoryless Discrete-Time Signal Detection in Long-Range Dependent Noise

The problem of designing optimum memoryless detectors for known signals in long-range dependent (LRD) noise is considered. In particular, under the performance criterion of asymptotic relative efficiency (ARE), optimum memoryless detection in LRD noise is investigated by exploiting the Hermite expansions. The detectors considered have the form of a nonlinearity followed by an accumulator and threshold comparator. It is shown that when the noise is LRD and Gaussian, all nonlinearities with Hermite rank one are asymptotically equivalent in terms of efficiency and are most powerful. Moreover, the optimum nonlinearities with Hermite rank greater than one are given by the corresponding Hermite polynomials. The case of non-Gaussian LRD noise that can be derived by nonlinear transformation of Gaussian noise is also considered. In this case, the globally optimum nonlinearity is very difficult to obtain in general. Instead, we proposed a suboptimal nonlinearity, which is given by a linear combination of the corresponding locally optimum detector and the inverse of the transform function generating the noise. Simulations show that the proposed detector outperforms the locally optimal detector for LRD non-Gaussian noise.

Cartesian authentication codes from functions with optimal nonlinearity

In this paper, we present several classes of authentication codes using functions with perfect nonlinearity and optimum nonlinearity. Some of the authentication codes are optimal. On the other hand, these authentication codes are easy to implement due to their simple algebraic structures.

Adaptive Filters with Error Nonlinearities: Mean-Square Analysis and Optimum Design

This paper develops a unified approach to the analysis and design of adaptive filters with error nonlinearities. In particular, the paper performs stability and steady-state analysis of this class of filters under weaker conditions than what is usually encountered in the literature, and without imposing any restriction on the color or statistics of the input. The analysis results are subsequently used to derive an expression for the optimum nonlinearity, which turns out to be a function of the probability density function of the estimation error. Some common nonlinearities are shown to be approximations to the optimum nonlinearity. The framework pursued here is based on energy conservation arguments.

General formula for conditional meanusing higher order statistics

A general formula is given for conditional mean, which is the well-known minimum variance estimator, in terms of cross-cumulants and locally optimum nonlinearities. This formula will greatly facilitate the solution of various problems emerging in nonlinear system identification, non-Gaussian estimation theory, and nonlinear signal processing.

A neural-network approach to nonparametric and robust classification procedures

In this paper algorithms of neural-network type are introduced for solving estimation and classification problems when assumptions about independence, Gaussianity, and stationarity of the observation samples are no longer valid. Specifically, the asymptotic normality of several nonparametric classification tests is demonstrated and their implementation using a neural-network approach is presented. Initially, the neural nets train themselves via learning samples for nominal noise and alternative hypotheses distributions resulting in near optimum performance in a particular stochastic environment. In other than the nominal environments, however, high efficiency is maintained by adapting the optimum nonlinearities to changing conditions during operation via parallel networks, without disturbing the classification process. Furthermore, the superiority in performance of the proposed networks over more traditional neural nets is demonstrated in an application involving pattern recognition.

Robust sequential MIPA array processor with dependent sampling

The problem of detecting small stochastic signals in poorly specified noise environments, often encountered in underwater acoustics, is considered. The array processing detectors are designed to exhibit a near‐optimum performance for nominal distributions and maintain high efficiency to changing noise environments by adapting their optimum nonlinearities using an m‐interval polynomial approximation (MIPA) of them. Sequential operation of the detectors is used to reduce the average time to decision. Furthermore, the assumption of independent samples is relaxed while temporal and spatial noise dependences are considered separately. A formulation is given that predicts the thresholds under the time‐dependent sampling in order to maintain the same error probabilities as in the independent sampling case. For the space‐dependent noise process, the thresholds remain constant for fixed error probabilities. Instead, the cross‐scores need to be adjusted to compensate for space dependence. The performance of these d...

Normalized data nonlinearities for LMS adaptation

Properly designed nonlinearly-modified LMS algorithms, in which various quantities in the stochastic gradient estimate are operated upon by memoryless nonlinearities, have been shown to perform better than the LMS algorithm in system identification-type problems. The authors investigate one such algorithm given by W/sub k+l/=W/sub k/+/spl mu/(d/sub k/-W/sub k//sup t/X/sub k/)X/sub k/f(X/sub k/) in which the function f(X/sub k/) is a scalar function of the sum of the squares of the N elements of the input data vector X/sub k/. This form of algorithm generalizes the so-called normalized LMS (NLMS) algorithm. They evaluate the expected behavior of this nonlinear algorithm for both independent input vectors and correlated Gaussian input vectors assuming the system identification model. By comparing the nonlinear algorithm's behavior with that of the LMS algorithm, they then provide a method of optimizing the form of the nonlinearity for the given input statistics. In the independent input case, they show that the optimum nonlinearity is a single-parameter version of the NLMS algorithm with an additional constant in the denominator and show that this algorithm achieves a lower excess mean-square error (MSE) than the LMS algorithm with an equivalent convergence rate. Additionally, they examine the optimum step size sequence for the optimum nonlinear algorithm and show that the resulting algorithm performs better and is less complex to implement than the optimum step size algorithm derived for another form of the NLMS algorithm. Simulations verify the theory and the predicted performance improvements of the optimum normalized data nonlinearity algorithm. >

Stochastic gradient adaptation under general error criteria

Examines a family of adaptive filter algorithms of the form W/sub k+1/=W/sub k/+/spl mu/f(d/sub k/-W/sub k//sup t/X/sub k/)X/sub k/ in which f(/spl middot/) is a memoryless odd-symmetric nonlinearity acting upon the error. Such algorithms are a generalization of the least-mean-square (LMS) adaptive filtering algorithm for even-symmetric error criteria. For this algorithm family, the authors derive general expressions for the mean and mean-square convergence of the filter coefficients For both arbitrary stochastic input data and Gaussian input data. They then provide methods for optimizing the nonlinearity to minimize the algorithm misadjustment for a given convergence rate. Using the calculus of variations, it is shown that the optimum nonlinearity to minimize misadjustment near convergence under slow adaptation conditions is independent of the statistics of the input data and can be expressed as -p'(x)/p(x), where p(x) is the probability density function of the uncorrelated plant noise. For faster adaptation under the white Gaussian input and noise assumptions, the nonlinearity is shown to be x/{1+/spl mu//spl lambda/x/sup 2///spl sigma//sub k//sup 2/}, where /spl lambda/ is the input signal power and /spl sigma//sub k//sup 2/ is the conditional error power. Thus, the optimum stochastic gradient error criterion for Gaussian noise is not mean-square. It is shown that the equations governing the convergence of the nonlinear algorithm are exactly those which describe the behavior of the optimum scalar data nonlinear adaptive algorithm for white Gaussian input. Simulations verify the results for a host of noise interferences and indicate the improvement using non-mean-square error criteria. >

Nonparametric density estimation and detection in impulsive interference channels. II. Detectors

For pt. I see ibid., vol.42, no.2-4, p.1684-1697 (1994). Nonlinear processing significantly enhances detector performance in nongaussian noise relative to that of linear detectors. Several nonparametric detection schemes for impulsive noise channels are formulated using the nonparametric probability density estimators developed in Part I. The likelihood ratio test and the small-signal (locally optimum) nonlinearity provide the basis for the formulation of these nonparametric detection schemes. Several modifications to these basic strategies are used to compensate for inaccuracies in the density estimates. In particular, for the problem of detecting a known signal in impulsive noise, two modifications to the standard likelihood ratio test are considered: the first is adapted from robust statistics, whereas the second, the "L/sub 1/-error-based" detector is specifically formulated for use with density estimates. Both schemes are found to perform close to the optimum likelihood ratio detector for a wide variety of impulsive noise densities. From the merits of these two tests, a new detection scheme that approximates the locally optimum nonlinearity is then developed. This detector, which uses the nonparametric density estimators developed in Part I, is shown to perform very well for the wide variety of impulsive and heavy tailed densities considered in the study. This nonparametric-density-estimate-based detector is also shown to outperform more conventional nonparametric detectors in impulsive noise.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonparametric density estimation and detection in impulsive interference channels. I. Estimators

For pt. I see ibid., vol.42, no.2-4, p.1684-1697 (1994). Nonlinear processing significantly enhances detector performance in nongaussian noise relative to that of linear detectors. Several nonparametric detection schemes for impulsive noise channels are formulated using the nonparametric probability density estimators developed in Part I. The likelihood ratio test and the small-signal (locally optimum) nonlinearity provide the basis for the formulation of these nonparametric detection schemes. Several modifications to these basic strategies are used to compensate for inaccuracies in the density estimates. In particular, for the problem of detecting a known signal in impulsive noise, two modifications to the standard likelihood ratio test are considered: the first is adapted from robust statistics, whereas the second, the L/sub 1/-error-based detector is specifically formulated for use with density estimates. Both schemes are found to perform close to the optimum likelihood ratio detector for a wide variety of impulsive noise densities. From the merits of these two tests, a new detection scheme that approximates the locally optimum nonlinearity is then developed. This detector, which uses the nonparametric density estimators developed in Part I, is shown to perform very well for the wide variety of impulsive and heavy tailed densities considered in the study. This nonparametric-density-estimate-based detector is also shown to outperform more conventional nonparametric detectors in impulsive noise. >