Classical Gaussian Noise Research Articles

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

Given a sequence \((\xi _n)\) of standard i.i.d complex Gaussian random variables, Peres and Virág (in the paper “Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process” Acta Math. (2005) 194, 1-35) discovered the striking fact that the zeros of the random power series \(f(z) = \sum _{n=1}^\infty \xi _n z^{n-1}\) in the complex unit disc \({\mathbb {D}}\) constitute a determinantal point process. The study of the zeros of the general random series f(z), where the restriction of independence is relaxed upon the random variables \((\xi _n)\) is an important open problem. This paper proves that if \((\xi _n)\) is an infinite sequence of complex Gaussian random variables, such that their covariance matrix is invertible and its inverse is a Toeplitz matrix, then the zero set of f(z) constitutes a determinantal point process with the same distribution as the case of i.i.d variables studied by Peres and Virág. The arguments are based on some interplays between Hardy spaces and reproducing kernels. Illustrative examples are constructed from classical Toeplitz matrices and the classical fractional Gaussian noise.

Physical resilience to insider attacks in IoT networks: Independent cryptographically secure sequences for DSSS anti-jamming

Wireless communication is a key technology for the Internet of Things (IoT). Due to its open nature, the physical layer of wireless systems is a high-priority target for an adversary whose goal is to disrupt the normal behavior of the system. In particular, jamming attacks are one of the most straightforward and effective types of attacks: information flow of the system is stopped or severely disturbed. In this paper, we propose a method to improve the jamming resilience of IoT systems based on Direct-Sequence Spread-Spectrum (DSSS) techniques. Our proposal is inspired by the Moving Target Defense (MTD) paradigm. MTD strategies randomize components of a system, increasing the effort an attacker needs to compromise the system. We use state-of-the-art Cryptographically Secure Pseudo-Random Number Generators outputs as spreading sequences for DSSS. The sequences of the proposed system are generated in an ad-hoc, independent, and distributed way. We show probabilistically that the generated sequences have robust cross-correlation properties. We define a multi-user system model to evaluate the Bit-Error-Rate of our proposal in the presence of two types of jammers: a classical band-limited Gaussian noise jammer, and an insider smart jammer with knowledge of one spreading sequence used in the system. Our proposal proactively mitigates the insider jammer attack. We quantify the insider smart jammer resilience of a system implementing our proposal, as a function of the length of the spreading sequences and the jammer power.

Random Fourier series with Dependent Random Variables

Given a sequence of independent standard Gaussian variables (Zn), the classical Pisier algebra P is the class of all continuous functions f on the unit circle T such that for each t ∈ 𝕋, the random Fourier series $$ {\sum}_{n\in \mathrm{\mathbb{Z}}}{Z}_n\hat{f}(n)\times \exp \left(2\pi \mathrm{i} nt\right) $$ converges in L2 and the corresponding sums constitute a Gaussian process that admits a continuous version. It was constructed by Pisier in 1979 to answer a long-standing question raised by Katznelson. In this paper, we consider the general random Fourier series $$ {\sum}_{n\in \mathrm{\mathbb{Z}}}{Z}_n\hat{f}(n)\times \exp \left(2\pi \mathrm{i} nt\right) $$ where ξ = (ξn) is a discrete Gaussian process of standard Gaussian random variables but with the restriction of independence relaxed and study the corresponding class P(ξ) of continuous functions f on 𝕋. We obtain sufficient conditions (based on some spectral properties of the covariance matrix of (ξn)) for each of the relations 𝒫 ⊂ (ξ), 𝒫(ξ) ⊂ 𝒫, and 𝒫 = 𝒫(ξ). We illustrate these results by the classical fractional Gaussian noise. Whether in general (ξ) is also a Banach algebra is an open problem.

Zeros of Gaussian power series with dependent random variables

Peres and Virág in 2005 studied the zeros of the random power series ∑ n = 1 ∞ a n z n − 1 in the unit disc whose coefficients ( a n ) are independent and identically distributed symmetrical Gaussian complex random variables. They discovered that its zeros constitute a determinant point process and derived many other interesting related properties. In this paper, we study the general Gaussian power series where the independence requirement is relaxed by taking f ( z ) = ∑ n = 1 ∞ Δ n z n , where ( Δ n ) is the classical fractional Gaussian noise of index 0 ≤ H < 1 . The intensity of the point process of the zeros of f is used to derive important properties. As for the case of independent variables (corresponding to H = 1 / 2 ), the zeros cluster around the unit circle, but in any domain inside the open unit disc, the dependent case (that is, H ≠ 1 / 2 ) yields fewer zeros than the classical case of independent random variables. This implies that the clustering around the unit circle is faster than in the independent case. Moreover, in contrast to independent random variables, the zeros of f are generally asymmetrically distributed in the unit circle. However, asymptotically, the two point processes are equivalent except near the sole point z = 1 . An interesting open problem is whether this point process is also a determinantal process.

Moment-Based Spectrum Sensing Under Generalized Noise Channels

A new spectrum sensing detector is proposed and analytically studied, when it operates under generalized noise channels. Particularly, the McLeish distribution is used to model the underlying noise, which is suitable for both non-Gaussian (impulsive) as well as classical Gaussian noise modeling. The introduced detector adopts a moment-based approach, whereas it is not required to know the transmit signal and channel fading statistics (i.e., blind detection). Important performance metrics are presented in closed forms, such as the false-alarm probability, detection probability and decision threshold. Analytical and simulation results are cross-compared validating the accuracy of the proposed approach. Finally, it is demonstrated that the proposed approach outperforms the conventional energy detector in the practical case of noise uncertainty, yet introducing a comparable computational complexity.

Generalized Energy Detection Under Generalized Noise Channels

Generalized energy detection (GED) is analytically studied when operates under fast-faded channels and in the presence of generalized noise. For the first time, the McLeish distribution is used to model the underlying noise, which is suitable for both non-Gaussian (impulsive) as well as classical Gaussian noise channels. Important performance metrics are presented in closed forms, such as the false-alarm and detection probabilities as well as the decision threshold. Analytical and simulation results are cross-compared validating the accuracy of the proposed approach in the entire signal-to-noise ratio regime. Finally, useful outcomes are extracted with respect to GED system settings under versatile noise environments and when noise uncertainty is present.

The maximum likelihood climate change for global warming under the influence of greenhouse effect and Lévy noise.

An abrupt climatic transition could be triggered by a single extreme event, and an α-stable non-Gaussian Lévy noise is regarded as a type of noise to generate such extreme events. In contrast with the classic Gaussian noise, a comprehensive approach of the most probable transition path for systems under α-stable Lévy noise is still lacking. We develop here a probabilistic framework, based on the nonlocal Fokker-Planck equation, to investigate the maximum likelihood climate change for an energy balance system under the influence of greenhouse effect and Lévy fluctuations. We find that a period of the cold climate state can be interrupted by a sharp shift to the warmer one due to larger noise jumps with low frequency. Additionally, the climate change for warming 1.5°C under an enhanced greenhouse effect generates a steplike growth process. These results provide important insights into the underlying mechanisms of abrupt climate transitions triggered by a Lévy process.

Landau–Zener transitions in coupled qubits: Effects of coloured noise

Abstract The feasibility of stochastic Hamiltonians for interacting qubits to deliver consistent quantum speedups in quantum adiabatic optimisation remains an open question. This paper studies theoretically the noisy Landau–Zener model of a system of two interacting qubits. Exact analytic results are obtained for all transition probabilities via nonstationary perturbation technique, even where semiclassical trajectories interfere. Where the system is driven by slow classical Gaussian noise, destructive interference of the trajectories results in forbidden transitions. However, in the fast noise regime, these transitions have non-zero probabilities. We elucidate the roles of noise parameters on the decoherence of coupled qubits.

Noise-induced multilevel Landau–Zener transitions: Density matrix investigation

The generalised multilevel Landau–Zener problem is solved by applying the density matrix technique within the framework of nonstationary perturbation theory. The exact survival probability is achieved as a proof of the Brundobler–Elzer hypothesis (Brundobler and Elzer (1993) [38]). The effect of classical Gaussian noise is investigated by averaging the solution over the noise realisation. A generalised formula for slow noise-induced transition probability is obtained and found to agree exactly with all known results. Exact results are reported for the Demkov–Osherov model in the slow and fast noise limits. Thermal transition probabilities are obtained via the activation Arrhenius law and observed to tailor a qubit from thermal decoherence.

Decoherence of two entangled spin qubits coupled to an interacting sparse nuclear spin bath: Application to nitrogen vacancy centers

We consider pure dephasing of Bell states of electron spin qubits interacting with a sparse bath of nuclear spins. Using the newly developed two-qubit generalization of cluster correlation expansion method, we calculate the spin echo decay of $|\Psi\rangle$ and $|\Phi\rangle$ states for various interqubit distances. Comparing the results with calculations in which dephasing of each qubit is treated independently, we identify signatures of influence of common part of the bath on the two qubits. At large interqubit distances, this common part consists of many nuclei weakly coupled to both qubits, so that decoherence caused by it can be modeled by considering multiple uncorrelated sources of noise (clusters of nuclei), each of them weakly affecting the qubits. Consequently, the resulting genuinely two-qubit contribution to decoherence can be described as being caused by classical Gaussian noise. On the other hand, for small interqubit distances the common part of the environment contains clusters of spins that are strongly coupled to both qubits, and their contribution to two-qubit dephasing has visibly non-Gaussian character. We show that one van easily obtain information about non-Gaussianity of environmental noise affecting the qubits from the comparison of dephasing of $|\Psi\rangle$ and $|\Phi\rangle$ Bell states. Numerical results are obtained for two nitrogen vacancy centers interacting with a bath of $^{13}$C nuclei of natural concentration, for which we obtain that Gaussian description of correlated part of environmental noise starts to hold for centers separated by about 3 nm.

Robust Calibration of Radio Interferometers in Non-Gaussian Environment

The development of new phased array systems in radio astronomy, as the low frequency array (LOFAR) and the square kilometre array (SKA), formed of a large number of small and flexible elementary antennas, has led to significant challenges. Among them, model calibration is a crucial step in order to provide accurate and thus meaningful images and requires the estimation of all the perturbation effects introduced along the signal propagation path, for a specific source direction and antenna position. Usually, it is common to perform model calibration using the a priori knowledge regarding a small number of known strong calibrator sources but under the assumption of Gaussianity of the noise. Nevertheless, observations in the context of radio astronomy are known to be affected by the presence of outliers which are due to several causes, e.g., weak non-calibrator sources or man made radio frequency interferences. Consequently, the classical Gaussian noise assumption is violated leading to severe degradation in performances. In order to take into account the outlier effects, we assume that the noise follows a spherically invariant random distribution. Based on this modeling, a robust calibration algorithm is presented in this paper. More precisely, this new scheme is based on the design of an iterative relaxed concentrated maximum likelihood estimation procedure which allows to obtain closed-form expressions for the unknown parameters with a reasonable computational cost. This is of importance as the number of estimated parameters depends on the number of antenna elements, which is large for the new generation of radio interferometers. Numerical simulations reveal that the proposed algorithm outperforms the state-of-the-art calibration techniques.

Classical counterparts of quantum attractors in generic dissipative systems.

In the context of dissipative systems, we show that for any quantum chaotic attractor a corresponding classical chaotic attractor can always be found. We provide a general way to locate them, rooted in the structure of the parameter space (which is typically bidimensional, accounting for the forcing strength and dissipation parameters). In cases where an approximate pointlike quantum distribution is found, it can be associated with exceptionally large regular structures. Moreover, supposedly anomalous quantum chaotic behavior can be very well reproduced by the classical dynamics plus Gaussian noise of the size of an effective Planck constant ℏ_{eff}. We give support to our conjectures by means of two paradigmatic examples of quantum chaos and transport theory. In particular, a dissipative driven system becomes fundamental in order to extend their validity to generic cases.

Non-Markovian dynamics of quantum open systems embedded in a hybrid environment

Quantum systems of interest are typically coupled to several quantum channels (more generally environments). In this paper, we develop an exact stochastic Schrödinger equation for an open quantum system coupled to a hybrid environment containing both bosonic and fermionic particles. Such a stochastic differential equation may be obtained directly from a microscopic model through employing a classical complex Gaussian noise and a non-commutative fermionic noise to simulate the hybrid bath. As an immediate application of our developed stochastic approach, we show that the evolution of the reduced density matrix can be derived by taking the average over both the bosonic noise and the fermionic noise. Three specific examples are given in this paper to illustrate that the hybrid quantum trajectory is fully consistent with the standard quantum mechanics. Our examples also shed new light on the special features exhibited by the fermionic bath and bosonic bath.

Bayesian inference of earthquake parameters from buoy data using a polynomial chaos-based surrogate

This work addresses the estimation of the parameters of an earthquake model by the consequent tsunami, with an application to the Chile 2010 event. We are particularly interested in the Bayesian inference of the location, the orientation, and the slip of an Okada-based model of the earthquake ocean floor displacement. The tsunami numerical model is based on the GeoClaw software while the observational data is provided by a single DARTⓇ buoy. We propose in this paper a methodology based on polynomial chaos expansion to construct a surrogate model of the wave height at the buoy location. A correlated noise model is first proposed in order to represent the discrepancy between the computational model and the data. This step is necessary, as a classical independent Gaussian noise is shown to be unsuitable for modeling the error, and to prevent convergence of the Markov Chain Monte Carlo sampler. Second, the polynomial chaos model is subsequently improved to handle the variability of the arrival time of the wave, using a preconditioned non-intrusive spectral method. Finally, the construction of a reduced model dedicated to Bayesian inference is proposed. Numerical results are presented and discussed.

The Foraging Brain: Evidence of Lévy Dynamics in Brain Networks.

In this research we have analyzed functional magnetic resonance imaging (fMRI) signals of different networks in the brain under resting state condition. To such end, the dynamics of signal variation, have been conceived as a stochastic motion, namely it has been modelled through a generalized Langevin stochastic differential equation, which combines a deterministic drift component with a stochastic component where the Gaussian noise source has been replaced with α-stable noise. The parameters of the deterministic and stochastic parts of the model have been fitted from fluctuating data. Results show that the deterministic part is characterized by a simple, linear decreasing trend, and, most important, the α-stable noise, at varying characteristic index α, is the source of a spectrum of activity modes across the networks, from those originated by classic Gaussian noise (α = 2), to longer tailed behaviors generated by the more general Lévy noise (1 ≤ α < 2). Lévy motion is a specific instance of scale-free behavior, it is a source of anomalous diffusion and it has been related to many aspects of human cognition, such as information foraging through memory retrieval or visual exploration. Finally, some conclusions have been drawn on the functional significance of the dynamics corresponding to different α values.

Classical nature of nuclear spin noise near clock transitions of Bi donors in silicon

Whether a quantum bath can be approximated as classical noise is a fundamental issue in central spin decoherence and also of practical importance in designing noise-resilient quantum control. Spin qubits based on bismuth donors in silicon have tunable interactions with nuclear spin baths and are first-order insensitive to magnetic noise at so-called clock-transitions (CTs). This system is therefore ideal for studying the quantum/classical nature of nuclear spin baths since the qubit-bath interaction strength determines the back-action on the baths and hence the adequacy of a classical noise model. We develop a Gaussian noise model with noise correlations determined by quantum calculations and compare the classical noise approximation to the full quantum bath theory. We experimentally test our model through dynamical decoupling sequence of up to 128 pulses, finding good agreement with simulations and measuring electron spin coherence times approaching one second - notably using natural silicon. Our theoretical and experimental study demonstrates that the noise from a nuclear spin bath is analogous to classical Gaussian noise if the back-action of the qubit on the bath is small compared to the internal bath dynamics, as is the case close to CTs. However, far from the CTs, the back-action of the central spin on the bath is such that the quantum model is required to accurately model spin decoherence.

Effects of colored noise on Landau-Zener transitions: Two- and three-level systems

We investigate the Landau-Zener transition in two- and three- level systems subject to a classical Gaussian noise. Two complementary limits of the noise being fast and slow compared to characteristic Landau-Zener tunnel times are discussed. The analytical solution of a density matrix (Bloch) equation is given for a long time asymptotic of transition probability. It is demonstrated that the transition probability induced/assisted by the fast noise can be obtained through a procedure of {\it Bloch's equation averaging} with further reduction it to a master equation. In contrast to the case of fast noise, the transition probability for LZ transition induced/assisted by the slow classical noise can be obtained by averaging a {\it solution} of Bloch's equation over the noise realization. As a result, the transition probability is described by the activation Arrhenius law. The approximate solution of the Bloch's equation at finite times is written in terms of Fresnel's integrals and interpreted in terms of interference pattern. We discuss consequences of a local isomorphism between SU(2) and SO(3) groups and connections between Schr\"odinger and Bloch descriptions of spin dynamics. Based on this isomorphism we establish the relations between S=1/2 and S=1 transition probabilities influenced by the noise. A possibility to use the slow noise as a probe for tunnel time is discussed.

The non validity of the gaussian approximation for multi-user interference in ultra wide band impulse radio: from an inconvenience to an advantage - transactions papers

The huge potential of ultra wide band impulse radio (UWB IR) with multiple access was initially evaluated using the standard Gaussian approximation which assumed that multi user interference could be approximated as classical Gaussian noise. Later the standard Gaussian approximation has been proved to be invalid in several cases. Thus the performance of the classical UWB IR correlation receiver dramatically reduces. We show in this paper how the non Gaussianity of the multi user interference can be exploited, turning the non Gaussianity into an advantage, increasing the performance with respect to the Gaussian interference case which was supposed to be too optimistic until now.

Multiple-Access Interference Plus Noise-Constrained Least Mean Square (MNCLMS) Algorithm for CDMA Systems

Since multiuser code-division multiple-access (CDMA) communications systems suffer significantly from multiple-access interference (MAI) and from classical white Gaussian noise, it is therefore necessary to consider their impact on the performance of these systems. It is well known that the learning speed of any adaptive filtering algorithm is increased by adding a constraint to it. In this paper, a constrained least-mean-square (LMS) algorithm, which incorporates the knowledge of the number of users, spreading sequence length, and additive noise variance, is developed subject to the new combined constraint comprising the MAI and noise variance for a synchronous downlink direct-sequence CDMA system. The novelty of this constraint resides in the fact that the MAI variance was never used as a constraint. In our approach, a Robbins-Monro algorithm is used to minimize the conventional mean-square-error criterion subject to the variance of the new constraint (MAI plus noise). This constrained optimization technique results in an (MAI plus noise)-constrained LMS (MNCLMS) algorithm. The MNCLMS algorithm is a type of variable step-size LMS algorithm where the step-size rule arises naturally from the constraints on MAI and noise variance. Convergence and tracking analysis of the proposed algorithm are carried out in the presence of MAI. Finally, a number of simulations are conducted to compare the performance of the MNCLMS algorithm with other adaptive algorithms.

Measuring a linear approximation to weakly nonlinear MIMO systems

The paper addresses the problem of preserving the same LTI approximation of a nonlinear MIMO (multiple-input multiple-output) system. It is shown that when a nonlinear MIMO system is modeled by a multidimensional Volterra series, periodic noise and random multisines are equivalent excitations to the classical Gaussian noise, in a sense that they yield in the limit, as the number of the harmonics M → ∞ , the same linear approximation to the nonlinear MIMO system. This result extends previous results derived for nonlinear SISO (single-input single-output) systems. Based upon the analysis of the variability of the measured FRF (frequency response function) due to the presence of the nonlinearities and the randomness of the excitations, a new class of equivalent input signals is proposed, allowing for a lower variance of the nonlinear FRF measurements, while the same linear approximation is retrieved.