Wide-sense Stationary Stochastic Processes Research Articles

Spectral Differential Privacy: Application to Smart Meter Data

We present Spectral Differential Privacy (SpDP), a novel form of differential privacy designed to protect the frequency content of time series data that come from wide sense stationary stochastic processes. This notion is motivated by privacy needs in applications with time series data over unbounded time, such as smart meters. First, a notion of differential privacy on the space of (discretized) spectral densities is introduced. A Gaussian-like mechanism for SpDP is then presented that provides differential privacy to the spectral density. Next, a novel streaming implementation is developed to enable real-time use of the proposed mechanism. The privacy guarantee provided by SpDP is independent of the time duration over which data is collected or shared. In contrast, time-domain trajectory-level differential privacy (TrDP) will require noise with large variance to provide privacy over an extended time duration. The technique is numerically evaluated using smart meter data from a single home to compare the utility of SpDP to that of time-domain trajectory-level differential privacy. The noise added by SpDP is substantially smaller than that added by time-domain TrDP, particularly when privacy over long time horizons is sought by TrDP.

Assessment of Wide-Sense Stationarity of an Underwater Acoustic Channel Based on a Pseudo-Random Binary Sequence Probe Signal

The performances of Underwater Acoustic Communication (UAC) systems are strongly related to the specific propagation conditions of the underwater channel. Designing the physical layer of a reliable data transmission system requires a knowledge of channel characteristics in terms of the specific parameters of the stochastic model. The Wide-Sense Stationary Uncorrelated Scattering (WSSUS) assumption simplifies the stochastic description of the channel, and thus the estimation of its transmission parameters. However, shallow underwater channels may not meet the WSSUS assumption. This paper proposes a method for testing the Wide-Sense Stationary (WSS) part of the WSSUS feature of a UAC channel on the basis of the complex envelope of a received probe Pseudo-Random Binary Sequence (PRBS) signal. Two correlation coefficients are calculated that can be interpreted, together, as a measure that determines whether the channel is WSS or not. A similar wide-sense stationarity assessment can be performed on the basis of the Time-Varying Impulse Response (TVIR) of a UAC channel. However, the method proposed in this paper requires fewer computational operations in the receiver of a UAC system. PRBS signal transmission tests were conducted in the UAC channel simulator and in real conditions during an inland water experiment. The correlation coefficient values obtained using the method based on the envelope of a probe signal and the method of analysing the TVIR estimates are compared. The results are similar, and thus, it is possible to assess if the UAC channel can be modelled as a WSS stochastic process without the need for TVIR estimation.

Covariance-based dissimilarity measures applied to clustering wide-sense stationary ergodic processes

We introduce a new unsupervised learning problem: clustering wide-sense stationary ergodic stochastic processes. A covariance-based dissimilarity measure together with asymptotically consistent algorithms is designed for clustering offline and online datasets, respectively. We also suggest a formal criterion on the efficiency of dissimilarity measures, and discuss an approach to improve the efficiency of our clustering algorithms, when they are applied to cluster particular type of processes, such as self-similar processes with wide-sense stationary ergodic increments. Clustering synthetic data and real-world data are provided as examples of applications.

A Unified Frequency-Domain Model for Automatic Generation Control Assessment Under Wind Power Uncertainty

High frequency uncertainty of wind power is highly stochastic and unpredictable. It can have considerable impacts on automatic generation control (AGC) for power systems. This paper proposes a novel frequency-domain model to evaluate AGC performances under wind power uncertainty in frequency domain. High frequency volatilities are accurately described by the frequency model of wide-sense stationary stochastic process. The newly introduced frequency tools enable the variations of power/energy units and network variables to be represented by the generalized transfer functions of the independent or joint stochastic processes of wind power fluctuations. The time-domain operation constraints and performance criteria of CPS1 and CPS2 are successfully transformed into frequency domain on basis of a theorem and its corollary related to the calculation of cross correlation functions proved in this paper. In general, this paper proposes a unified way of modeling volatility sources, system operations, and AGC performance assessments in frequency domain. Case studies on an isolated system and IEEE 118 interconnected system demonstrate the effectiveness and the conveniences of the proposed frequency domain model.

Assessment of sub-Nyquist deterministic and random data sampling techniques for operational modal analysis

This article assesses numerically the potential of two different spectral estimation approaches supporting non-uniform-in-time data sampling at sub-Nyquist average rates (i.e. below the Nyquist frequency) to reduce data transmission payloads in wireless sensor networks for operational modal analysis of civil engineering structures. This consideration relaxes transmission bandwidth constraints in wireless sensor networks and prolongs sensor battery life since wireless transmission is the most energy-hungry on-sensor operation. Both the approaches assume acquisition of sub-Nyquist structural response acceleration measurements and transmission to a base station without on-sensor processing. The response acceleration power spectral density matrix is estimated directly from the sub-Nyquist measurements, and structural mode shapes are extracted using the frequency-domain decomposition algorithm. The first approach relies on the compressive sensing theory to treat sub-Nyquist randomly sampled data assuming that the acceleration signals are sparse/compressible in the frequency domain (i.e. have a small number of Fourier coefficients with significant magnitude). The second approach is based on a power spectrum blind sampling technique considering periodic deterministic sub-Nyquist “multi-coset” sampling and treating the acceleration signals as wide-sense stationary stochastic processes without posing any sparsity conditions. The modal assurance criterion is adopted to quantify the quality of mode shapes derived by the two approaches at different sub-Nyquist compression rates using computer-generated signals of different sparsity and field-recorded stationary data pertaining to an overpass in Zurich, Switzerland. It is shown that for a given compression rate, the performance of the compressive sensing–based approach is detrimentally affected by signal sparsity, while the power spectrum blind sampling–based approach achieves modal assurance criterion >0.96 independently of signal sparsity for compression ratios as low as 22% the Nyquist rate. It is concluded that the power spectrum blind sampling–based approach reduces effectively data transmission requirements in wireless sensor networks for operational modal analysis, without being limited by signal sparsity and without requiring a priori assumptions or knowledge of signal sparsity.

Interrelation between the convergence rates in von Neumann’s and Birkhoff’s ergodic theorems

In the Lp spaces, 1 < p < ∞, we prove some inequalities for discrete and continuous times that make it possible to obtain the convergence rate in Birkhoff’s theorem in the presence of bounds on the convergence rate in von Neumann’s ergodic theorem belonging to a sufficiently large rate range. The exact operator analogs of these inequalities for contraction semigroups in Lp are given. These results also have the obvious exact analogs in the class of wide-sense stationary stochastic processes.

Constants in the estimates of the convergence rate in the Birkhoff ergodic theorem with continuous time

For the case of continuous time, we prove inequalities that make it possible, in the presence of power-type estimates of the convergence rate in the von Neumann ergodic theorem, to obtain estimates for the convergence rate in the Birkhoff ergodic theorem. These results have obvious exact analogs in the class of wide-sense stationary stochastic processes.

On the Prediction of a Class of Wide-Sense Stationary Random Processes

We prove that under suitable conditions, a multi-band wide sense stationary stochastic process can be linearly predicted at time with arbitrarily small error using past samples taken at uniform rate. This result generalizes previous similar results for band-limited signals. Moreover we prove that the prediction problem from uniform past samples is equivalent to a disjoint translates condition on the spectrum together with the divergence of a logarithmic integral. We also show that, for the band-limited case, under similar conditions, non uniform samples can be taken.

Approximation of Wide-Sense Stationary Stochastic Processes by Shannon Sampling Series

In this paper, the convergence behavior of the symmetric and the nonsymmetric Shannon sampling series is analyzed for bandlimited continuous-time wide-sense stationary stochastic processes that have absolutely continuous spectral measure. It is shown that the nonsymmetric sampling series converges in the mean-square sense uniformly on compact subsets of the real axis if and only if the power spectral density of the process fulfills a certain integrability condition. Moreover, if this condition is not fulfilled, then the pointwise mean-square approximation error of the nonsymmetric sampling series and the supremum of the mean-square approximation error over the real axis of the symmetric sampling series both diverge. This shows that there is a significant difference between the convergence behavior of the symmetric and the nonsymmetric sampling series.

On the constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem

We study the rate of convergence in von Neumann’s ergodic theorem. We obtain constants connecting the power rate of convergence of ergodic means and the power singularity at zero of the spectral measure of the corresponding dynamical system (these concepts are equivalent to each other). All the results of the paper have obvious exact analogs for wide-sense stationary stochastic processes.

Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series

The convergence of Laplace transforms on time scales is generalized to the bilateral case. The bilateral Laplace transform of a signal on a time scale subsumes the continuous time bilateral Laplace transform, and the discrete time bilateral z-transform as special cases. As in the unilateral case, the regions of convergence (ROCs) time scale Laplace transforms are determined by the time scale’s graininess. ROCs for the bilateral Laplace transforms of double sided time scale exponentials are determined by two modified Hilger circles. The ROC is the intersection of points external to modified Hilger circle determined by behavior for positive time and the points internal to the second modified Hilger circle determined by negative time. Since graininess lies between zero and infinity, there can exist conservative ROCs applicable for all time scales. For continuous time (ℝ) bilateral transforms, the circle radii become infinite and results in the familiar ROC between two lines parallel to the imaginary z axis. Likewise, on ℤ, the ROC is an annulus. For signals on time scales bounded by double sided exponentials, the ROCs are at least that of the double sided exponential. The Laplace transform is used to define the box minus shift through which time scale convolution can be defined. Generalizations of familiar properties of signals on ℝ and ℤ include identification of the identity convolution operator, the derivative theorem, and characterizations of wide sense stationary stochastic processes for an arbitrary time scales including autocorrelation and power spectral density expressions.

Convergence behavior of non-equidistant sampling series

The convergence of sampling series with non-equidistant sampling points cannot be guaranteed for the Paley–Wiener space PW π 1 if the class of sampling patterns is not restricted. In this paper we consider sampling patterns that are made of the zeros of sine-type functions and analyze the local and global convergence behavior of the sampling series. It is shown that oversampling is necessary for global uniform convergence. If no oversampling is used there exists for every sampling pattern a signal such that the peak value of the approximation error grows arbitrarily large. Furthermore, we use the findings to derive results about the mean-square convergence behavior of the sampling series for bandlimited wide-sense stationary stochastic processes. Finally, a procedure is given to construct functions of sine type and possible sampling patterns.

On the Relationship Between Correlation Length and Rate of Fluctuation of Random Fields

We show that the correlation length of a random field, modelled as a real wide-sense stationary stochastic process, defines a lower limit on the average squared rate of fluctuation of the field. For a sample-continuous process, the uncertainty relation between the correlation length and the spectral bandwidth exhibits a lower limit that depends explicitly on the selected correlation threshold level and implicitly on the functional form of the correlation function. We discuss implications of this result on how a selected correlation threshold relates to the estimated levels of field distortion and maximum amplitude of mode-stirred reverberation fields.

Desempenho do Algoritimo LMS na Identificação de Canais Variantes no Tempo e seu Emprego em Esquemas de Recepção MJSE-PSP

An analysis of the steady-state mean square error (MSE) performance of the Least Mean Square (LMS) algorithm in the identification of time-varying channels is presented. It is based on a set hypotheses usually adopted in the mobile communications context. The channel impulse response is modelled as time-varying transversal filter whose coefficients are wide-sense stationary stochastic processes with known power (Doppler) spectra. A generic expression of the steady state MSE as a function of the LMS step-size parameter is obtained. Two specific Doppler spectrum models are considered for which the optimization of the LMS step-size parameter in the sense of MSE minimization is also addressed. Close match between numerical and analytical results is shown in several examples. Besides, the impact of the LMS step-size parameter optimization on the performance of an adaptive MLSE (Maximum Likelihood Sequence Estimation) receiver is also addressed by Monte Carlo simulation.

On the existence of L1(R) solutions to wiener-Hopf type equations

By means of the Fourier transforms of distributions we find necessary and sufficient conditions for the existence of L1(R) solutions to Wiener-Hopf type integral equations. Thus we establish general criteria for the existence of L1(R) filters operating on the observed signal to best approximate the true signal. The theorems apply to wide sense stationary stochastic processes.

Estimation - Correlation, Modeling and Identification in Adaptive Array Processors

This paper studies the system modeling and identification issues that arise in implementation of maximum likelihood detectors and adaptive array processors of signals that have propagated over randomly time-varying propagation and scattering channels. A multidimensional estimator correlator processor is used to combine the ideas of system modeling and identification with detection theory and adaptive array processing. Part of the estimator structure can be obtained from received data by the usual adaptive array processing techniques (estimation and inversion of noise covariance matrices), but the channel output estimator requires prior knowledge of the signal part of the channel output covariance matrix. This requires modeling and identification of deterministic and stochastic propagation and scattering operators. A number of convenient matrix representations for these operators have been developed. Identification of stochastic operators is based on the vector Karhunen-Loeve expansion. It is shown that in the special case of wide-sense stationary stochastic processes and periodic signal vector, orthonormal basis and eigenvalues for channel modeling and identification can be determined by discrete Fourier transform and singular value decomposition.

Minimum information spectral analysis

The problem of obtaining an adequate spectral estimate based on the information provided by only a finite number of observations regarded as an incomplete realization of a discrete wide sense stationary stochastic process is considered. From an a priori choice in favour of either the univariate autoregressive (AR), moving average (MA) or the mixed ARMA parameterization and the availability of some estimated autocovariance lags, both recursive and non-recursive estimators are obtained by maximizing the spectral entropy. These estimators preserve minimum information subject to the unknown autocovariance lags. Moreover, the approach of maximizing the spectral entropy is shown to be equivalent to the application of a rectangular window to the cepstrum.

Walsh power spectrum for wide-sense stationary stochastic processes (Corresp.)

Existing concepts of Walsh power spectra for wide-sense stationary stochastic processes are restricted to the case of autopower spectra because they are based on real Walsh functions. In this paper a Walsh power spectrum is developed which is based on a system of complex Walsh functions and thus applies to auto and cross power spectra as well. It is shown that this Walsh power spectrum is related to the Fourier power spectrum by a linear transformation. This fact makes it possible to calculate Fourier power spectrum estimates from corresponding Walsh power spectrum estimates. An estimation algorithm for the Walsh power spectrum is given in the paper.

An algorithm for computing the cumulative distribution function for magnitude squared coherence estimates

A simple and efficient algorithm is developed for computing the conditional cumulative distribution function for estimates of the magnitude-squared coherence between two wide-sense stationary real zero-mean Gaussian stochastic processes. This algorithm is applied to computing the confidence bounds for the estimates.

On the Inverse of Some Covariance Matrices of Toeplitz Type

A matrix is said to be of Toeplitz type if it has equal elements along diagonals. These matrices, with the additional property of symmetry, arise frequently in statistical work, as covariance matrices of wide-sense stationary stochastic processes, in nonparametric theory, etc. The inverse is often of interest, and a method, is developed to find its components in close form when the $T \times T$ matrix has only $2m + 1$ nonvanishing (central) diagonals $( {1\leqq m < T} )$. The method consists in posing difference equations for the components of the inverse, and solving them explicitly. The resulting procedures reproduce known results when $m = 1$, provide an expression for the inverse when $m = 2$, provide approximations in these two important cases and an interpretation for another approximation for general m, and are shown to be particularly suited for two methods of estimation in moving average models of time series. The inverse of a related matrix is also studied.