Wide-sense Stationary Process Research Articles

Wide-sense stationary discrete-time random processes: Analyses in the discrete-Fresnel and discrete-frequency domains

This article investigates the expectation and autocorrelation functions of random processes in the discrete-Fresnel and the discrete-frequency domains, originating from wide-sense stationary (WSS) random processes in the discrete-time domain. Through mathematical deduction, it is demonstrated that WSS random processes in the discrete-time domain exhibit the same expectation and autocorrelation functions as transformed to the discrete-Fresnel domain. In contrast, this property is not observed in the discrete-frequency domain. White, pink, and brown random processes are considered to illustrate these behaviors in the discrete-Fresnel and discrete-frequency domains. Furthermore, this study numerically compares data communication systems based on single-carrier with cyclic prefix, orthogonal chirp-division multiplexing, and orthogonal frequency-division multiplexing schemes, which are respectively designed in the discrete-time, discrete-Fresnel, and discrete-frequency domains. The numerical results show that orthogonal chirp-division multiplexing and single-carrier with cyclic prefix schemes achieve comparable communication performance under any additive noise modeled by a wide-sense stationary random process, regardless of the chosen linear equalization or the absence of it.

Characterization of minimal network structures modeling stochastic processes

Identifying the underlying structure of a network from observed data is an important problem across various disciplines. Given the general ill-posed nature of the problem, since in many cases, multiple plausible network models can explain the data, this article concentrates on characterizing classes of models providing possible explanations. Specifically, we explore linear models that can account for observed data in the form of wide-sense stationary processes accommodating the potential presence of feedback loops and direct feedthroughs. To achieve this, we leverage key insights from the theory of graphical models. In particular, we extensively employ Pearl-Verma Theorem in causal discovery which allows one to recover all minimal network structures compatible with the observed data. We adapt such a result to deal with stochastic processes and reinterpret it as a Gram-Schmidt orthogonalization procedure in a suitable Hilbert space. This reinterpretation allows us to characterize all minimal networks explaining a set of data, which have the property of not having any algebraic loops.

Combining the global trends of DFA or CDFA of different orders

When dealing with the detrended fluctuation analysis (DFA) and its variants such as the higher-order DFA and the continuous DFA (CDFA), removing the mean, integrating the signal and detrending it amount to applying an equivalent linear filter characterized by its frequency response when processing a wide-sense stationary random process. By studying the frequency response of each variant, one can better understand its behavior. Thus, in this paper, we first compare the frequency responses of two DFAs whose orders are different. Then, by deriving the higher-order CDFA using Lagrange multipliers, we can show how the frequency responses of the equivalent filters of the CDFA and the DFA are related to each other. Finally, we propose to combine the global trends obtained with the DFA or the CDFA of different orders to derive a new variant. Once again, the behavior is studied using the frequency response of the equivalent filter. Illustrations and comments on ARFIMA processes and Weierstrass functions are also given to evaluate their long range dependence.

AP-frames and stationary random processes

It is known that, in general, an AP-frame is an L 2 ( R ) -frame and conversely. Here, in part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for a Gabor system { g ( t − k ) e i l ( t − k ) , l ∈ L = ω 0 Z , k ∈ K = t 0 Z } to be an L 2 ( R ) -Frame in terms of Gaussian stationary random processes. In addition, if X = ( X ( t ) ) t ∈ R is a wide sense stationary random process, we study density conditions for the associated stationary sequences { 〈 X , g k , l 〉 , l ∈ L , k ∈ K } .

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.

Measure for degree of time variance and measure for degree of non-stationarity: application to discrete LPTV systems

Processing and analysis of time-varying systems and associated non-stationary signals are of great practical importance. The issue of quantifying the amount of time variance and amount of non-stationarity has gained importance recently. This work is also an effort in that direction. In particular, in this paper, we define two measures, one for the degree of time variance (MDTV) in the context of linear periodically time-varying systems (LPTV) and second for the degree of non-stationarity (MDNOST) of cyclostationary processes as they naturally arise when a wide-sense stationary (WSS) process passes through an LPTV system. While MDTV is a system property, on the other hand, MDNOST is a signal property. This paper also tries to find a relationship between the amount of time variation of a system with the amount of non-stationarity it generates. In this light, two concepts are analyzed for two different scenarios. First, we consider a system consisting of an upsampler followed by a linear time-invariant (LTI) filter and second a dual rate system (DRS). Both these systems are LPTV. We have shown that the MDTV generated by both LPTV systems is the same as the MDNOST of their output process through simulations and rigorous mathematical proofs.

A New Physically Motivated Clutter Model With Applications to Nondestructive Ultrasonic Testing.

In this article, we construct a new physically motivated model for the grain scattering noise (clutter). We assume the clutter is the output of a random linear time-invariant (LTI) filter, the impulse response of which is considered to be a realization of a Gaussian wide sense stationary (WSS) random process, when the transmitted ultrasonic pulse is at the input. In an ultrasonic non-destructive testing (NDT) session, the clutter noise, which is signal-dependent and caused by the microstructure of the material, presents a substantial challenge in identifying defects in the material under testing. The model is used to aid in the detection of a defect in the material. The model incorporates, explicitly, many important physical characteristics of the generated clutter, such as the average grain size, the random shape, and orientation of the grains, and emphasizes the single scatterer assumption (Rayleigh region). The statistical properties of the model are formulated and derived. A comparison with the usual matched filter to indicate the model advantage is performed. An application to real ultrasonic data is conducted with excellent results. Furthermore, we explored the effect of the choice of the model parameters, and the model shows robustness toward parameter misspecification. We then tested the performance under a deviation from the single scatterer assumption, for a more complex target, using simulated noise and obtained promising results.

ENF Signal Enhancement in Audio Recordings

In electric network frequency (ENF) based audio forensics, the ENF signal captured in a questioned audio recording is estimated and analyzed for authentication purposes. However, the captured ENF signal is usually contaminated by very strong noise and interference. In this paper, we propose a robust filtering algorithm (RFA) for ENF signal enhancement in audio recordings, which could effectively suppress the additive noise and facilitate subsequent ENF estimation, especially in practical low signal-to-noise ratio (SNR) situations. The proposed algorithm encodes the time domain expression of the preprocessed audio signal (ENF signal plus noise) as the instantaneous frequencies (IFs) of an analytical sinusoidal frequency modulated (SFM) signal. Then, a kernel function is utilized to generate a sinusoidal time-frequency distribution (STFD) whose peaks correspond to the IFs of the analytical signal, i.e., the denoised ENF signal. It is then proven that finding the STFD peaks is equivalent to finding the averaged phases of the kernel function if the additive noise is a zero mean wide sense stationary (WSS) process. The RFA serves as a noise reduction mechanism yielding improved SNR at the filter output. Combined with generic frequency estimation methods, ENF extraction accuracy could be substantially improved with the use of the proposed RFA than without using it. Both synthetic and experimental results are provided to illustrate the effectiveness of our proposal. Reliable ENF extraction could be achieved under noise level down to −20 dB SNR, and the RFA is suitable for a wide range of ENF-based forensic applications.

A modified version of the Pisarenko method to estimate the power spectral density of any asymptotically wide sense stationary vector process

The Pisarenko method estimates the power spectral density (PSD) of Gaussian wide sense stationary (WSS) 1-dimensional (scalar) processes when the PSD is Lipschitz. In this paper we modify the Pisarenko method to estimate the generating function of a sequence of block Toeplitz matrices from another sequence of matrices when both sequences are asymptotically equivalent. This modified version of the Pisarenko method allows us to estimate the PSD of any asymptotically WSS (AWSS) multidimensional (vector) process even if the process is not Gaussian and even if the PSD is continuous, but not Lipschitz.

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.

Investigating Validity of Wide-Sense Stationary Assumption in Millimeter Wave Radio Channels

In this paper, impact of antenna directivity and bandwidth on the small scale fading statistics have been analyzed for millimeter-wave (mmWave) radio channels. For this purpose, small-scale fading measurements at the mmWave frequency band (58-62 GHz) are carried out using transmit and receive antennas with different antenna directivities (emulated beamforming gains). Measurements emulate a non line-of-sight scenario when the communication between transmit and receive antennas is possible only through a single multipath cluster. In order to compare results, measurements in a line-of-sight scenario with omni-directional antennas are also carried out for reference purpose. Considering two main mmWave system features i.e., high antenna directivity and higher system bandwidth, we report the following results: 1) Randomness/fading in the received signal magnitude vanishes with an increase in bandwidth. 2) The channel impulse response h (t, τ) does not remain a wide-sense stationary (WSS) random process in the slow-time domain i.e., along t, where, the fast-time domain refers to the dimension along τ. 3) Measured channels are WSS in the frequency domain and the coherence bandwidth increases when propagation channels are illuminated with high gain antennas.

On the Papoulis Sampling Theorem: Some General Conditions

Some general conditions for multichannel sampling are established for wide sense stationary sequences with spectral density. First, necessary and sufficient conditions are given for these processes so that they are linearly determined by the samples obtained from a multichannel sampling scheme. Some results are studied for stationary sequences and then applied to the problem of sampling, not necessarily band limited, wide sense stationary processes. Conditions are also given for the existence of a frame sequence of the samples. In the case of frames, the condition that the spectral measure is absolutely continuous is proved to be necessary.

Rate Distortion Function of Gaussian Asymptotically WSS Vector Processes.

In this paper, we obtain an integral formula for the rate distortion function (RDF) of any Gaussian asymptotically wide sense stationary (AWSS) vector process. Applying this result, we also obtain an integral formula for the RDF of Gaussian moving average (MA) vector processes and of Gaussian autoregressive MA (ARMA) AWSS vector processes.

Some extensions of linear approximation and prediction problems for stationary processes

Let (B(t))t∈Θ with Θ=Z or Θ=R be a wide sense stationary process with discrete or continuous time. The classical linear prediction problem consists of finding an element in span¯{B(s),s≤t} providing the best possible mean square approximation to the variable B(τ) with τ>t.In this article we investigate this and some other similar problems where, in addition to prediction quality, optimization takes into account other features of the objects we search for. One of the most motivating examples of this kind is an approximation of a stationary process B by a stationary differentiable process X taking into account the kinetic energy that X spends in its approximation efforts.

Jeffrey's divergence between ARFIMA processes

The symmetric Kullback–Leibler divergence known as Jeffrey's divergence (JD) has found applications in signal and image processing, from radar clutter modeling to texture analysis. Recently, several studies were done on the JD between ergodic wide-sense stationary autoregressive (AR) and/or moving average (MA) processes. It was shown that the so-called asymptotic JD increment can be useful to compare ergodic wide-sense stationary ARMA processes. An interpretation of the asymptotic JD increment was also proposed. It consists in calculating the power of the first process filtered by the inverse filter associated with the second process, and conversely. However, in some biomedical applications, econometrics and other areas, long-memory processes have rather to be studied. Therefore, this paper aims at addressing the JD between ergodic wide-sense stationary autoregressive fractionally integrated moving average (ARFIMA) processes. More particularly, we study the influence of the ARFIMA parameters on the value of the asymptotic JD increment. Then, we analyze if the interpretation of the asymptotic JD increment based on inverse filtering is still valid for this type of process. Finally, some simulation results illustrate the theoretical analysis.

On the Time-Varying Non-Linearity Effects Over the Spectrum of MIMO-OFDM Wireless Communications Systems

This article presents analytical expressions that can be used in the evaluation of the effects of a non-linear mobile (multi-path) channel on the spectrum of MIMO-OFDM communications systems, such as 4G systems (LTE, WiMax) and probably the future 5G systems. The channel is modeled by a time-varying Volterra Series and the MIMO-OFDM signal by a complex, proper, wide-sense stationary Gaussian random process with zero mean. The resulting mathematical expressions are applied to a specific situation, allowing a quantitative evaluation of the out-band spectrum regrowth of MIMO-OFDM wireless communications systems produced by the use of non-linear devices in mobile channels.

Upper Bounds for the Rate Distortion Function of Finite-Length Data Blocks of Gaussian WSS Sources

In this paper, we present upper bounds for the rate distortion function (RDF) of finite-length data blocks of Gaussian wide sense stationary (WSS) sources and we propose coding strategies to achieve such bounds. In order to obtain those bounds, we previously derive new results on the discrete Fourier transform (DFT) of WSS processes.

Knowledge-based wideband radar target detection in the heterogeneous environment

In this paper, we address wideband radar target detection in the heterogeneous environment. Firstly, a linear model of the wideband radar target return with the steering vector dispersion is established. Secondly, the heterogeneous clutter is modeled as a two-dimensional wide-sense stationary (WSS) process with inverse complex Wishart distributed random covariance matrices in the time-space and frequency domain. Then, several generalized likelihood ratio test (GLRT) based detectors are designed, some of which integrate the prior knowledge of the clutter covariance matrix with the Bayesian approach, while the others are with the heuristic approach. Finally, the performance of the detectors is evaluated by simulations, and the results show that the detectors based on the Bayesian approach outperform the other detectors.

Estimation of Optimum Number of Poles for Random Signal by Yule-Walker Method

The Yule-Walker method is an effective method to estimate the system response or spectrum for random signal. Hence most of the noise and spurious signals are random in nature, so it is very convenient to estimate their spectrum by Yule-Walker method successfully. The Yule-Walker method is an autoregressive process to estimate the poles and errors also based on the number of poles for Wide Sense Stationary (WSS) process as well. Moreover the value of zero will be correspondingly calculated based on the poles in case of all poles model. The main concern of this paper is to analyze the Yule-Walker method and estimate the poles and zero along with the error based on the number of poles for a random signal. Moreover analyze the results to find out the optimum number of poles for least possible error.