A summary of many of the new techniques developed in the last two decades for spectrum analysis of discrete time series is presented in this tutorial. An examination of the underlying time series model assumed by each technique serves as the common basis for understanding the differences among the various spectrum analysis approaches. Techniques discussed include the classical periodogram, classical Blackman-Tukey, autoregressive (maximum entropy), moving average, autotegressive-moving average, maximum likelihood, Prony, and Pisarenko methods. A summary table in the text provides a concise overview for all methods, including key references and appropriate equations for computation of each spectral estimate.

Spectrum analysis&#8212;A modern perspective

By exploiting the communication infrastructure among the sensors, actuators, and control systems, attackers may compromise the security of smart-grid systems, with techniques such as denial-of-service (DoS) attack, random attack, and data-injection attack. In this paper, we present a mathematical model of the system to study these pitfalls and propose a robust security framework for the smart grid. Our framework adopts the Kalman filter to estimate the variables of a wide range of state processes in the model. The estimates from the Kalman filter and the system readings are then fed into the $\chi^{2}$ -detector or the proposed Euclidean detector. The $\chi^{2}$ -detector is a proven effective exploratory method used with the Kalman filter for the measurement of the relationship between dependent variables and a series of predictor variables. The $\chi^{2}$ -detector can detect system faults/attacks, such as DoS attack, short-term, and long-term random attacks. However, the studies show that the $\chi^{2}$ -detector is unable to detect the statistically derived false data-injection attack. To overcome this limitation, we prove that the Euclidean detector can effectively detect such a sophisticated injection attack.

Detection of Faults and Attacks Including False Data Injection Attack in Smart Grid Using Kalman Filter

This paper presents a tutorial review of lattice structures and their use for adaptive prediction of time series Lattice filters associated with stationary covariance sequences and their properties are discussed The least squares prediction problem is defined for the given data case, and it is shown that many of the currently used lattice methods are actually approximations to the stationary least squares solution The recently developed class of adaptive least squares lattice algorithms are described in detail, both in their unnormalized and normalized forms The performance of the adaptive least squares lattice algorithm is compared to that of some gradient adaptive methods Lattice forms for ARMA processes, for joint process estimation, and for the sliding-window covariance case are presented The use of lattice structures for efficient factorization of covariance matrices and solution of Toeplitz sets of equations is briefly discussed

Lattice filters for adaptive processing

Time-Invariant Multivariable Systems with Unmeasurable Arbitrary Disturbances"

In this paper, a study of adaptive lattice algorithms as applied to channel equalization is presented. The orthogonalization properties of the lattice algorithms make them appear promising for equalizing channels which exhibit heavy amplitude distortion. Furthermore, unlike the majority of other orthogonalization algorithms, the number of operations per update for the adaptive lattice equalizers is linear with respect to the number of equalizer taps.

Channel Equalization Using Adaptive Lattice Algorithms

A test for real-time detection of soft failures in navigation systems using Kalman filters has been proposed by Kerr. The test is based on the overlap between the confidence regions associated with two estimates, one obtained from a Kalman filter using on-line measurements, and the other based solely on a priori information. An alternate computational technique is presented which is based on constructing a chi-square test statistic from the difference between the two estimates and comparing it to a precomputed threshold. The chi-square test avoids the iterative computations required by the two-ellipsoid method for dimensions of two and higher.

A Chi-square test for fault-detection in Kalman filters

An algorithm for the sequential identification of the parameters of a stationary process described by an autoregressive (AR) model is presented. A set of parameters obtained by a transformation of the AR model is introduced which leads to certain computational advantages. An example is presented to illustrate the use of the algorithms.

M. Srinath

Papers

A Chi-square test for fault-detection in Kalman filters

Sequential algorithm for identification of parameters of an autoregressive process