On the behavior of large empirical autocovariance matrices between the past and the future

The asymptotic behavior of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated sequence is studied. Using Gaussian tools, it is established that the distribution behaves as a deterministic probability measure whose support [Formula: see text] is characterized. It is also established that the squared singular values are almost surely located in a neighborhood of [Formula: see text].

The Structure of Autocovariance Matrix of Discrete Time Subfractional Brownian Motion

This article explores the structure of autocovariance matrix of discrete time subfractional Brownian motion and obtains an approximation theorem and a structure theorem to the autocovariance matrix of this stochastic process. Moreover, we give an expression to the unique time varying eigenvalue of the autocovariance matrix in asymptotic means and prove that the increments of subfractional Brownian motion are asymptotic stationary processes. At last, we illustrate these results with numerical experiments and give some probable applications in finite impulse response filter.

Characterization and Simulation of Gunfire with Karhunen-Loeve Expansion

Gunfire is used as an example to illustrate how the Karhunen-Loeve (K-L) expansion can be used to characterize and simulate nonstationary random events. This paper will develop a method to describe the nonstationary random process in terms of a K-L expansion. The gunfire record is broken up into a sequence of transient waveforms, each representing the response to the firing of a single round. First, the mean is estimated and subtracted from each waveform. The mean is an estimate of the deterministic part of the gunfire. The autocovariance matrix is estimated from the matrix of these single-round gunfire records. Each column is a realization of the firing of a single round. The gunfire is characterized with the K-L expansion of the autocovariance matrix. The gunfire is simulated by generating realizations of records of a single-round firing from the expansion and the mean waveform. The individual realizations are then assembled into a realization of a time history of many rounds firing. The method is straightforward and easy to implement, and produces a simulated record very much like the original measured gunfire record.

Analysis of the Nonstationary Response of Vehicles With Multiple Wheels

Vehicles moving on rough surfaces are subject to inputs which are often conveniently regarded as random processes. In general, the excitation process is “perceived” by the vehicle as a nonstationary random process either due to inhomogeneity in the ground profile or variations in the vehicle’s velocity, or both, Hitherto this second case has not been tractable analytically due to the time variable delay between inputs. In this paper this difficulty is overcome and expressions are derived for the propagation of the mean vector and zero-lag autocovariance matrix. An example of a vehicle modelled by a bicycle configuration is discussed.