Nonstationary Input Research Articles

Detectable and undetectable input signals for the integrate-and-fire model

We consider the integrate-and-fire model with non-stationary, stochastic inputs and address the following issue: what are the conditions on the input currents that make the input signal undetectable? A novel theoretical approach to tackle the problem for the model with non-stationary inputs is introduced. When the noise strength is independent of the deterministic component of the synaptic input, an expression for the critical input signal is given. If the input signal is weaker than the critical input signal, the neuron ultimately stops firing, i.e. is not able to detect the input signal; otherwise it fires with probability one. Similar results are established for Poisson type inputs where the strength of the noise is proportional to the deterministic component of the synaptic input.

Gaussian and non-Gaussian stochastic sensitivity analysis of discrete structural system

The derivatives of the response of a structural system with respect to the system parameters are termed sensitivities. They play an important role in assessing the effect of uncertainties in the mathematical model of the system and in predicting changes of the response due to changes of the design parameters. In this paper, a time domain approach for evaluating the sensitivity of discrete structural systems to deterministic, as well as to Gaussian or non-Gaussian stochastic input is presented. In particular, in the latter case, the stochastic input has been assumed to be a delta-correlated process and, by using Kronecker algebra extensively, cumulant sensitivities of order higher than two have been obtained by solving sets of algebraic or differential equations for stationary and non-stationary input, respectively. The theoretical background is developed for the general case of multi-degrees-of-freedom (MDOF) primary system with an attached secondary single-degree-of-freedom (SDOF) structure. However, numerical examples for the simple case of an SDOF primary–secondary structure, in order to explore how variations of the system parameters influence the system, are presented. Finally, it should be noted that a study of the optimal placement of the secondary system within the primary one should be conducted on an MDOF structure.

A robust, parallelizable, O(m), a posteriori recursive least squares algorithm for efficient adaptive filtering

This article presents a new recursive least squares (RLS) adaptive algorithm. The proposed computational scheme uses a finite window by means of a lemma for the system matrix inversion that is, for the first time, stated and proven here. The new algorithm has excellent tracking capabilities. Moreover, its particular structure allows for stabilization by means of a quite simple method. Its stabilized version performs very well not only for a white noise input but also for nonstationary inputs as well. It is shown to follow music, speech, environmental noise, etc., with particularly good tracking properties. The new algorithm can be parallelized via a simple technique. Its parallel form is very fast when implemented with four processors.

Improved Representation-burden Conservation Network for LearningNon-stationary VQ

In a recent publication [1], it was shown that a biologically plausible RCN (Representation-burden Conservation Network) in which conservation is achieved by bounding the summed representation-burden of all neurons at constant 1, is effective in learning stationary vector quantization. Based on the conservation principle, a new approach for designing a dynamic RCN for processing both stationary and non-stationary inputs is introduced in this paper. We show that, in response to the input statistics changes, dynamic RCN improves its original counterpart in incremental learning capability as well as in self-organizing the network structure. Performance comparisons between dynamic RCN and other self-development models are also presented. Simulation results show that dynamic RCN is very effective in training a near-optimal vector quantizer in that it manages to keep a balance between the equiprobable and equidistortion criterion.

TRANSVERSE VIBRATION OF MULTI MASS LOADED VARIABLE SECTION BEAMS UNDER RANDOM EXCITATION

Dynamic analysis of non-uniform section beams with general end conditions having arbitrary mass and stiffness distributions and proportional viscous damping distribution with several arbitrarily located point mass loadings has been presented for non-stationary random input excitation. Modal analysis has been employed to obtain expressions for second order response statistics after decoupling the system equations. The natural frequencies and mode shape functions for such beams are obtained in closed form as an intermediate step in the outlined approach. Theoretical results for natural frequencies have been compared with experimental evaluations and with data available in the literature. The complete approach is illustrated with an example of a cantilever beam subjected to support excitation.

Identification of non-minimum phase systems using evolutionary spectral theory

In this paper we present an application of the Wold-Cramer representation of non-stationary processes to the identification of non-minimum phase linear time-invariant (LTI) systems and modeling of non-stationary signals. According to the Wold-Cramer representation, a non-stationary signal can be expressed as an infinite sum of sinusoids with time-varying random magnitudes and phases. For the identification of LTI systems with non-stationary input and output, we relate the Wold-Cramer representations of the system's input and output to obtain estimates of the magnitude and phase frequency responses of the system. Our procedure uses estimators from the evolutionary periodogram recently introduced, permits the identification of non-minimum phase systems, and provides also rational estimates. Furthermore, we show that when the system output is corrupted by additive stationary noise it is possible to avoid the effects of the noise in the identification. If the noise is non-stationary and Gaussian, we show that it can be removed by considering the evolutionary bispectrum, recently introduced. The analysis also provides a model for a non-stationary signal, as the output of the cascade of a linear time-varying (LTV) and an LTI systems with white noise as input. To illustrate the performance of our procedure, we present simulations of the identification of non-minimum phase LTI systems with noisy outputs.

Block transmission over dispersive channels: transmit filter optimization and realization, and MMSE-DFE receiver performance

Optimal transmit filters for packet-based data transmission on dispersive Gaussian-noise linear time-invariant channels are derived by maximizing the mutual information, subject to a fixed input power budget. A quasi-stationary approximation to the optimal nonstationary input covariance process is derived and shown to exhibit negligible mutual information loss from the optimal case, for situations of most practical interest. Moreover, this quasi-stationary approximation results in efficiently computed lattice or pole-zero implementations of the transmit filter. By considering the popular finite-impulse-response minimum-mean-square-error decision-feedback equalizer (FIR MMSE-DFE) as a receiver structure, we show that transmitter optimization results in an appreciable improvement in the decision-point signal-to-noise ratio. Finally, we show that, as the output blocklength becomes infinite, the optimum finite-dimensional nonstationary input covariance process converges to a stationary process whose power spectrum obeys the well-known water-pouring distribution.

Sensitivity to the Service-Time Distribution in the Nonstationary Erlang Loss Model

The stationary Erlang loss model is a classic example of an insensitive queueing system: the steady-state distribution of the number of busy servers depends on the service-time distribution only through its mean. However, when the arrival process is a nonstationary Poisson process, the insensitivity property is lost. We develop a simple, effective numerical algorithm for the Mt/PH/s/0 model with two service phases and a nonhomogeneous Poisson arrival process, and apply it to show that the time-dependent blocking probability with nonstationary input can be strongly influenced by the service-time distribution beyond the mean. With sinusoidal arrival rates, the peak blocking probability typically increases as the service-time distribution gets less variable. The influence of the service-time distribution, including this seemingly anomalous behavior, can be understood and predicted from the modified-offered-load and stationary-peakedness approximations, which exploit exact results for related infinite-server models.

Behavior of the partial correlation coefficients of a least squares lattice filter in the presence of a nonstationary chirp input

This paper studies the performance of the a posteriori recursive least squares lattice filter in the presence of a nonstationary chirp signal. The forward and backward partial correlation (PARCOR) coefficients for a Wiener-Hopf optimal filter are shown to be complex conjugates for the general case of a nonstationary input with constant power. Such an optimal filter is compared to a minimum mean square error based least squares lattice adaptive filter. Expressions are found for the behavior of the first stage of the adaptive filter based on the least squares algorithm. For the general nth stage, the PARCOR coefficients of the previous stages are assumed to have attained Wiener-Hopf optimal steady state. The PARCOR coefficients of such a least squares adaptive filter are compared with the optimal coefficients for such a nonstationary input. The optimal lattice fitter is seen to track a chirp input without any error, and the tracking lag in such an adaptive filter is due to the least squares update procedure. The expression for the least squares based PARCOR coefficients are found to contain two terms: a decaying convergence term due to the weighted estimation procedure and a tracking component that asymptotically approaches the optimal coefficient value. The rate of convergence is seen to depend inversely on the forgetting factor. The tracking lag of the filter is derived as a function of the rate of nonstationarity and the forgetting factor. It is shown that for a given chirp rate there is a threshold adaptation constant below which the total tracking error is negligible. For forgetting factors above this threshold, the error increases nonlinearly. Further, this threshold forgetting factor decreases with increasing chirp rate. Simulations are presented to validate the analysis. >

Blind Deconvolution of Linear Systems with Multilevel Nonstationary Inputs

A method is proposed to deal with the problem of blind deconvolution of a special non-Gaussian linear process, in which the input to the linear system is a real- or complex-valued multilevel random sequence that satisfies certain regularity conditions. The gist of the method is to apply a linear filter to the observed process and adjust the filter until a multilevel output is obtained. It is shown that the deconvolution problem can be solved (with only scale/rotation and shift ambiguities) if the output sequence of the filter contains a subsequence that converges weakly to a multilevel random variable. A cost function is proposed so that any minimizer of the cost function provides a solution to the deconvolution problem. Moreover, when the process is parametric, it is shown that a consistent estimator of the parameter can be obtained by minimizing an empirical criterion. The estimation accuracy is shown to depend on the tail behavior of the inverse system, which in many cases decays exponentially as the sample size grows. Special consideration is given to applications in the equalization of digital communications systems.

The nonstationary effects on a softening duffing oscillator

This paper presents a numerical study for the bifurcations of a softening Duffing oscillator subjected to stationary and nonstationary excitation. The nonstationary inputs used are linear functions of time. The bifurcations are the results of either a single control parameter or two control parameters that are constrained to vary in a selected direction on the plane of forcing amplitude and forcing frequency. The results indicate: 1. Delay (memory, penetration) of nonstationary bifurcations relative to stationary bifurcations may occur. 2. The nonstationary trajectories jump into the neighboring stationary trajectories with possible overshoots, while the stationary trajectories transit smoothly. 3. The nonstationary penetrations (delays) are compressed to zero with an increasing number of iterations. 4. The nonstationary responses converge through a period-doubling sequence to a nonstationary limit motion that has the characteristics of chaotic motion. The Duffing oscillator has been used as an example of the existence of broad effects of nonstationary (time dependent) and codimensional (control parameter variations in the bifurcation region) inputs which markedly modify the dynamical behavior of dynamical systems.

Fault Detection of Dynamical Systems with Nonstationary Inputs

The paper presents a practical fault detection method for dynamical systems with some arbitrary and nonstationary inputs. The method implies an impulse invariant transformation for obtaining the equivalent system having as input a stationary uncorrelated sequence, and as output the response for this input. The transformation uses the AR filters of the quasi-stationary data blocks of the original input, resulted by a segmentation procedure. A similar procedure is used to segment the processed output in quasi-stationary data blocks. Each transition from one data block to another marks a change in the dynamic properties of the system. The method is applied for detection of changes in dynamic characteristics of a vibrating structure subject to a transient natural excitation.

An on-line unsupervised learning machine for adaptive feature extraction

Adaptive feature extraction is useful in many information processing systems. This paper proposes a learning machine implemented via a neural network to perform such a task using the tool principal component analysis. This machine (1) is adaptive to nonstationary input, (2) is based on an unsupervised learning concept and requires no knowledge of if, or when, the input changes statistically, and (3) performs online computation that requires little memory or data storage. Associated with this machine, the authors propose a learning algorithm (LEAP), whose convergence properties are theoretically analyzed and whose performance is evaluated via computer simulations. Two major contributions of this paper are: (1) Under appropriate conditions, the authors prove that the algorithm will extract multiple principal components, when the learning rate is constant; and (2) they identify a near optimal domain of attraction.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

ESTIMATION AND BLIND DECONVOLUTION OF AUTOREGRESSIVE SYSTEMS WITH NONSTATIONARY BINARY INPUTS

Abstract. The problem of parameter estimation and blind deconvolution of auto‐regressive (AR) systems with independent nonstationary binary inputs is considered. The estimation procedure consists of applying a moving‐average filter (equalizer) to the observed data and adjusting the parameters of the filter so as to minimize a criterion that measures the binariness of its output. The output sequence itself serves as an estimate of the unobservable binary input of the AR system. Without assuming stationarity of the inputs, it is shown that the proposed method produces a consistent estimator of the AR system not only in the sense of converging to the true parameter as the sample size increases, but also in the sense of attaining the true parameter of the AR system for a sufficiently large sample size. For noisy data, the estimation criterion is modified on the basis of an asymptotic analysis of the effect of the noise. It is shown that the modified criterion is also consistent (in the usual sense) and its variability depends upon the filtered noise. Some simulation results are presented to demonstrate the performance of the proposed method for parameter estimation as well as for blind deconvolution.

Nonstationary Response of Structures with Closely Spaced Frequencies

The response statistics are examined for linear structures subjected to nonstationary stochastic input. Modal expansions are used and attention is on the mean square, mean zero‐upcrossing rate, and bandwidth of the response. The analysis begins with an exact derivation of the covariances of the response and its Hilbert transform in terms of stationary statistics. The resulting expressions are found to be lengthy double integrals. Then, approximations for these statistics are developed for lightly damped structures with closely spaced natural frequencies. These approximations, which are in terms of simple convolution integrals, are generalizations of previous results for oscillator response. It is found that the expressions for the modal cross‐covariances have a different form from those of the modal auto‐covariances. It is also shown that the nonstationary responses can have time‐varying mean zero‐upcrossing rates and mean‐square responses that are highly dependent on the modal participation factors. A co...

Blind identification and deconvolution of linear systems driven by binary random sequences

The problem of blind identification and deconvolution of linear systems with independent binary inputs is addressed. To solve the problem, a linear system is applied to the observed data and adjusted so as to produce binary outputs. It is proved that the system coincides with the inverse of the unknown system (with scale and shift ambiguities), whether it is minimum or nonminimum phase. These results are derived for nonstationary independent binary inputs of infinite or finite length. Based on these results, an identification method is proposed for parametric linear systems. It is shown that under some mild conditions, a consistent estimator of the parameter can be obtained by minimizing a binariness criterion for the output data. Unlike many other blind identification and deconvolution methods, this criterion handles nonstationary signals and does not utilize any moment information of the inputs. Three numerical examples are presented to demonstrate the effectiveness of the proposed method. >

Synchronised linear almost periodically time-varying adaptive filter

Conventional adaptive filters only suit stationary and slowly time-varying nonstationary inputs. Based on the previous work on linear almost periodically time-varying adaptive filtering, the paper develops such an adaptive filter adopting the output-modulator structure. In addition to being capable of dealing with almost cyclostationary inputs (an important class of nonstationary signals), this scheme is also able, by means of adaptive frequency tracking, to remain synchronous with the varying input statistics.

A Simple Optimization Technique of Vibration Controller for Mechanical Structures Subjected to Nonstationary Input.

In this study, the optimization problem of dynamical characteristic of vibration controllers for mechanical structures subjected to such a nonstationary input as seismic wave is dealt with. A simple optimization technique based upon the stationary design and the modal analysis is presented. In order to take into account the nonstationary influence in the response, the stationary criterion is modified in each mode by the ratio of the response variance of each mode to the square of the response spectrum: By applying this technique to the optimization problem of the allocation and the dynamical characteristics of a dynamical vibration absorber for a piping system, its usefulness is verified.

Cross-correlation of system outputs-ambiguity function

Outputs of disjoint linear systems may be correlated for a nonstationary input process. For a nonstationary white input, the output cross-correlation function is simply characterized in terms of the generalized ambiguity function. An example is given for passive radiowave measurements with multiple individual sensors, in which case the ambiguity function is the spatial inverse Fourier transform of the aperture field cross-correlation. >

Sensitivity analysis of yield level for seismic response of elastoplastic structure.

From the past research, it is known that the maximum rms relative displacement of an elastoplastic structure subjected to a nonstationary input has a minimum value at an appropriate yielding level owing to the energy absorption. In order to obtain the optimal yielding level, this paper presents a sensitivity analysis of the yielding level for the seismic response of elasto plastic structures. Herein the sensitivity analysis is to find the sensitivity coefficient of the yielding level to the maximum rms relative displacement. By using these sensitivity coefficients, the probability distribution of the rms relative displacement can be evaluated from that of the yielding level of the structure. The seismic response of the structure was studied assuming that the restoring force characteristics of structure are represented by a simplified Clough model with the stiffness degradation, and the covariance of responses were computed by using an equivalent linearization technique on the assumption of narrow band response. The theoretical results were compared with those of digital computer simulation.