On causal extrapolation of sequences with applications to forecasting

The paper suggests a method of extrapolation of notion of one-sided semi-infinite sequences representing traces of two-sided band-limited sequences; this features ensure uniqueness of this extrapolation and possibility to use this for forecasting. This lead to a forecasting method for more general sequences without this feature based on minimization of the mean square error between the observed path and a predicable sequence. These procedure involves calculation of this predictable path; the procedure can be interpreted as causal smoothing. The corresponding smoothed sequences allow unique extrapolations to future times that can be interpreted as optimal forecasts.Comment: arXiv admin note: substantial text overlap with arXiv:1111.670

Sampling from a system-theoretic viewpoint: Part II - Noncausal solutions

This paper puts to use concepts and tools introduced in Part I to address a wide spectrum of noncausal sampling and reconstruction problems. Particularly, we follow the system-theoretic paradigm by using systems as signal generators to account for available information and system norms (L2 and L∞) as performance measures. The proposed optimization-based approach recovers many known solutions, derived hitherto by different methods, as special cases under different assumptions about acquisition or reconstructing devices (e.g., polynomial and exponential cardinal splines for fixed samplers and the Sampling Theorem and its modifications in the case when both sampler and interpolator are design parameters). We also derive new results, such as versions of the Sampling Theorem for downsampling and reconstruction from noisy measurements, the continuous-time invariance of a wide class of optimal sampling-and-reconstruction circuits, etcetera

Sampling from a system-theoretic viewpoint

This paper studies a system-theoretic approach to the problem of reconstructing an analog signal from its samples. The idea, borrowed from earlier treatments in the control literature, is to address the problem as a hybrid model-matching problem in which performance is measured by system norms. \ud \ud The paper is split into three parts. In Part I we present the paradigm and revise the lifting technique, which is our main technical tool. In Part II optimal samplers and holds are designed for various analog signal reconstruction problems. In some cases one component is fixed while the remaining are designed, in other cases all three components are designed simultaneously. No causality requirements are imposed in Part II, which allows to use frequency domain arguments, in particular the lifted frequency response as introduced in Part I. In Part III the main emphasis is placed on a systematic incorporation of causality constraints into the optimal design of reconstructors. We consider reconstruction problems, in which the sampling (acquisition) device is given and the performance is measured by the $L^2$-norm of the reconstruction error. The problem is solved under the constraint that the optimal reconstructor is $l$-causal for a given $l\geq 0,$ i.e., that its impulse response is zero in the time interval $(-\infty,-l h),$ where $h$ is the sampling period. We derive a closed-form state-space solution of the problem, which is based on the spectral factorization of a rational transfer function

Generalizations of the sampling theorem: Seven decades after Nyquist

The sampling theorem is one of the most basic and fascinating topics in engineering sciences. The most well-known form is Shannon's uniform-sampling theorem for bandlimited signals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also well-known. The connection between such extensions and the theory of filter banks in DSP has been well established. This paper presents some of the less known aspects of sampling, with special emphasis on non bandlimited signals, pointwise stability of reconstruction, and reconstruction from nonuniform samples. Applications in multiresolution computation and in digital spline interpolation are also reviewed

Bilinear modeling and nonlinear estimation

New methods are illustrated for online nonlinear estimation applied to the lateral deflection of an elastic beam on board measurements of angular rates and angular accelerations. The development of the filter equations, together with practical issues of their numerical solution as developed from global linearization by nonlinear output injection are contrasted with the usual method of the extended Kalman filter (EKF). It is shown how nonlinear estimation due to gyroscopic coupling can be implemented as an adaptive covariance filter using off-the-shelf Kalman filter algorithms. The effect of the global linearization by nonlinear output injection is to introduce a change of coordinates in which only the process noise covariance is to be updated in online implementation. This is in contrast to the computational approach which arises in EKF methods arising by local linearization with respect to the current conditional mean. Processing refinements for nonlinear estimation based on optimal, nonlinear interpolation between observations are also highlighted. In these methods the extrapolation of the process dynamics between measurement updates is obtained by replacing a transition matrix with an operator spline that is optimized off-line from responses to selected test inputs