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

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

On strict passivity and its application to interpolation and Hl control

The authors introduce the L2-system and derive necessary and sufficient conditions for these systems to be strictly passive. Strictly passive L2-systems are characterized as having a representation in terms of a co-J-lossless matrix. A state space proof is developed and provides a Riccati equation characterization of a strictly passive L 2-system, as well as a formula for the co-J-lossless matrix representation. Applications to Nevanlinna-Pick interpolation and an H∞ filtering problem are considere

Sampled signal reconstruction via H2 optimization

In this paper the sampled signal reconstruction problem is formulated and solved as the sampled-data H2 smoothing problem. Both infinite (non-causal reconstructor) and finite (reconstructor with relaxed causality) preview cases are considered. The optimal reconstructors are in the form of the cascade of a discrete-time smoother and a generalized hold (interpolator). In the particular case of reconstructing polynomial signals with infinite preview, the proposed procedure recovers the cardinal B-spline reconstructors

Sampling from a system-theoretic viewpoint: Part I - Concepts and tools

This paper is first in a series of papers studying 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. In this paper we present the paradigm and revise underlying technical tools, such as the lifting technique and some topics of the operator theory. This material facilitates a systematic and unified treatment of a wide range of sampling and reconstruction problems, recovering many hitherto considered different solutions and leading to new results. Some of these applications are discussed in the second part

Optimal signal reconstruction from a series of recurring delayed measurements

The modern sampled-data approach provides a general methodology for signal reconstruction. This paper discusses some implications for optimal signal reconstruction when a series of recurring measurements, some delayed, are available for the reconstruction.\ud \u

Optimal relaxed causal sampler using sampled-date system theory

This paper studies the design of an optimal relaxed causal sampler using sampled data system theory. A lifted frequency domain approach is used to obtain the existence conditions and the optimal sampler. A state space formulation of the results is also provided. The resulting optimal relaxed causal sampler is a cascade of a linear continuous time system followed by a generalized sampler and a discrete system