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

signal reconstruction via h infty sampled data control theory beyond the shannon paradigm this paper presents a new method for signal reconstruction by leveraging sampled data control theory we formulate the signal reconstruction problem in terms of an analog performance optimization problem using a stable discrete time filter the proposed h performance criterion naturally takes inter sample behavior into account reflecting the energy distributions of the signal we present methods for computing optimal solutions which are guaranteed to be stable and causal detailed comparisons to alternative methods are provided we discuss some applications in sound and image reconstruction

paper zeros of sampled systems the zeros of the discrete time system obtained when sampling a continuous time system are explored theorems for the limiting zeros for large and small sampling periods are given conditions which guarantee that the sampled system has stable zeros are also presented

generalized sampling stability and performance analysis generalized sampling provides a general mechanism for recovering an unknown input function f x spl epsiv spl hscr from the samples of the responses of m linear shift invariant systems sampled at 1 mth the reconstruction rate the system can be designed to perform a projection of f x onto the reconstruction subspace v spl phi span spl phi x k sub k spl epsiv z for example the family of bandlimited signals with spl phi x sinc x this implies that the reconstruction will be perfect when the input signal is included in v spl phi the traditional framework of papoulis 1977 generalized sampling theory otherwise one recovers a signal approximation f x spl epsiv v spl phi that is consistent with f x in the sense that it produces the same measurements to characterize the stability of the algorithm we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a riesz basis of v spl phi and we use the corresponding riesz bounds to define the condition number of the system we then use these results to analyze the stability of various instances of interlaced and derivative sampling next we consider the issue of performance which becomes pertinent once we have extended the applicability of the method to arbitrary input functions that is when spl hscr is considerably larger than v spl phi and the reconstruction is no longer exact we show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation we then perform a detailed analysis for the case in which the analysis filters are in l sub 2 and determine all relevant bound constants explicitly finally we use an interlaced sampling example to illustrate these various calculations

l mathrm 2 transverse conformal killing forms on complete foliated manifolds in this article we study the l 2 transverse conformal killing forms on complete foliated riemannian manifolds and prove some vanishing theorems also we study the same problems on kahler foliations with a complete bundle like metric

cardinal exponential splines part i theory and filtering algorithms causal exponentials play a fundamental role in classical system theory starting from those elementary building blocks we propose a complete and self contained signal processing formulation of exponential splines defined on a uniform grid we specify the corresponding b spline basis functions and investigate their reproduction properties green function and exponential polynomials we also characterize their stability riesz bounds we show that the exponential b spline framework allows an exact implementation of continuous time signal processing operators including convolution differential operators and modulation by simple processing in the discrete b spline domain we derive efficient filtering algorithms for multiresolution signal extrapolation and approximation extending earlier results for polynomial splines finally we present a new asymptotic error formula that predicts the magnitude and the nth order decay of the l sub 2 approximation error as a function of the knot spacing t

h infty optimal fractional delay filters fractional delay filters are digital filters to delay discrete time signals by a fraction of the sampling period since the delay is fractional the intersample behavior of the original analog signal becomes crucial in contrast to the conventional designs based on the shannon sampling theorem with the band limiting hypothesis the present paper proposes a new approach based on the modern sampled data h optimization that aims at restoring the intersample behavior beyond the nyquist frequency by using the lifting transform or continuous time blocking the design problem is equivalently reduced to a discrete time h optimization which can be effectively solved by numerical computation softwares moreover a closed form solution is obtained under an assumption on the original analog signals design examples are given to illustrate the advantage of the proposed method

cardinal exponential splines part ii think analog act digital by interpreting the green function reproduction property of exponential splines in signal processing terms we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts the link is that the latter are the b spline coefficients of the former which happen to be exponential splines motivated by this observation we introduce an extended family of cardinal splines the generalized e splines to generalize the concept for all convolution operators with rational transfer functions we construct the corresponding compactly supported b spline basis functions which are characterized by their poles and zeros thereby establishing an interesting connection with analog filter design techniques we investigate the properties of these new b splines and present the corresponding signal processing calculus which allows us to perform continuous time operations such as convolution differential operators and modulation by simple application of the discrete version of these operators in the b spline domain in particular we show how the formalism can be used to obtain exact discrete implementations of analog filters finally we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real world analog to digital a to d and d to a conversion systems

a function space approach to sampled data control systems and tracking problems this paper presents a new framework for hybrid sampled data control systems instead of considering the state only at sampling instants this paper introduces a function piece during the sampling period as the state and gives an infinite dimensional model with such a state space this gives the advantage that sampled data systems with built in intersample behavior can be regarded as linear time invariant discrete time systems as a result the approach makes it possible to introduce such time invariant concepts as transfer functions poles and zeros to the sampled data systems even with the presence of the intersample behavior in particular tracking problems can be studied in this setting in a simple and unified way and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions this leads to the internal model principle for sampled data systems

on the degenerations of solvable leibniz algebras abstract the present paper is devoted to the description of rigid solvable leibniz algebras in particular we prove that solvable leibniz algebras under some conditions on the nilradical are rigid and we describe four dimensional solvable leibniz algebras with three dimensional rigid nilradical we show that the grunewald o halloran s conjecture any n dimensional nilpotent lie algebra is a degeneration of some algebra of the same dimension holds for leibniz algebras of dimensions less than four the algebra of level one which is omitted in the 1991 gorbatsevich s paper is indicated

mimo linear equalization with an h infty criterion in this paper we study the problem of linearly equalizing the multiple input multiple output mimo communications channels from an h infty point of view h infty estimation theory has been recently introduced as a method for designing filters that have acceptable performance in the face of model uncertainty and lack of statistical information on the exogenous signals in this paper we obtain a closed form solution to the square mimo linear h infty equalization problem and parameterize all possible h infty optimal equalizers in particular we show that for minimum phase channels a scaled version of the zero forcing equalizer is h infty optimal the results also indicate an interesting dichotomy between minimum phase and nonminimum phase channels for minimum phase channels the best causal equalizer performs as well as the best noncausal equalizer whereas for nonminimum phase channels causal equalizers cannot reduce the estimation error bounds from their a priori values our analysis also suggests certain remedies in the nonminimum phase case namely allowing for finite delay oversampling or using multiple sensors for example we show that h infty equalization of nonminimum phase channels requires a time delay of at least l units where l is the number of nonminimum phase zeros of the channel

technical communique approximation of frequency response for sampled data control systems this paper proves that the frequency response gains of fast sample fast hold approximations of a sampled data system converge to that of the original system as the sampling rate gets faster while this may appear to hold trivially there is a serious technical difficulty and the proof is indeed nontrivial it is also guaranteed that this convergence is uniform on the total frequency range the latter property is necessary to guarantee that a single approximant can be used for frequency response computation for the overall frequency range

a new approach to the discretization of continuous time controllers a new method for discretizing continuous time controllers is derived using principles of controller approximation it focuses on the closed loop use of the discrete time controller the resulting approximation criterion is a measure for stability of the control system provides an upper bound on the sampling time for which stability can be guaranteed and because it is based on continuous time controller approximation it indicates the cost of discretization in terms of performance degradation the discrete time controller is obtained through minimization of this criterion which can be performed with standard software used in h sub infinity controller design

design of multirate filter banks by spl hscr sub spl infin optimization a procedure is developed to design iir synthesis filters in a multirate filter bank the filters minimize the l sub 2 induced norm of the error system between the multirate filter bank and a desired pure time delay system this criterion is reduced to one of spl hscr sub spl infin optimization for which there is ready made software

frequency domain min max optimization of noise shaping delta sigma modulators this paper proposes a min max design of noise shaping delta sigma modulators we first characterize the all stabilizing loop filters for a linearized modulator model by this characterization we formulate the design problem of lowpass bandpass and multi band modulators as minimization of the maximum magnitude of the noise transfer function ntf in fixed frequency band s we show that this optimization minimizes the worst case reconstruction error and hence improves the snr signal to noise ratio of the modulator the optimization is reduced to an optimization with a linear matrix inequality lmi via the generalized kyp kalman yakubovich popov lemma the obtained ntf is an fir finite impulse response filter which is favorable in view of implementation we also derive a stability condition for the nonlinear model of delta sigma modulators with general quantizers including uniform ones this condition is described as an norm condition which is reduced to an lmi via the kyp lemma design examples show advantages of our design

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

yet another h sup spl infin discretization a new approach to the conversion of the sampled data h sup spl infin problem to an equivalent pure discrete time lumped variables counterpart is presented the approach is independent of controller causality constraints in particular it is applicable to problems such as preview tracking and fixed lag smoothing where existing reduction methods fail we demonstrate this by presenting a way to solve the open problem no 51 from the book open problems in mathematical systems and control theory

a minimum squared error framework for generalized sampling we treat the problem of reconstructing a signal from its nonideal samples where the sampling and reconstruction spaces as well as the class of input signals can be arbitrary subspaces of a hilbert space our formulation is general and includes as special cases reconstruction from finitely many samples as well as uniform sampling of continuous time signals which are not necessarily bandlimited to obtain a good approximation of the signal in the reconstruction space from its samples we suggest two design strategies that attempt to minimize the squared norm error between the signal and its reconstruction the approaches we propose differ in their assumptions on the input signal if the signal is known to lie in an appropriately chosen subspace then we propose a method that achieves the minimal squared error on the other hand when the signal is not restricted we show that the minimal norm reconstruction cannot generally be obtained instead we suggest minimizing the worst case squared error between the reconstructed signal and the best possible but usually unattainable approximation of the signal within the reconstruction space we demonstrate both theoretically and through simulations that the suggested methods can outperform the consistent reconstruction approach previously proposed for this problem

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

classification of orbit closures of 4 dimensional complex lie algebras abstract let l n c be the variety of complexn dimensional lie algebras the groupgln c acts on it via change of basis an orbito under this action consists of all structures isomorphic to the aim of this paper is to give a complete classification of orbit closures of 4 dimensional lie algebras i e determining all o where l 4 c starting with a classification of complex lie algebras of dimensionn 4 we study the behavior of several lie algebra invariants under degeneration i e under transition to the orbit closure as a corollary we will show that all degenerations in l 3 c can be realized via a one parameter subgroup but this is not the case in l 4 c

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution